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Mechanics of Materials

Ferdinand P. Beer, E. Russell Johnston, Jr., John T. DeWolf

Chapter 3

Torsion - all with Video Answers

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Chapter Questions

01:08

Problem 1

Determine the torque $\mathbf{T}$ that causes a maximum shearing stress of $70 \mathrm{MPa}$ in the steel cylindrical shaft shown.

Narayan Hari
Narayan Hari
Numerade Educator
01:08

Problem 2

For the cylindrical shaft shown, determine the maximum shearing stress caused by a torque of magnitude $T=800 \mathrm{N} \cdot \mathrm{m}$.

Narayan Hari
Narayan Hari
Numerade Educator
04:55

Problem 3

A 1.75 -kN.m torque is applied to the solid cylinder shown. Determine $(a)$ the maximum shearing stress, $(b)$ the percent of the torque carried by the inner 25 -mm-diameter core.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:18

Problem 4

(a) Determine the maximum shearing stress caused by a 40 -kip.in. torque $\mathbf{T}$ in the 3 -in.- -diameter solid aluminum shaft shown. $(b)$ Solve part $a,$ assuming that the solid shaft has been replaced by a hollow shaft of the same outer diameter and of 1 -in. inner diameter.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:01

Problem 5

(a) For the 3 -in.-diameter solid cylinder and loading shown, determine the maximum shearing stress. ( $b$ ) Determine the inner diameter of the 4 -in.-diameter hollow cylinder shown, for which the maximum stress is the same as in part $a$.

Ajay Singhal
Ajay Singhal
Numerade Educator
View

Problem 6

(a) For the hollow shaft and loading shown, determine the maximum shearing stress. ( $b$ ) Determine the diameter of a solid shaft for which the maximum shearing stress under the loading shown is the same as in part $(a)$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:18

Problem 7

The solid spindle $A B$ is made of a steel with an allowable shearing stress of 12 ksi, and sleeve $C D$ is made of a brass with an allowable shearing stress of 7 ksi. Determine ( $a$ ) the largest torque $\mathbf{T}$ that can be applied at $A$ if the allowable shearing stress is not to be exceeded in sleeve $C D,(b)$ the corresponding required value of the diameter $d_{s}$ of spindle $A B$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:18

Problem 8

The solid spindle $A B$ has a diameter $d_{s}=1.5$ in. and is made of a steel with an allowable shearing stress of 12 ksi, while sleeve $C D$ is made of a brass with an allowable shearing stress of 7 ksi. Determine the largest torque $\mathbf{T}$ that can be applied at $A$.
The torques shown are exerted on pulleys $A, B,$ and $C .$ The diameter of shaft $A B$ is 1.3 in. and that of $B C$ is 1.8 in. Knowing that both shafts are solid, determine the maximum shearing stress in (a) shaft (b) shaft $B C$ $A B$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
05:15

Problem 9

The shafts of the pulley assembly shown are to be redesigned. Knowing that the allowable shearing stress in each shaft is $8.5 \mathrm{ksi}$ determine the smallest allowable diameter of $(a)$ shaft $A B$ (b) shaft $B C$.

Narayan Hari
Narayan Hari
Numerade Educator
05:15

Problem 10

The shafts of the pulley assembly shown are to be redesigned. Knowing that the allowable shearing stress in each shaft is $8.5 \mathrm{ksi}$ determine the smallest allowable diameter of $(a)$ shaft $A B$ (b) shaft $B C$.

Narayan Hari
Narayan Hari
Numerade Educator
08:04

Problem 11

The torques shown are exerted on pulleys $A$ and $B$. Knowing that both shafts are solid, determine the maximum shearing stress in $(a) \operatorname{shaft} A B$, (b) shaft $B C$.

Narayan Hari
Narayan Hari
Numerade Educator
02:36

Problem 12

To reduce the total mass of the assembly of Prob. $3.11,$ a new design is being considered in which the diameter of shaft $B C$ will be smaller. Determine the smallest diameter of shaft $B C$ for which the maximum value of the shearing stress in the assembly will not increase.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:54

Problem 13

Under normal operating conditions, the electric motor exerts a torque of $2.4 \mathrm{kN} \cdot \mathrm{m}$ on shaft $A B .$ Knowing that each shaft is solid determine the maximum shearing stress in $(a)$ shaft $A B$ (b) shaft $B C,(c)$ shaft $C D$

Satpal Satpal
Satpal Satpal
Numerade Educator
02:36

Problem 14

To reduce the total mass of the assembly of Prob. $3.13,$ a new design is being considered in which the diameter of shaft $B C$ will be smaller. Determine the smallest diameter of shaft $B C$ for which the maximum value of the shearing stress in the assembly will not be increased.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:18

Problem 15

The allowable shearing stress is 15 ksi in the 1.5 -in.-diameter steel rod $A B$ and 8 ksi in the 1.8 -in.-diameter brass rod $B C .$ Neglecting the effect of stress concentrations, determine the largest torque T that can be applied at $A$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:09

Problem 16

The allowable shearing stress is 15 ksi in the steel rod $A B$ and 8 ksi in the brass rod $B C .$ Knowing that a torque of magnitude $T=10$ kip.in. is applied at $A,$ determine the required diameter of $(a) \operatorname{rod} A B$ $(b) \operatorname{rod} B C$

Satpal Satpal
Satpal Satpal
Numerade Educator
02:18

Problem 17

The solid shaft shown is formed of a brass for which the allowable shearing stress is 55 MPa. Neglecting the effect of stress concentrations, determine the smallest diameters $d_{A B}$ and $d_{B C}$ for which the allowable shearing stress is not exceeded.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
06:11

Problem 18

Solve Prob. 3.17 assuming that the direction of $\mathbf{T}_{C}$ is reversed.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:18

Problem 19

Shaft $A B$ is made of a steel with an allowable shearing stress of $90 \mathrm{MPa},$ and shaft $B C$ is made of an aluminum with an allowable shearing stress of 60 MPa. Knowing that the diameter of shaft $B C$ is $50 \mathrm{mm}$ and neglecting the effect of stress concentrations, determine $(a)$ the largest torque $\mathbf{T}$ that can be applied at $A$ if the allowable stress is not to be exceeded in shaft $B C,(b)$ the corresponding required diameter of shaft $A B$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:18

Problem 20

Shaft $A B$ has a 30 -mm diameter and is made of a steel with an allowable shearing stress of 90 MPa; shaft $B C$ has a 50 -mm diameter and is made of an aluminum alloy with an allowable shearing stress of 60 MPa. Neglecting the effect of stress concentrations, determine the largest torque $\mathbf{T}$ that can be applied at $A$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
08:04

Problem 21

Two solid steel shafts are connected by the gears shown. A torque of magnitude $T=900 \mathrm{N} \cdot \mathrm{m}$ is applied to shaft $A B .$ Knowing that the allowable shearing stress is $50 \mathrm{MPa}$ and considering only stresses due to twisting, determine the required diameter of $(a)$ shaft $A B,(b)$ shaft $C D$.

Narayan Hari
Narayan Hari
Numerade Educator
02:18

Problem 22

Shaft $C D$ is made from a 66 -mm-diameter rod and is connected to the 48 -mm-diameter shaft $A B$ as shown. Considering only stresses due to twisting and knowing that the allowable shearing stress is 60 MPa for each shaft, determine the largest torque $\mathbf{T}$ that can be applied.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
05:15

Problem 23

Under normal operating conditions a motor exerts a torque of magnitude $T_{F}$ at $F$. The shafts are made of a steel for which the allowable shearing stress is $12 \mathrm{ksi}$ and have diameters $d_{C D E}=0.900$ in. and $d_{F G H}=0.800$ in. Knowing that $r_{D}=6.5$ in. and $r_{G}=4.5$ in. determine the largest allowable value of $T_{F}$

Narayan Hari
Narayan Hari
Numerade Educator
02:36

Problem 24

Under normal operating conditions a motor exerts a torque of magnitude $T_{F}=1200$ lb\cdotin. at $F .$ Knowing that $r_{D}=8$ in., $r_{G}=3$ in. and the allowable shearing stress is $10.5 \mathrm{ksi}$ in each shaft, determine the required diameter of $(a)$ shaft $C D E$ $(b)$ shaft $F G H$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
05:15

Problem 25

The two solid shafts are connected by gears as shown and are made of a steel for which the allowable shearing stress is 7000 psi. Knowing the diameters of the two shafts are, respectively, $d_{B C}=1.6$ in. and $d_{E F}=1.25$ in., determine the largest torque $\mathbf{T}_{C}$ that $\operatorname{can}$ be applied at $C$.

Narayan Hari
Narayan Hari
Numerade Educator
05:15

Problem 26

The two solid shafts are connected by gears as shown and are made of a steel for which the allowable shearing stress is 8500 psi. Knowing that a torque of magnitude $T_{C}=5$ kip.in. is applied at $C$ and that the assembly is in equilibrium, determine the required diameter of $(a)$ shaft $B C,(b)$ shaft $E F$.

Narayan Hari
Narayan Hari
Numerade Educator
06:19

Problem 27

For the gear train shown, the diameters of the three solid shafts are:
\[d_{A B}=20 \mathrm{mm} \quad d_{C D}=25 \mathrm{mm} \quad d_{E F}=40 \mathrm{mm}\]
Knowing that for each shaft the allowable shearing stress is $60 \mathrm{MPa}$ determine the largest torque $\mathbf{T}$ that can be applied.

Ajay Singhal
Ajay Singhal
Numerade Educator
06:19

Problem 28

A torque $T=900 \mathrm{N} \cdot \mathrm{m}$ is applied to shaft $A B$ of the gear train shown. Knowing that the allowable shearing stress is $80 \mathrm{MPa}$ $(b) \operatorname{shaft} C D$ determine the required diameter of $(a)$ shaft $A B$ $(c)$ shaft $E F$

Ajay Singhal
Ajay Singhal
Numerade Educator
03:16

Problem 29

While the exact distribution of the shearing stresses in a hollowcylindrical shaft is as shown in Fig. $\mathrm{P} 3.29 a$, an approximate value can be obtained for $\tau_{\max }$ by assuming that the stresses are uniformly distributed over the area $A$ of the cross section, as shown in Fig. $\mathrm{P} 3.29 b,$ and then further assuming that all of the elementary shearing forces act at a distance from $O$ equal to the mean radius $\frac{1}{2}\left(c_{1}+c_{2}\right)$ of the cross section. This approximate value is $\tau_{0}=T / A r_{m},$ where $T$ is the applied torque. Determine the ratio $\tau_{\max } / \tau_{0}$ of the true value of the maximum shearing stress and its approximate value $\tau_{0}$ for values of $c_{1} / c_{2}$ respectively equal to 1.00 $0.95,0.75,0.50,$ and 0.

Chai Santi
Chai Santi
Numerade Educator
03:58

Problem 30

(a) For a given allowable shearing stress, determine the ratio $T / w$ of the maximum allowable torque $T$ and the weight per unit length $w$ for the hollow shaft shown. (b) Denoting by $(T / w)_{0}$ the value of this ratio for a solid shaft of the same radius $c_{2},$ express the ratio $T / w$ for the hollow shaft in terms of $(T / w)_{0}$ and $c_{1} / c_{2}$.

Chai Santi
Chai Santi
Numerade Educator
03:22

Problem 31

While an oil well is being drilled at a depth of $2500 \mathrm{m},$ it is observed that the top of the 200 -mm-diameter steel drill pipe $(G=77.2 \mathrm{GPa})$ rotates through 2.5 revolutions before the drilling bit starts to operate. Determine the maximum shearing stress caused in the pipe by torsion.

Mayukh Banik
Mayukh Banik
Numerade Educator
00:48

Problem 32

(a) Determine the angle of twist caused by a 40 -kip.in. torque $\mathbf{T}$ in the 3 -in.-diameter solid aluminum shaft shown $\left(G=3.7 \times 10^{6} \mathrm{psi}\right)$ (b) Solve part (a), assuming that the solid shaft has been replaced by a hollow shaft of the same outer diameter and a 1 -in. inner diameter.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:04

Problem 33

Determine the smallest allowable diameter of a 10 -ft-long steel rod $\left(G=11.2 \times 10^{6} \mathrm{psi}\right)$ if the rod is to be twisted through $90^{\circ}$ without exceeding a shearing stress of 15 ksi.

Surendra Kumar
Surendra Kumar
Numerade Educator
00:43

Problem 34

(a) For the aluminum pipe shown $(G=27 \mathrm{GPa}),$ determine the torque $\mathbf{T}_{0}$ causing an angle of twist of $2^{\circ} .(b)$ Determine the angle of twist if the same torque $\mathbf{T}_{0}$ is applied to a solid cylindrical shaft of the same length and cross-sectional area.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:13

Problem 35

The electric motor exerts a $500-\mathrm{N} \cdot \mathrm{m}$ torque on the aluminum shaft $A B C D$ when it is rotating at a constant speed. Knowing that $G=27 \mathrm{GPa}$ and that the torques exerted on pulleys $B$ and $C$ are as shown, determine the angle of twist between $(a) B$ and $C$ (b) $B$ and $D$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
14:42

Problem 36

The torques shown are exerted on pulleys $A$ and $B$. Knowing that the shafts are solid and made of steel $(G=77.2 \mathrm{GPa})$, determine the angle of twist between $(a) A$ and $B,(b) A$ and $C$.
The aluminum rod $B C(G=26 \mathrm{GPa})$ is bonded to the brass rod $A B$ $(G=39 \text { GPa }) .$ Knowing that each rod is solid and has a diameter of $12 \mathrm{mm}$, determine the angle of twist $(a)$ at $B,(b)$ at $C$.

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:41

Problem 37

The aluminum rod $A B(G=27 \mathrm{GPa})$ is bonded to the brass rod $B D$ $(G=39 \mathrm{GPa}) .$ Knowing that portion $C D$ of the brass rod is hollow and has an inner diameter of $40 \mathrm{mm}$, determine the angle of twist at $A$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:41

Problem 38

The aluminum $\operatorname{rod} A B(G=27 \mathrm{GPa})$ is bonded to the brass rod $B D$ $(G=39 \mathrm{GPa}) .$ Knowing that portion $C D$ of the brass rod is hollow and has an inner diameter of $40 \mathrm{mm}$, determine the angle of twist at $A$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:13

Problem 39

The solid spindle $A B$ has a diameter $d_{s}=1.75$ in. and is made of a steel with $G=11.2 \times 10^{6}$ psi and $\tau_{\text {all }}=12 \mathrm{ksi},$ while sleeve $C D$ is made of a brass with $G=5.6 \times 10^{6} \mathrm{psi}$ and $\tau_{\text {all }}=7$ ksi. Determine $(a)$ the largest torque $\mathbf{T}$ that can be applied at $A$ if the given allowable stresses are not to be exceeded and if the angle of twist of sleeve $C D$ is not to exceed $0.375^{\circ}$ (b) the corresponding angle through which end $A$ rotates.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
03:13

Problem 40

The solid spindle $A B$ has a diameter $d_{s}=1.5$ in. and is made of a steel with $G=11.2 \times 10^{6}$ psi and $\tau_{\text {all }}=12 \mathrm{ksi},$ while sleeve $C D$ is made of a brass with $G=5.6 \times 10^{6} \mathrm{psi}$ and $\tau_{\text {all }}=7 \mathrm{ksi}$. Determine the largest angle through which end $A$ can be rotated.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
05:15

Problem 41

Two shafts, each of $\frac{7}{8}$ -in. diameter, are connected by the gears shown. Knowing that $G=11.2 \times 10^{6}$ psi and that the shaft at $F$ is fixed, determine the angle through which end $A$ rotates when a 1.2 -kip.in. torque is applied at $A$.

Narayan Hari
Narayan Hari
Numerade Educator
10:09

Problem 42

The angle of rotation of end $A$ of the gear-and-shaft system shown must not exceed $4^{\circ} .$ Knowing that the shafts are made of a steel for which $\tau_{\text {all }}=65 \mathrm{MPa}$ and $G=77.2 \mathrm{GPa}$, determine the largest torque T that can be safely applied at end $A$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
13:14

Problem 43

A coder $F,$ used to record in digital form the rotation of shaft $A,$ is connected to the shaft by means of the gear train shown, which consists of four gears and three solid steel shafts each of diameter $d .$ Two of the gears have a radius $r$ and the other two a radius $n r .$ If the rotation of the coder $F$ is prevented, determine in terms of $T, l, G, J,$ and $n$ the angle through which end $A$ rotates.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:25

Problem 44

For the gear train described in Prob. 3.43 determine the angle through which end $A$ rotates when $T=5$ lb\cdotin., $l=2.4$ in., $d=\frac{1}{16}$ in. $, G=11.2 \times 10^{6} \mathrm{psi},$ and $n=2$.

Uma Kumari
Uma Kumari
Numerade Educator
03:21

Problem 45

The design specifications of a 1.2 -m-long solid circular transmission shaft require that the angle of twist of the shaft not exceed $4^{\circ}$ when a torque of $750 \mathrm{N} \cdot \mathrm{m}$ is applied. Determine the required diameter of the shaft, knowing that the shaft is made of a steel with an allowable shearing stress of $90 \mathrm{MPa}$ and a modulus of rigidity of $77.2 \mathrm{GPa}$.

Anand Jangid
Anand Jangid
Numerade Educator
01:35

Problem 46

The design specifications for the gear-and-shaft system shown require that the same diameter be used for both shafts and that the angle through which pulley $A$ will rotate when subjected to a 2-kip.in. torque $\mathbf{T}_{A}$ while pulley $D$ is held fixed will not exceed $7.5^{\circ} .$ Determine the required diameter of the shafts if both shafts are made of a steel with $G=11.2 \times 10^{6} \mathrm{psi}$ and $\tau_{\mathrm{all}}=12 \mathrm{ksi}$.

Chai Santi
Chai Santi
Numerade Educator
03:13

Problem 47

Solve Prob. $3.46,$ assuming that both shafts are made of a brass with $G=5.6 \times 10^{6} \mathrm{psi}$ and $\tau_{\text {all }}=8 \mathrm{ksi}$.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
02:31

Problem 48

The design of the gear-and-shaft system shown requires that steel shafts of the same diameter be used for both $A B$ and $C D .$ It is further required that $\tau_{\max } \leq 60$ MPa and that the angle $\phi_{D}$ through which end $D$ of shaft $C D$ rotates not exceed $1.5^{\circ} .$ Knowing that $G=77.2 \mathrm{GPa},$ determine the required diameter of the shafts.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:31

Problem 49

The electric motor exerts a torque of $800 \mathrm{N} \cdot \mathrm{m}$ on the steel shaft $A B C D$ when it is rotating at a constant speed. Design specifications require that the diameter of the shaft be uniform from $A$ to $D$ and that the angle of twist between $A$ and $D$ not exceed $1.5^{\circ} .$ Knowing that $\tau_{\max } \leq 60$ MPa and $G=77.2 \mathrm{GPa}$, determine the minimum diameter shaft that can be used.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:18

Problem 50

A hole is punched at $A$ in a plastic sheet by applying a 600 -N force $\mathbf{P}$ to end $D$ of lever $C D,$ which is rigidly attached to the solid cylindrical shaft $B C$. Design specifications require that the displacement of $D$ should not exceed $15 \mathrm{mm}$ from the time the punch first touches the plastic sheet to the time it actually penetrates it. Determine the required diameter of shaft $B C$ if the shaft is made of a steel with $G=77.2 \mathrm{GPa}$ and $\tau_{\text {all }}=80 \mathrm{MPa}$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
View

Problem 51

The solid cylinders $A B$ and $B C$ are bonded together at $B$ and are attached to fixed supports at $A$ and $C .$ Knowing that the modulus of rigidity is $3.7 \times 10^{6}$ psi for aluminum and $5.6 \times 10^{6}$ psi for brass, determine the maximum shearing stress $(a)$ in cylinder $A B,(b)$ in cylinder $B C$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:31

Problem 52

Solve Prob. 3.51 , assuming that cylinder $A B$ is made of steel, for which $G=11.2 \times 10^{6} \mathrm{psi}$.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:39

Problem 53

A torque $T=40$ kip.in. is applied at end $A$ of the composite shaft shown. Knowing that the modulus of rigidity is $11.2 \times 10^{6}$ psi for steel and $4 \times 10^{6}$ psi for aluminum, determine $(a)$ the maximum shearing stress in the steel core, $(b)$ the maximum stress in the aluminum shell, $(c)$ the angle of twist at $A$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:39

Problem 54

The composite shaft shown is to be twisted by applying a torque $\mathbf{T}$ at end $A .$ Knowing that the modulus of rigidity is $11.2 \times 10^{6}$ psi for steel and $4 \times 10^{6}$ psi for aluminum, determine the largest angle through which end $A$ may be rotated, if the following allowable stresses are not to be exceeded: $\tau_{\text {steel }}=8000$ psi and $\tau_{\text {aluminum }}=6000$ psi.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
View

Problem 55

Two solid steel shafts $(G=77.2 \mathrm{GPa})$ are connected to a coupling disk $B$ and to fixed supports at $A$ and $C .$ For the loading shown, determine $(a)$ the reaction at each support, (b) the maximum shearing stress in shaft $A B,(c)$ the maximum shearing stress in shaft $B C$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:34

Problem 56

Solve Prob. $3.55,$ assuming that the shaft $A B$ is replaced by a hollow shaft of the same outer diameter and 25 -mm inner diameter.

Chai Santi
Chai Santi
Numerade Educator
10:09

Problem 57

At a time when rotation is prevented at the lower end of each shaft, a $50-\mathrm{N} \cdot \mathrm{m}$ torque is applied to end $A$ of shaft $A B .$ Knowing that $G=77.2$ GPa for both shafts, determine $(a)$ the maximum shearing stress in shaft $C D,(b)$ the angle of rotation at $A$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
04:14

Problem 58

Solve Prob. $3.57,$ assuming that the 80 -N.m torque is applied to end $C$ of shaft $C D$.

Narayan Hari
Narayan Hari
Numerade Educator
02:11

Problem 59

The steel jacket $C D$ has been attached to the 40 -mm-diameter steel shaft $A E$ by means of rigid flanges welded to the jacket and to the rod. The outer diameter of the jacket is $80 \mathrm{mm}$ and its wall thickness is $4 \mathrm{mm} .$ If 500 -N.m torques are applied as shown, determine the maximum shearing stress in the jacket.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
00:43

Problem 60

A torque $\mathbf{T}$ is applied as shown to a solid tapered shaft $A B .$ Show by integration that the angle of twist at $A$ is
\[\phi=\frac{7 T L}{12 \pi G c^{4}}\]

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:55

Problem 61

The mass moment of inertia of a gear is to be determined experimentally by using a torsional pendulum consisting of a $6-\mathrm{ft}$ steel wire. Knowing that $G=11.2 \times 10^{6}$ psi, determine the diameter of the wire for which the torsional spring constant will be 4.27 lb $\cdot \mathrm{ft} / \mathrm{rad}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
00:55

Problem 62

A solid shaft and a hollow shaft are made of the same material and are of the same weight and length. Denoting by $n$ the ratio $c_{1} / c_{2}$ show that the ratio $T_{s} / T_{h}$ of the torque $T_{s}$ in the solid shaft to the torque $T_{h}$ in the hollow shaft is $(a) \sqrt{\left(1-n^{2}\right)} /\left(1+n^{2}\right)$ if the $\max$ imum shearing stress is the same in each shaft, $(b)\left(1-n^{2}\right) /\left(1+n^{2}\right)$ if the angle of twist is the same for each shaft.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:51

Problem 63

An annular plate of thickness $t$ and modulus $G$ is used to connect shaft $A B$ of radius $r_{1}$ to tube $C D$ of radius $r_{2} .$ Knowing that a torque $\mathbf{T}$ is applied to end $A$ of shaft $A B$ and that end $D$ of tube $C D$ is fixed, $(a)$ determine the magnitude and location of the maximum shearing stress in the annular plate, $(b)$ show that the angle through which end $B$ of the shaft rotates with respect to end $C$ of the tube is
\[\phi_{B C}=\frac{T}{4 \pi G t}\left(\frac{1}{r_{1}^{2}}-\frac{1}{r_{2}^{2}}\right)\]

Satpal Satpal
Satpal Satpal
Numerade Educator
01:55

Problem 64

Using an allowable shearing stress of $5.4 \mathrm{ksi}$, design a solid steel shaft to transmit 16 hp at a speed of (a) 1200 $\mathrm{rpm}$ (b) 2400 rpm.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:24

Problem 65

Using an allowable shearing stress of 58 MPa, design a solid steel shaft to transmit $18 \mathrm{kW}$ at a frequency of $(a) 30 \mathrm{Hz},(b) 60 \mathrm{Hz}$.

Chai Santi
Chai Santi
Numerade Educator
02:29

Problem 66

Determine the maximum shearing stress in a solid shaft of 1.4-in. diameter as it transmits 66 hp at a speed of $(a) 750 \mathrm{rpm}$ (b) 1500 rpm.

Anand Jangid
Anand Jangid
Numerade Educator
06:06

Problem 67

Determine the maximum shearing stress in a solid shaft of 18 -mm diameter as it transmits $3.4 \mathrm{kW}$ at a frequency of $(a) 25 \mathrm{Hz},(b) 50 \mathrm{Hz}$

Satpal Satpal
Satpal Satpal
Numerade Educator
00:58

Problem 68

While a steel shaft of the cross section shown rotates at $120 \mathrm{rpm}$ a stroboscopic measurement indicates that the angle of twist is $2^{\circ}$ in a $4-\mathrm{m}$ length. Using $G=77.2 \mathrm{GPa}$, determine the power being transmitted.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
00:47

Problem 69

Determine the required thickness of the 50 -mm tubular shaft of Concept Application $3.7,$ if it is to transmit the same power while rotating at a frequency of $30 \mathrm{Hz}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:29

Problem 70

A hollow steel drive shaft $\left(G=11.2 \times 10^{6} \mathrm{psi}\right)$ is $8 \mathrm{ft}$ long and its outer and inner diameters are respectively equal to 2.50 in. and 1.25 in. Knowing that the shaft transmits 200 hp while rotating at 1500 rpm, determine $(a)$ the maximum shearing stress, $(b)$ the angle of twist of the shaft.

Anand Jangid
Anand Jangid
Numerade Educator
02:31

Problem 71

The hollow steel shaft shown $\left(G=77.2 \mathrm{GPa}, \tau_{\text {all }}=50 \mathrm{MPa}\right)$ rotates at $240 \mathrm{rpm} .$ Determine $(a)$ the maximum power that can be transmitted, (b) the corresponding angle of twist of the shaft.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:29

Problem 72

One of two hollow drive shafts of a cruise ship is 125 ft long, and its outer and inner diameters are 16 in. and 8 in., respectively. The shaft is made of a steel for which $\tau_{\text {all }}=8500$ psi and $G=11.2 \times$ $10^{6}$ psi. Knowing that the maximum speed of rotation of the shaft is $165 \mathrm{rpm}$, determine ( $a$ ) the maximum power that can be transmitted by the one shaft to its propeller, (b) the corresponding angle of twist of the shaft.

Anand Jangid
Anand Jangid
Numerade Educator
02:29

Problem 73

The design of a machine element calls for a 40 -mm-outer-diameter shaft to transmit $45 \mathrm{kW}$. (a) If the speed of rotation is $720 \mathrm{rpm}$ determine the maximum shearing stress in shaft $a .(b)$ If the speed of rotation can be increased $50 \%$ to $1080 \mathrm{rpm}$, determine the largest inner diameter of shaft $b$ for which the maximum shearing stress will be the same in each shaft.

Anand Jangid
Anand Jangid
Numerade Educator
05:32

Problem 74

Three shafts and four gears are used to form a gear train that will transmit power from the motor at $A$ to a machine tool at $F$. (Bearings for the shafts are omitted in the sketch.) The diameter of each shaft is as follows: $d_{A B}=16 \mathrm{mm}, d_{C D}=20 \mathrm{mm}, d_{E F}=28 \mathrm{mm} .$ Knowing that the frequency of the motor is $24 \mathrm{Hz}$ and that the allowable shearing stress for each shaft is $75 \mathrm{MPa}$, determine the maximum power that can be transmitted.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:32

Problem 75

Three shafts and four gears are used to form a gear train that will transmit power from the motor at $A$ to a machine tool at $F$. (Bearings for the shafts are omitted in the sketch.) The diameter of each shaft is as follows: $d_{A B}=16 \mathrm{mm}, d_{C D}=20 \mathrm{mm}, d_{E F}=28 \mathrm{mm} .$ Knowing that the frequency of the motor is $24 \mathrm{Hz}$ and that the allowable shearing stress for each shaft is $75 \mathrm{MPa}$, determine the maximum power that can be transmitted.
The two solid shafts and gears shown are used to transmit 16 hp from the motor at $A$ operating at a speed of $1260 \mathrm{rpm}$, to a machine tool at $D$. Knowing that each shaft has a diameter of 1 in., determine the maximum shearing stress $(a)$ in shaft $A B,(b)$ in shaft $C D$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:15

Problem 76

The two solid shafts and gears shown are used to transmit $16 \mathrm{hp}$ from the motor at $A$ operating at a speed of $1260 \mathrm{rpm}$ to a machine tool at $D$. Knowing that the maximum allowable shearing stress is $8 \mathrm{ksi}$, determine the required diameter of (a) shaft $A B,(b)$ shaft $C D$.

Narayan Hari
Narayan Hari
Numerade Educator
05:15

Problem 77

The two solid shafts and gears shown are used to transmit $16 \mathrm{hp}$ from the motor at $A$ operating at a speed of $1260 \mathrm{rpm}$ to a machine tool at $D$. Knowing that the maximum allowable shearing stress is $8 \mathrm{ksi}$, determine the required diameter of (a) shaft $A B,(b)$ shaft $C D$.

Narayan Hari
Narayan Hari
Numerade Educator
02:31

Problem 78

The shaft-disk-belt arrangement shown is used to transmit 3 hp from point $A$ to point $D .(a)$ Using an allowable shearing stress of 9500 psi, determine the required speed of shaft $A B$ $(b)$ Solve part $(a),$ assuming that the diameters of shafts $A B$ and $C D$ are, respectively, 0.75 in. and 0.625 in.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:29

Problem 79

A 1.5 -in.-diameter steel shaft of length $4 \mathrm{ft}$ will be used to transmit 60 hp between a motor and a pump. Knowing that $G=11.2$ $\times 10^{6} \mathrm{psi},$ determine the lowest speed of rotation at which the stress does not exceed 8500 psi and the angle of twist does not exceed $2^{\circ}$

Anand Jangid
Anand Jangid
Numerade Educator
03:21

Problem 80

A 2.5 -m-long steel shaft of 30 -mm diameter rotates at a frequency of $30 \mathrm{Hz}$. Determine the maximum power that the shaft can transmit, knowing that $G=77.2 \mathrm{GPa}$, that the allowable shearing stress is $50 \mathrm{MPa},$ and that the angle of twist must not exceed $7.5^{\circ}$.

Anand Jangid
Anand Jangid
Numerade Educator
00:38

Problem 81

A steel shaft must transmit $150 \mathrm{kW}$ at a speed of $360 \mathrm{rpm} .$ Knowing that $G=77.2 \mathrm{GPa}$, design a solid shaft so that the maximum shearing stress will not exceed $50 \mathrm{MPa}$ and the angle of twist in a $2.5-\mathrm{m}$ length must not exceed $3^{\circ}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:05

Problem 82

A 1.5 -m-long tubular steel shaft $(G=77.2 \mathrm{GPa})$ of 38 -mm outer diameter $d_{1}$ and $30-\mathrm{mm}$ inner diameter $d_{2}$ is to transmit $100 \mathrm{kW}$ between a turbine and a generator. Knowing that the allowable shearing stress is $60 \mathrm{MPa}$ and that the angle of twist must not exceed $3^{\circ},$ determine the minimum frequency at which the shaft can rotate.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:50

Problem 83

A 1.5 -m-long tubular steel shaft of 38 -mm outer diameter $d_{1}$ is to be made of a steel for which $\tau_{\text {all }}=65$ MPa and $G=77.2$ GPa. Knowing that the angle of twist must not exceed $4^{\circ}$ when the shaft is subjected to a torque of $600 \mathrm{N} \cdot \mathrm{m}$, determine the largest inner diameter $d_{2}$ that can be specified in the design.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:20

Problem 84

The stepped shaft shown must transmit $40 \mathrm{kW}$ at a speed of $720 \mathrm{rpm}$ Determine the minimum radius $r$ of the fillet if an allowable stress of 36 MPa is not to be exceeded.

Narayan Hari
Narayan Hari
Numerade Educator
01:14

Problem 85

The stepped shaft shown rotates at 450 rpm. Knowing that $r=0.2$ in. determine the largest torque $\mathbf{T}$ that can be transmitted without exceeding an allowable shearing stress of 7500 psi.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:29

Problem 86

Knowing that the stepped shaft shown transmits a torque of magnitude $T=2.50$ kip $\cdot$ in. $,$ determine the maximum shearing stress in the shaft when the radius of the fillet is $(a) r=\frac{1}{8}$ in. (b) $r=\frac{3}{16}$ in.

Chai Santi
Chai Santi
Numerade Educator
02:20

Problem 87

Knowing that the stepped shaft shown must transmit $45 \mathrm{kW}$ at a speed of 2100 rpm, determine the minimum radius $r$ of the fillet if an allowable shearing stress of $50 \mathrm{MPa}$ is not to be exceeded.

Narayan Hari
Narayan Hari
Numerade Educator
02:20

Problem 88

The stepped shaft shown must transmit $45 \mathrm{kW}$. Knowing that the allowable shearing stress in the shaft is $40 \mathrm{MPa}$ and that the radius of the fillet is $r=6 \mathrm{mm}$, determine the smallest permissible speed of the shaft.

Narayan Hari
Narayan Hari
Numerade Educator
04:29

Problem 89

A torque of magnitude $T=200$ lb -in. is applied to the stepped shaft shown, which has a full quarter-circular fillet. Knowing that $D=1$ in. determine the maximum shearing stress in the shaft when $(a) d=0.8$ in. (b) $d=0.9$ in.

Chai Santi
Chai Santi
Numerade Educator
06:19

Problem 90

In the stepped shaft shown, which has a full quarter-circular fillet, the allowable shearing stress is 80 MPa. Knowing that $D=30 \mathrm{mm}$ determine the largest allowable torque that can be applied to the shaft if $(a) d=26 \mathrm{mm}$ $(b) d=24 \mathrm{mm}$

Ajay Singhal
Ajay Singhal
Numerade Educator
02:29

Problem 91

In the stepped shaft shown, which has a full quarter-circular fillet, $D=1.25$ in. and $d=1$ in. Knowing that the speed of the shaft is $2400 \mathrm{rpm}$ and that the allowable shearing stress is $7500 \mathrm{psi}$, determine the maximum power that can be transmitted by the shaft.

Anand Jangid
Anand Jangid
Numerade Educator
02:20

Problem 92

The solid circular shaft shown is made of a steel that is assumed to be elastoplastic with $\tau_{Y}=145 \mathrm{MPa}$. Determine the magnitude $T$ of the applied torques when the plastic zone is $(a) 16 \mathrm{mm}$ deep, (b) $24 \mathrm{mm}$ deep.

Chai Santi
Chai Santi
Numerade Educator
01:44

Problem 93

A 1.25 -in. diameter solid rod is made of an elastoplastic material with $\tau_{Y}=5$ ksi. Knowing that the elastic core of the rod is 1 in. in diameter, determine the magnitude of the applied torque $\mathbf{T}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:36

Problem 94

A 2 -in.-diameter solid shaft is made of a mild steel that is assumed to be elastoplastic with $\tau_{Y}=20$ ksi. Determine the maximum shearing stress and the radius of the elastic core caused by the application of a torque of magnitude $(a) 30$ kip $\cdot$ in. $(b)$ 40 kip -in.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:21

Problem 95

The solid shaft shown is made of a mild steel that is assumed to be elastoplastic with $G=77.2 \mathrm{GPa}$ and $\tau_{Y}=145 \mathrm{MPa}$. Determine the maximum shearing stress and the radius of the elastic core caused by the application of a torque of magnitude $(a) T=600 \mathrm{N} \cdot \mathrm{m}$ $(b) T=1000 \mathrm{N} \cdot \mathrm{m}$.

Anand Jangid
Anand Jangid
Numerade Educator
03:21

Problem 96

The solid shaft shown is made of a mild steel that is assumed to be elastoplastic with $\tau_{Y}=145$ MPa. Determine the radius of the elastic core caused by the application of a torque equal to $1.1 T_{Y},$ where $T_{Y}$ is the magnitude of the torque at the onset of yield.

Anand Jangid
Anand Jangid
Numerade Educator
03:00

Problem 97

It is observed that a straightened paper clip can be twisted through several revolutions by the application of a torque of approximately $60 \mathrm{N} \cdot \mathrm{m} .$ Knowing that the diameter of the wire in the paper clip is $0.9 \mathrm{mm},$ determine the approximate value of the yield stress of the steel.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
03:21

Problem 98

The solid shaft shown is made of a mild steel that is assumed to be elastoplastic with $G=77.2 \mathrm{GPa}$ and $\tau_{Y}=145 \mathrm{MPa}$. Determine the angle of twist caused by the application of a torque of magnitude $(a) T=600 \mathrm{N} \cdot \mathrm{m}$, $(b) T=1000 \mathrm{N} \cdot \mathrm{m}$

Anand Jangid
Anand Jangid
Numerade Educator
00:48

Problem 99

For the solid circular shaft shown, determine the angle of twist caused by the application of a torque of magnitude (a) $T=80$ kip-in. (b) $T=130$ kip.in.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:21

Problem 100

A torque $\mathbf{T}$ is applied to the 20 -mm-diameter steel rod $A B$. Assuming the steel to be elastoplastic with $G=77.2 \mathrm{GPa}$ and $\tau_{Y}=145 \mathrm{MPa}$ determine ( $a$ ) the torque $T$ when the angle of twist at $A$ is $25^{\circ},(b)$ the corresponding diameter of the elastic core of the shaft.

Anand Jangid
Anand Jangid
Numerade Educator
02:36

Problem 101

A 3 -ft-long solid shaft has a diameter of 2.5 in. and is made of a mild steel that is assumed to be elastoplastic with $\tau_{Y}=21 \mathrm{ksi}$ and $G=11.2 \times 10^{6}$ psi. Determine the torque required to twist the shaft (b) $5^{\circ}$ through an angle of $(a) 2.5^{\circ}, 3$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:21

Problem 102

An 18 -mm-diameter solid circular shaft is made of a material that is assumed to be elastoplastic with $\tau_{Y}=145 \mathrm{MPa}$ and $G=77.2 \mathrm{GPa}$ For a 1.2 -m length of the shaft, determine the maximum shearing stress and the angle of twist caused by a 200 -N.m torque.

Anand Jangid
Anand Jangid
Numerade Educator
03:21

Problem 103

A 1.25 -in.-diameter solid circular shaft is made of a material that is assumed to be elastoplastic with $\tau_{Y}=18 \mathrm{ksi}$ and $G=11.2 \times 10^{6} \mathrm{psi}$ For an 8 -ft length of the shaft, determine the maximum shearing stress and the angle of twist caused by a 7.5 -kip.in. torque.

Anand Jangid
Anand Jangid
Numerade Educator
03:21

Problem 104

The shaft $A B$ is made of a material that is elastoplastic with $\tau_{Y}=90 \mathrm{MPa}$ and $G=30$ GPa. For the loading shown, determine (a) the radius of the elastic core of the shaft, $(b)$ the angle of twist at end $B$.

Anand Jangid
Anand Jangid
Numerade Educator
06:51

Problem 105

A solid circular rod is made of a material that is assumed to be elastoplastic. Denoting by $T_{Y}$ and $\phi_{Y}$, respectively, the torque and the angle of twist at the onset of yield, determine the angle of twist if the torque is increased to $(a) T=1.1 T_{Y},$ (b) $T=1.25 T_{Y},$ $(c) T=1.3 T_{Y}$.

David González Cornejo
David González Cornejo
Numerade Educator
03:21

Problem 106

The hollow shaft shown is made of steel which is assumed to be elastoplastic with $\tau_{Y}=145 \mathrm{MPa}$ and $G=77.2$ GPa. Determine the magnitude $T$ of the torque and the corresponding angle of twist (a) at the onset of yield, ( $b$ ) when the plastic zone is $10 \mathrm{mm}$ deep.

Anand Jangid
Anand Jangid
Numerade Educator
00:43

Problem 107

For the shaft of Prob. $3.106,$ determine $(a)$ angle of twist at which the section first becomes fully plastic, ( $b$ ) the corresponding magnitude $T$ of the applied torque. Sketch the $T-\phi$ curve for the shaft.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:08

Problem 108

A shaft of mild steel is machined to the shape shown and then twisted by torques of magnitude 40 kip.in. Assuming the steel to be elastoplastic with $\tau_{Y}=21$ ksi, determine $(a)$ the thickness of the plastic zone in portion $C D$ of the shaft, $(b)$ the length of portion $B E$ that remains fully elastic.

Chai Santi
Chai Santi
Numerade Educator
00:43

Problem 109

The magnitude of the torque $\mathbf{T}$ applied to the tapered shaft of Prob. 3.108 is slowly increased. Determine $(a)$ the largest torque that may be applied to the shaft, $(b)$ the length of portion $B E$ that remains fully elastic.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:42

Problem 110

A hollow shaft of outer and inner diameters respectively equal to 0.6 in. and 0.2 in. is fabricated from an aluminum alloy with the stress-strain diagram shown. Determine the torque required to twist a 9 -in. length of the shaft through $10^{\circ}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:10

Problem 111

Using the stress-strain diagram shown, determine $(a)$ the torque that causes a maximum shearing stress of 15 ksi in a 0.8 -in.-diameter solid rod, (b) the corresponding angle of twist in a 20 -in. length of the rod.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:05

Problem 112

A 50 -mm-diameter cylinder is made of a brass for which the stressstrain diagram is as shown. Knowing that the angle of twist is $5^{\circ}$ in a 725 -mm length, determine by approximate means the magnitude $T$ of the torque applied to the shaft.

Ajay Singhal
Ajay Singhal
Numerade Educator
02:54

Problem 113

Three points on the nonlinear stress-strain diagram used in Prob. 3.112 $\operatorname{are}(0,0),(0.0015,55 \mathrm{MPa}),$ and $(0.003,80 \mathrm{MPa}) .$ By fitting the polynomial $T=A+B \gamma+C \gamma^{2}$ through these points, the following approximate relation has been obtained.
\[T=46.7 \times 10^{9} \gamma-6.67 \times 10^{12} \gamma^{2}\]
Solve Prob. 3.112 using this relation, Eq. (3.2), and Eq. (3.23).

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:21

Problem 114

The solid circular drill rod $A B$ is made of a steel that is assumed to be elastoplastic with $\tau_{Y}=22 \mathrm{ksi}$ and $G=11.2 \times 10^{6} \mathrm{psi}$ Knowing that a torque $T=75$ kip.in. is applied to the rod and then removed, determine the maximum residual shearing stress in the rod.

Anand Jangid
Anand Jangid
Numerade Educator
04:41

Problem 115

In Prob. 3.114 , determine the permanent angle of twist of the rod.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:21

Problem 116

The solid shaft shown is made of a steel that is assumed to be elastoplastic with $\tau_{Y}=145 \mathrm{MPa}$ and $G=77.2 \mathrm{GPa}$. The torque is increased in magnitude until the shaft has been twisted through $6^{\circ}$ the torque is then removed. Determine $(a)$ the magnitude and location of the maximum residual shearing stress, (b) the permanent angle of twist.

Anand Jangid
Anand Jangid
Numerade Educator
12:55

Problem 117

After the solid shaft of Prob. 3.116 has been loaded and unloaded as described in that problem, a torque $\mathbf{T}_{1}$ of sense opposite to the original torque $\mathbf{T}$ is applied to the shaft. Assuming no change in the value of $\phi_{Y},$ determine the angle of twist $\phi_{1}$ for which yield is initiated in this second loading and compare it with the angle $\phi_{Y}$ for which the shaft started to yield in the original loading.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:21

Problem 118

The hollow shaft shown is made of a steel that is assumed to be elastoplastic with $\tau_{Y}=145 \mathrm{MPa}$ and $G=77.2 \mathrm{GPa}$. The magnitude $T$ of the torques is slowly increased until the plastic zone first reaches the inner surface of the shaft; the torques are then removed. Determine the magnitude and location of the maximum residual shearing stress in the rod.

Anand Jangid
Anand Jangid
Numerade Educator
09:19

Problem 119

In Prob. 3.118 , determine the permanent angle of twist of the rod.

Jonathan Ibarra
Jonathan Ibarra
Numerade Educator
07:05

Problem 120

A torque $\mathbf{T}$ applied to a solid rod made of an elastoplastic material is increased until the rod is fully plastic and then removed. ( $a$ ) Show that the distribution of residual shearing stresses is as represented in the figure. (b) Determine the magnitude of the torque due to the stresses acting on the portion of the rod located within a circle of radius $c_{0}.$

Henry York
Henry York
Numerade Educator
00:40

Problem 121

Determine the smallest allowable square cross section of a steel shaft of length $20 \mathrm{ft}$ if the maximum shearing stress is not to exceed $10 \mathrm{ksi}$ when the shaft is twisted through one complete revolution. Use $G=11.2 \times 10^{6}$ psi.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:19

Problem 122

Determine the smallest allowable length of a stainless steel shaft of $\frac{3}{8} \times \frac{3}{4}$ -in. cross section if the shearing stress is not to exceed 15 ksi when the shaft is twisted through $15^{\circ} .$ Use $G=11.2 \times 10^{6}$ psi.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:54

Problem 123

Using $\tau_{\text {all }}=70 \mathrm{MPa}$ and $G=27 \mathrm{GPa}$, determine for each of the aluminum bars shown the largest torque $\mathbf{T}$ that can be applied and the corresponding angle of twist at end $B$.

Naman Kumar
Naman Kumar
Numerade Educator
02:54

Problem 124

Knowing that the magnitude of the torque $\mathbf{T}$ is $200 \mathrm{N} \cdot \mathrm{m}$ and that $G=27$ GPa, determine for each of the aluminum bars shown the maximum shearing stress and the angle of twist at end $B$.

Naman Kumar
Naman Kumar
Numerade Educator
03:13

Problem 125

Using $\tau_{\text {all }}=7.5 \mathrm{ksi}$ and knowing that $G=5.6 \times 10^{6} \mathrm{psi}$, determine for each of the cold-rolled yellow brass bars shown the largest torque $\mathbf{T}$ that can be applied and the corresponding angle of twist at end $B$.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
03:13

Problem 126

Knowing that $T=7$ kip.in. and that $G=5.6 \times 10^{6} \mathrm{psi}$, determine for each of the cold-rolled yellow brass bars shown the maximum shearing stress and the angle of twist at end $B$.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
01:38

Problem 127

The torque $\mathbf{T}$ causes a rotation of $0.6^{\circ}$ at end $B$ of the aluminum bar shown. Knowing that $b=15 \mathrm{mm}$ and $G=26 \mathrm{GPa}$, determine the maximum shearing stress in the bar.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
06:07

Problem 128

The torque $\mathbf{T}$ causes a rotation of $2^{\circ}$ at end $B$ of the stainless steel bar shown. Knowing that $b=20 \mathrm{mm}$ and $G=75 \mathrm{GPa}$, determine the maximum shearing stress in the bar.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
05:15

Problem 129

Two shafts are made of the same material. The cross section of shaft $A$ is a square of side $b$ and that of shaft $B$ is a circle of diameter $b$ Knowing that the shafts are subjected to the same torque, determine the ratio $\tau_{A} / \tau_{B}$ of maximum shearing stresses occurring in the shafts.

Narayan Hari
Narayan Hari
Numerade Educator
00:40

Problem 130

Shafts $A$ and $B$ are made of the same material and have the same cross-sectional area, but $A$ has a circular cross section and $B$ has a square cross section. Determine the ratio of the maximum torques $T_{A}$ and $T_{B}$ when the two shafts are subjected to the same maximum shearing stress $\left(\tau_{A}=\tau_{B}\right) .$ Assume both deformations to be elastic.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
00:40

Problem 131

Shafts $A$ and $B$ are made of the same material and have the same length and cross-sectional area, but $A$ has a circular cross section and $B$ has a square cross section. Determine the ratio of the maximum values of the angles $\phi_{A}$ and $\phi_{B}$ when the two shafts are subjected to the same maximum shearing stress $\left(\tau_{A}=\tau_{B}\right) .$ Assume both deformations to be elastic.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:29

Problem 132

Shafts $A$ and $B$ are made of the same material and have the same crosssectional area, but $A$ has a circular cross section and $B$ has a square cross section. Determine the ratio of the maximum shearing stresses occurring in $A$ and $B$, respectively, when the two shafts are subjected to the same torque $\left(T_{A}=T_{B}\right)$. Assume both deformations to be elastic.

Chai Santi
Chai Santi
Numerade Educator
05:15

Problem 133

Each of the three steel bars is subjected to a torque as shown. Knowing that the allowable shearing stress is $8 \mathrm{ksi}$ and that $b=1.4$ in., determine the maximum torque $\mathbf{T}$ that can be applied to each bar.

Narayan Hari
Narayan Hari
Numerade Educator
02:54

Problem 134

Each of the three aluminum bars shown is to be twisted through an angle of $2^{\circ} .$ Knowing that $b=30 \mathrm{mm}, \tau_{\text {all }}=50 \mathrm{MPa},$ and $G=27$ GPa, determine the shortest allowable length of each bar.

Naman Kumar
Naman Kumar
Numerade Educator
02:09

Problem 135

A 1.25 -m-long steel angle has an $\mathrm{L} 127 \times 76 \times 6.4$ cross section. From Appendix E we find that the thickness of the section is $6.4 \mathrm{mm}$ and that its area is $1250 \mathrm{mm}^{2} .$ Knowing that $\tau_{\text {all }}=60 \mathrm{MPa}$ and that $G=77.2 \mathrm{GPa}$, and ignoring the effect of stress concentrations, determine $(a)$ the largest torque $\mathbf{T}$ that can be applied, $(b)$ the corresponding angle of twist.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:36

Problem 136

A $3000-1 b$ -in. torque is applied to a 6 -ft-long steel angle with an $\mathrm{L} 4 \times 4 \times \frac{3}{8}$ cross section. From Appendix $\mathrm{E}$ we find that the thickness of the section is $\frac{3}{8}$ in. and that its area is 2.86 in $^{2}$. Knowing that $G=11.2 \times 10^{6}$ psi, determine ( $a$ ) the maximum shearing stress along line $a-a,(b)$ the angle of twist.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:09

Problem 137

A $4-\mathrm{m}$ -long steel member has a $\mathrm{W} 310 \times 60$ cross section. Knowing that $G=77.2 \mathrm{GPa}$ and that the allowable shearing stress is $40 \mathrm{MPa},$ determine $(a)$ the largest torque $\mathrm{T}$ that can be applied, (b) the corresponding angle of twist. Refer to Appendix E for the dimensions of the cross section and neglect the effect of stress concentrations. (Hint: Consider the web and flanges separately and obtain a relation between the torques exerted on the web and a flange, respectively, by expressing that the resulting angles of twist are equal.)

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:36

Problem 138

An 8 -ft-long steel member with a $\mathrm{W} 8 \times 31$ cross section is subjected to a 5 -kip.in. torque. The properties of the rolled-steel section are given in Appendix E. Knowing that $G=11.2 \times 10^{6}$ psi, determine ( $a$ ) the maximum shearing stress along line $a-a,(b)$ the maximum shearing stress along line $b-b,(c)$ the angle of twist. (See hint of Prob. 3.137 .)

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:18

Problem 139

A 5-kip.ft torque is applied to a hollow aluminum shaft having the cross section shown. Neglecting the effect of stress concentrations, determine the shearing stress at points $a$ and $b$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:20

Problem 140

A torque $T=750 \mathrm{kN} \cdot \mathrm{m}$ is applied to the hollow shaft shown that has a uniform 8 -mm wall thickness. Neglecting the effect of stress concentrations, determine the shearing stress at points $a$ and $b$.

Chai Santi
Chai Santi
Numerade Educator
01:03

Problem 141

A $5.6-\mathrm{kN} \cdot \mathrm{m}$ torque is applied to a hollow shaft having the cross section shown. Neglecting the effect of stress concentrations, determine the shearing stress at points $a$ and $b$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:33

Problem 142

A hollow member having the cross section shown is formed from sheet metal of 2 -mm thickness. Knowing that the shearing stress must not exceed 3 MPa, determine the largest torque that can be applied to the member.

Naman Kumar
Naman Kumar
Numerade Educator
04:33

Problem 143

A hollow member having the cross section shown is formed from sheet metal of 2 -mm thickness. Knowing that the shearing stress must not exceed 3 MPa, determine the largest torque that can be applied to the member.

Naman Kumar
Naman Kumar
Numerade Educator
02:18

Problem 144

A $90-\mathrm{N} \cdot \mathrm{m}$ torque is applied to a hollow shaft having the cross section shown. Neglecting the effect of stress concentrations, determine the shearing stress at points $a$ and $b$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:19

Problem 145

A hollow member having the cross section shown is to be formed from sheet metal of $\frac{1}{16}$ -in. thickness. Knowing that a 3 -kip.in. torque will be applied to the member, determine the smallest dimension $d$ that can be used if the shearing stress is not to exceed 500 psi.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:17

Problem 146

A sheet-metal strip 6 in. wide and 0.12 in. thick is to be formed into a tube of rectangular cross section. Knowing that $\tau_{\text {all }}=4 \mathrm{ksi}$, determine the largest torque that can be applied to the tube when $(a) w=1.5$ in. $(b) w=1.2$ in. $(c) w=1$ in.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:02

Problem 147

A cooling tube having the cross section shown is formed from a sheet of stainless steel of 3 -mm thickness. The radii $c_{1}=150 \mathrm{mm}$ and $c_{2}=$ $100 \mathrm{mm}$ are measured to the center line of the sheet metal. Knowing that a torque of magnitude $T=3 \mathrm{kN} \cdot \mathrm{m}$ is applied to the tube, determine $(a)$ the maximum shearing stress in the tube, (b) the magnitude of the torque carried by the outer circular shell. Neglect the dimension of the small opening where the outer and inner shells are connected.

AP
Andreas Papavassiliou
Numerade Educator
04:29

Problem 148

A hollow cylindrical shaft was designed to have a uniform wall thickness of 0.1 in. Defective fabrication, however, resulted in the shaft having the cross section shown. Knowing that a 15 -kip.in. torque is applied to the shaft, determine the shearing stresses at points $a$ and $b$.

Chai Santi
Chai Santi
Numerade Educator
02:56

Problem 149

Equal torques are applied to thin-walled tubes of the same length $L$ same thickness $t,$ and same radius $c .$ One of the tubes has been slit lengthwise as shown. Determine $(a)$ the ratio $\tau_{b} / \tau_{a}$ of the maximum shearing stresses in the tubes, $(b)$ the ratio $\phi_{b} / \phi_{a}$ of the angles of twist of the tubes.

RZ
Rubeena Zulfiqar
Numerade Educator
09:49

Problem 150

A hollow cylindrical shaft of length $L,$ mean radius $c_{m},$ and uniform thickness $t$ is subjected to a torque of magnitude $T .$ Consider, on the one hand, the values of the average shearing stress $\tau_{\text {ave }}$ and the angle of twist $\phi$ obtained from the elastic torsion formulas developed in Secs. $3.1 \mathrm{C}$ and 3.2 and, on the other hand, the corresponding values obtained from the formulas developed in Sec. 3.10 for thin-walled shafts. $(a)$ Show that the relative error introduced by using the thin-walled-shaft formulas rather than the elastic torsion formulas is the same for $\tau_{\text {ave }}$ and $\phi$ and that the relative error is positive and proportional to the ratio $t / c_{m \cdot}(b)$ Compare the percent error corresponding to values of the ratio $t / c_{m}$ of 0.1,0.2 and 0.4.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:17

Problem 151

A steel pipe of 12 -in. outer diameter is fabricated from $\frac{1}{4}$ -in.-thick plate by welding along a helix that forms an angle of $45^{\circ}$ with a plane parallel to the axis of the pipe. Knowing that the maximum allowable tensile stress in the weld is 12 ksi, determine the largest torque that can be applied to the pipe.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
06:19

Problem 152

A torque of magnitude $T=120 \mathrm{N} \cdot \mathrm{m}$ is applied to shaft $A B$ of the gear train shown. Knowing that the allowable shearing stress is $75 \mathrm{MPa}$ in each of the three solid shafts, determine the required diameter of $(a)$ shaft $A B$ (b) shaft $C D$ $(c)$ shaft $E F.$

Ajay Singhal
Ajay Singhal
Numerade Educator
04:50

Problem 153

The solid cylindrical rod $B C$ is attached to the rigid lever $A B$ and to the fixed support at $C .$ The vertical force $P$ applied at $A$ causes a small displacement $\Delta$ at point $A$. Show that the corresponding max imum shearing stress in the rod is
\[\tau=\frac{G d}{2 L a} \Delta\]
where $d$ is the diameter of the rod and $G$ is its modulus of rigidity.

Satpal Satpal
Satpal Satpal
Numerade Educator
01:35

Problem 154

In the bevel-gear system shown, $\alpha=18.43^{\circ} .$ Knowing that the allowable shearing stress is $8 \mathrm{ksi}$ in each shaft and that the system is in equilibrium, determine the largest torque $\mathbf{T}_{A}$ that can be applied at $A$.

Chai Santi
Chai Santi
Numerade Educator
00:39

Problem 155

Three solid shafts, each of $\frac{3}{4}$ -in. diameter, are connected by the gears shown. Knowing that $G=11.2 \times 10^{6}$ psi, determine ( $a$ ) the angle through which end $A$ of shaft $A B$ rotates, $(b)$ the angle through which end $E$ of shaft $E F$ rotates.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:18

Problem 156

The composite shaft shown consists of a 5 -mm-thick brass jacket $\left(G_{\text {brass }}=39 \quad \mathrm{GPa}\right)$ bonded to a $40-\mathrm{mm}$ -diameter steel core $\left(G_{\text {steel }}=77.2 \mathrm{GPa}\right) .$ Knowing that the shaft is subjected to a $600-\mathrm{N} \cdot \mathrm{m}$ torque, determine $(a)$ the maximum shearing stress in the brass jacket, (b) the maximum shearing stress in the steel core, $(c)$ the angle of twist of $B$ relative to $A$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
08:04

Problem 157

Ends $A$ and $D$ of the two solid steel shafts $A B$ and $C D$ are fixed, while ends $B$ and $C$ are connected to gears as shown. Knowing that the allowable shearing stress is $50 \mathrm{MPa}$ in each shaft, determine the largest torque $\mathbf{T}$ that can be applied to gear $B$.

Narayan Hari
Narayan Hari
Numerade Educator
01:16

Problem 158

As the hollow steel shaft shown rotates at $180 \mathrm{rpm},$ a stroboscopic measurement indicates that the angle of twist of the shaft is $3^{\circ}$ Knowing that $G=77.2 \mathrm{GPa}$, determine ( $a$ ) the power being transmitted, (b) the maximum shearing stress in the shaft.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
00:56

Problem 159

Knowing that the allowable shearing stress is 8 ksi for the stepped shaft shown, determine the magnitude $T$ of the largest torque that can be transmitted by the shaft when the radius of the fillet is $(a) r=\frac{3}{16}$ in, (b) $r=\frac{1}{4}$ in.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:56

Problem 160

Equal torques are applied to thin-walled tubes of the same thickness $t$ and same radius $c .$ One of the tubes has been slit lengthwise as shown. Determine the ratio $\tau_{b} / \tau_{a}$ of the maximum shearing stresses in the tubes.

RZ
Rubeena Zulfiqar
Numerade Educator
03:30

Problem 161

Two solid brass rods $A B$ and $C D$ are brazed to a brass sleeve $E F$ Determine the ratio $d_{2} / d_{1}$ for which the same maximum shearing stress occurs in the rods and in the sleeve.

Chai Santi
Chai Santi
Numerade Educator
01:34

Problem 162

The shaft $A B$ is made of a material that is elastoplastic with $\tau_{Y}=12.5 \mathrm{ksi}$ and $G=4 \times 10^{6}$ psi. For the loading shown, determine $(a)$ the radius of the elastic core of the shaft, (b) the angle of twist of the shaft.

Chai Santi
Chai Santi
Numerade Educator