Engineering Mechanics: Statics

Russell C. Hibbeler

Chapter 11

Virtual Work - all with Video Answers

Educators

Chapter Questions

Problem 1

Use the method of virtual work to determine the tension in cable $A C .$ The lamp weighs $10 \mathrm{lb}.$

Ryan Pollard

Ryan Pollard

Numerade Educator

Problem 2

The scissors jack supports a load $\mathbf{P}$. Determine the axial force in the screw necessary for equilibrium when the jack is in the position $\theta$. Each of the four links has a length $L$ and is pin connected at its center. Points $B$ and $D$ can move horizontally.

Rachel Peterson

Rachel Peterson

Numerade Educator

Problem 3

If a force of $P=5$ lb is applied to the handle of the mechanism, determine the force the screw exerts on the cork of the bottle. The screw is attached to the pin at $A$ and passes through the collar that is attached to the bottle neck at $B.$

Anand Jangid

Numerade Educator

Problem 4

The disk has a weight of $10 \mathrm{lb}$ and is subjected to a vertical force $P=8$ lb and a couple moment $M=8 \mathrm{lb} \cdot \mathrm{ft}$ Determine the disk's rotation $\theta$ if the end of the spring wraps around the periphery of the disk as the disk turns. The spring is originally unstretched.

Gordon Ayadju

Numerade Educator

Problem 5

The punch press consists of the ram $R,$ connecting rod $A B,$ and a flywheel. If a torque of $M=75 \mathrm{N} \cdot \mathrm{m}$ is applied to the flywheel, determine the force $\mathbf{F}$ applied at the ram to hold the rod in the position $\theta=60^{\circ}$

Rachel Peterson

Rachel Peterson

Numerade Educator

Problem 6

$\begin{array}{l} \text { The } \text { flywheel } \text { is } \text { subjected } \text { to } \text { a } \text { torque } \text { of }\end{array}$ $M=75 \mathrm{N} \cdot \mathrm{m} .$ Determine the horizontal compressive force $F$ and plot the result of $F$ (ordinate) versus the equilibrium position $\theta(\text { abscissa })$ for $0^{\circ} \leq \theta \leq 180^{\circ}$

Gordon Ayadju

Numerade Educator

Problem 7

When $\theta=20^{\circ},$ the 50 -lb uniform block compresses the two vertical springs 4 in. If the uniform links $A B$ and $C D$ each weigh 10 Ib, determine the magnitude of the applied couple moments $\mathbf{M}$ needed to maintain equilibrium when $\theta=20^{\circ}.$

Gordon Ayadju

Numerade Educator

Problem 8

The bar is supported by the spring and smooth collar that allows the spring to be always perpendicular to the bar for any angle $\theta$. If the unstretched length of the spring is $l_{0},$ determine the force $P$ needed to hold the bar in the equilibrium position $\theta .$ Neglect the weight of the bar.

Rashmi Sinha

Numerade Educator

Problem 9

The $4-$ ft members of the mechanism are pin connected at their centers. If vertical forces $P_{1}=P_{2}=30 \mathrm{lb}$ act at $C$ and $E$ as shown, determine the angle $\theta$ for equilibrium. The spring is unstretched when $\theta=45^{\circ}$ Neglect the weight of the members.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 10

The thin rod of weight $W$ rests against the smooth wall and floor. Determine the magnitude of force $\mathbf{P}$ needed to hold it in equilibrium for a given angle $\theta.$

Gordon Ayadju

Numerade Educator

Problem 11

If each of the three links of the mechanism have a mass of $4 \mathrm{kg}$, determine the angle $\theta$ for cquilibrium. The spring, which always remains vertical, is unstretched when $\theta=0^{\circ}$

Gordon Ayadju

Numerade Educator

Problem 12

The disk is subjected to a couple moment $M$. Determine the disk's rotation $\theta$ required for equilibrium. The end of the spring wraps around the periphery of the disk as the disk turns. The spring is originally unstretched.

Gordon Ayadju

Numerade Educator

Problem 13

A 5 -kg uniform serving table is supported on each side by pairs of two identical links, $A B$ and $C D,$ and springs $C E .$ If the bowl has a mass of $1 \mathrm{kg}$, determine the angle $\theta$ where the table is in equilibrium. The springs each have a stiffness of $k=200 \mathrm{N} / \mathrm{m}$ and are unstretched when $\theta=90^{\circ} .$ Neglect the mass of the links.

Gordon Ayadju

Numerade Educator

Problem 14

A $5-\mathrm{kg}$ uniform serving table is supported on each side by two pairs of identical links, $A B$ and $C D,$ and springs $C E .$ If the bowl has a mass of $1 \mathrm{kg}$ and is in equilibrium when $\theta=45^{\circ},$ determine the stiffness $k$ of each spring. The springs are unstretched when $\theta=90^{\circ} .$ Neglect the mass of the links.

Gordon Ayadju

Numerade Educator

Problem 15

The service window at a fast-food restaurant consists of glass doors that open and close automatically using a motor which supplies a torque $\mathbf{M}$ to each door. The far ends, $A$ and $B,$ move along the horizontal guides. If a food tray becomes stuck between the doors as shown, determine the horizontal force the doors exert on the tray at the position $\theta.$

Gordon Ayadju

Numerade Educator

Problem 16

The members of the mechanism are pin connected. If a vertical force of $800 \mathrm{N}$ acts at $A,$ determine the angle $\theta$ for equilibrium. The spring is unstretched when $\theta=0^{\circ}$ Neglect the mass of the links.

Gordon Ayadju

Numerade Educator

Problem 17

When $\theta=30^{\circ},$ the 25 -kg uniform block compresses the two horizontal springs $100 \mathrm{mm}$. Determine the magnitude of the applicd couple moments $\mathbf{M}$ needed to maintain equilibrium. Take $k=3 \mathrm{kN} / \mathrm{m}$ and neglect the mass of the links.

Gordon Ayadju

Numerade Educator

Problem 18

The "Nuremberg scissors" is subjected to horizontal force of $P=600 \mathrm{N}$. Determine the angle $\theta$ for equilibrium. The spring has a stiffness of $k=15 \mathrm{kN} / \mathrm{m}$ and is unstretched when $\theta=15^{\circ}.$

Gordon Ayadju

Numerade Educator

Problem 19

The Nurcmberg } & \text { scissors" } is \ subjected to a horizontal force of $P=600 \mathrm{N}$. Determine the stiffness $k$ of the spring for equilibrium when $\theta=60^{\circ} .$ The spring is unstretched when $\theta=15^{\circ}.$

Gordon Ayadju

Numerade Educator

Problem 20

The crankshaft is subjected to a torque of $M=50 \mathrm{N} \cdot \mathrm{m} .$ Determine the horizontal compressive force $F$ applied to the piston for equilibrium when $\theta=60^{\circ}.$

Gordon Ayadju

Numerade Educator

Problem 21

The crankshaft is subjected to a torque of $M=50 \mathrm{N} \cdot \mathrm{m} .$ Determine the horizontal compressive force $F$ and plot the result of $F$ (ordinate) versus $\theta$ (abscissa) for $0^{\circ} \leq \theta \leq 90^{\circ}$

Vipender Yadav

Numerade Educator

Problem 22

The spring is unstretched when $\theta=0^{\circ} .$ If $P=8 \mathrm{lb}$ determine the angle $\theta$ for equilibrium. Due to the roller guide, the spring always remains vertical. Neglect the weight of the links.

Gordon Ayadju

Numerade Educator

Problem 23

\Determine the weight of block $G$ required to balance the differential lever when the 20 -lb load $F$ is placed on the pan. The lever is in balance when the load and block are not on the lever. Take $x=12$ in.

Gordon Ayadju

Numerade Educator

Problem 24

If the load $F$ weighs $20 \mathrm{lb}$ and the block $G$ weighs 2 Ib, determine its position $x$ for equilibrium of the differential lever. The lever is in balance when the load and block are not on the lever.

Gordon Ayadju

Numerade Educator

Problem 25

The dumpster has a weight $W$ and a center of gravity at $G .$ Determine the force in the hydraulic cylinder needed to hold it in the general position $\theta$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 26

The potential energy of a one-degree-of-freedom system is defined by $V=\left(20 x^{3}-10 x^{2}-25 x-10\right) \mathrm{ft} \cdot \mathrm{lb}$ where $x$ is in $\mathrm{ft}$. Determine the equilibrium positions and investigate the stability for each position.

Gordon Ayadju

Numerade Educator

Problem 27

If the potential function for a conservative onedegree-of-freedom system is $V=(12 \sin 2 \theta+15 \cos \theta) \mathrm{J}$ where $0^{\circ}<\theta<180^{\circ}$, determine the positions for equilibrium and investigate the stability at each of these positions.

Gordon Ayadju

Numerade Educator

Problem 28

If the potential function for a conservative onedegree-of-freedom system is $V=\left(8 x^{3}-2 x^{2}-10\right) \mathrm{J}$ where $x$ is given in meters, determine the positions for equilibrium and investigate the stability at each of these positions.

Gordon Ayadju

Numerade Educator

Problem 29

If the potential function for a conservative onedegree-of-freedom system is $V=(10 \cos 2 \theta+25 \sin \theta) \mathrm{J}$ where $0^{\circ}<\theta<180^{\circ}$, determine the positions for equilibrium and investigate the stability at each of these positions.

Gordon Ayadju

Numerade Educator

Problem 30

If the potential energy for a conservative onedegree-of-freedom system is expressed by the relation $V=\left(4 x^{3}-x^{2}-3 x+10\right) \mathrm{ft} \cdot \mathrm{lb},$ where $x$ is given in feet determine the equilibrium positions and investigate the stability at each position.

Gordon Ayadju

Numerade Educator

Problem 31

The uniform link $A B,$ has a mass of 3 kg and is pin connected at both of its ends. The rod $B D$, having negligible weight, passes through a swivel block at $C .$ If the spring has a stiffness of $k=100 \mathrm{N} / \mathrm{m}$ and is unstretched when $\theta=0^{\circ}$ determine the angle $\theta$ for equilibrium and investigate the stability at the equilibrium position. Neglect the size of the swivel block.

Gordon Ayadju

Numerade Educator

Problem 32

The spring of the scale has an unstretched length of $a .$ Determine the angle $\theta$ for equilibrium when a weight $W$ is supported on the platform. Neglect the weight of the members. What value $W$ would be required to keep the scale in neutral equilibrium when $\theta=0^{\circ} ?$

Gordon Ayadju

Numerade Educator

Problem 33

The uniform bar has a mass of $80 \mathrm{kg}$. Determine the angle $\theta$ for equilibrium and investigate the stability of the bar when it is in this position. The spring has an unstretched length when $\theta=90^{\circ}.$

Gordon Ayadju

Numerade Educator

Problem 34

The uniform bar $A D$ has a mass of $20 \mathrm{kg}$. If the attached spring is unstretched when $\theta=90^{\circ},$ determine the angle $\theta$ for equilibrium. Note that the spring always remains in the vertical position due to the roller guide. Investigate the stability of the bar when it is in the equilibrium position.

Gordon Ayadju

Numerade Educator

Problem 35

The two bars each have a weight of 8 lb. Determine the required stiffness $k$ of the spring so that the two bars are in equilibrium when $\theta=30^{\circ} .$ The spring has an unstretched length of $1 \mathrm{ft}$

Gordon Ayadju

Numerade Educator

Problem 36

Determine the angle $\theta$ for equilibrium and investigate the stability at this position. The bars each have a mass of $3 \mathrm{kg}$ and the suspended block $D$ has a mass of $7 \mathrm{kg} .$ Cord $D C$ has a total length of $1 \mathrm{m}.$

Gordon Ayadju

Numerade Educator

Problem 37

Determine the angle $\theta$ for equilibrium and investigate the stability at this position. The bars each have a mass of $10 \mathrm{kg}$ and the spring has an unstretched length of $100 \mathrm{mm}.$

Gordon Ayadju

Numerade Educator

Problem 38

The two bars cach have a mass of 8 kg. Determine the required stiffness $k$ of the spring so that the two bars are in equilibrium when $\theta=60^{\circ} .$ The spring has an unstretched length of $1 \mathrm{m}$. Investigate the stability of the system at the equilibrium position.

Gordon Ayadju

Numerade Educator

Problem 39

A spring with a torsional stiffness $k$ is attached to the hinge at $B$. It is unstretched when the rod assembly is in the vertical position. Determine the weight $W$ of the block that results in neutral equilibrium. Hint: Establish the potential energy function for a small angle $\theta,$ i.e. approximate $\sin \theta \approx 0,$ and $\cos \theta \approx 1-\theta^{2} / 2.$

Gordon Ayadju

Numerade Educator

Problem 40

A conical holc is drilled into the bottom of the cylinder, which is supported on the fulcrum at $A$. Determine the minimum distance $d$ in order for it to remain in stable equilibrium.

Saman Zulfiqar

Numerade Educator

Problem 41

The uniform rod has a mass of 100 kg. If the spring is unstretched when $\theta=60^{\circ},$ determine the angle $\theta$ for equilibrium and investigate the stability at the equilibrium position. The spring is always in the horizontal position duc to the roller guide at $B.$

Gordon Ayadju

Numerade Educator

Problem 42

Each bar has a mass per length of $m_{0} .$ Determine the angles $\theta$ and $\phi$ at which they are suspended in equilibrium. The contact at $A$ is smooth, and both are pin connected at $B.$

Gordon Ayadju

Numerade Educator

Problem 43

The truck has a mass of $20 \mathrm{Mg}$ and a mass center at $G .$ Determine the steepest grade $\theta$ along which it can park without overturning and investigate the stability in this position.

Gordon Ayadju

Numerade Educator

Problem 44

The small postal scale consists of a counterweight $W_{1},$ connected to the members having negligible weight. Determine the weight $W_{2}$ that is on the pan in terms of the angles $\theta$ and $\phi$ and the dimensions shown. All members are pin connected.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

A $3-1 b$ weight is attached to the end of rod $A B C$. If the rod is supported by a smooth slider block at $C$ and rod $B D$, determine the angle $\theta$ for equilibrium. Neglect the weight of the rods and the slider.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 46

If the uniform rod $O A$ has a mass of 12 kg. determine the mass $m$ that will hold the rod in equilibrium when $\theta=30^{\circ} .$ Point $C$ is coincident with $B$ when $O A$ is horizontal. Neglect the size of the pulley at $B$.

Gordon Ayadju

Numerade Educator

Problem 47

The cylinder is made of two materials such that it has a mass of $m$ and a center of gravity at point $G .$ Show that when $G$ lies above the centroid $C$ of the cylinder, the equilibrium is unstable.

Gordon Ayadju

Numerade Educator

Problem 48

The bent rod has a weight of $5 \mathrm{lb} / \mathrm{ft}$. A pivot is attached at its center $A$ and the rod is balanced as shown. Determine the length $L$ of its vertical segments so that it remains in neutral equilibrium. Neglect the thickness of the rod.

Gordon Ayadju

Numerade Educator

Problem 49

The triangular block of weight $W$ rests on the smooth corners which are a distance $a$ apart. If the block has three equal sides of length $d$, determine the angle $\theta$ for equilibrium.

Penny Riley

Numerade Educator