Engineering Mechanics: Statics and Dynamics

R. C. Hibbeler

Chapter 17

Planar Kinetics of a Rigid Body: Force and Acceleration - all with Video Answers

Educators

Chapter Questions

Problem 1

Determine the moment of inertia $I_{y}$ for the slender rod. The rod's density $\rho$ and cross-sectional area $A$ are constant. Express the result in terms of the rod's total mass $m$.

Ahmed Kamel

Ahmed Kamel

Numerade Educator

Problem 2

The solid cylinder has an outer radius $R$, height $h$ and is made from a material having a density that varies from its center as $\rho=k+a r^{2},$ where $k$ and $a$ are constants. Determine the mass of the cylinder and its moment of inertia about the $z$ axis.

Ahmed Kamel

Numerade Educator

Problem 3

Determine the moment of inertia of the thin ring about the $z$ axis. The ring has a mass $m$.

Ahmed Kamel

Numerade Educator

Problem 4

The paraboloid is formed by revolving the shaded area around the $x$ axis. Determine the radius of gyration $k_{x}$ The density of the material is $\rho=5 \mathrm{Mg} / \mathrm{m}^{3}$.

Ahmed Kamel

Numerade Educator

Problem 5

Determine the radius of gyration $k_{x}$ of the body. The specific weight of the material is $\gamma=380 \mathrm{lb} / \mathrm{ft}^{3}$.

Ahmed Kamel

Numerade Educator

Problem 6

The sphere is formed by revolving the shaded area around the $x$ axis. Determine the moment of inertia $I_{x}$ and express the result in terms of the total mass $m$ of the sphere. The material has a constant density $\rho$.

Ahmed Kamel

Numerade Educator

Problem 7

The frustum is formed by rotating the shaded area around the $x$ axis. Determine the moment of inertia $I_{x}$ and express the result in terms of the total mass $m$ of the frustum. The frustum has a constant density $\rho$.

Ahmed Kamel

Numerade Educator

Problem 8

The hemisphere is formed by rotating the shaded area around the $y$ axis. Determine the moment of inertia $I_{y}$ and express the result in terms of the total mass $m$ of the hemisphere. The material has a constant density $\rho$.

Ahmed Kamel

Numerade Educator

Problem 9

Determine the moment of inertia of the homogeneous triangular prism with respect to the $y$ axis. Express the result in terms of the mass $m$ of the prism. Hint:
For integration, use thin plate elements parallel to the $x-y$ plane and having a thickness $d z$.

Ahmed Kamel

Numerade Educator

Problem 10

The pendulum consists of a 4-kg circular plate and a 2 -kg slender rod. Determine the radius of gyration of the pendulum about an axis perpendicular to the page and passing through point $O$.

Ahmed Kamel

Numerade Educator

Problem 11

The assembly is made of the slender rods that have a mass per unit length of $3 \mathrm{kg} / \mathrm{m}$. Determine the mass moment of inertia of the assembly about an axis perpendicular to the page and passing through point $O$.

Ahmed Kamel

Numerade Educator

Problem 12

Determine the moment of inertia of the solid steel assembly about the $x$ axis. Steel has a specific weight of $\gamma_{s t}=490 \mathrm{lb} / \mathrm{ft}^{3}$.

Ahmed Kamel

Numerade Educator

Problem 13

The wheel consists of a thin ring having a mass of $10 \mathrm{kg}$ and four spokes made from slender rods and each having a mass of 2 kg. Determine the wheel's moment of inertia about an axis perpendicular to the page and passing through point $A$.

Ahmed Kamel

Numerade Educator

Problem 14

If the large ring, small ring and each of the spokes weigh $100 \mathrm{lb}, 15 \mathrm{lb},$ and $20 \mathrm{lb}$, respectively, determine the mass moment of inertia of the wheel about an axis perpendicular to the page and passing through point $A$.

Ahmed Kamel

Numerade Educator

Problem 15

Determine the moment of inertia about an axis perpendicular to the page and passing through the pin at $O$ The thin plate has a hole in its center. Its thickness is $50 \mathrm{mm}$ and the material has a density $\rho=50 \mathrm{kg} / \mathrm{m}^{3}$.

Ahmed Kamel

Numerade Educator

Problem 16

Determine the mass moment of inertia of the thin plate about an axis perpendicular to the page and passing through point $O .$ The material has a mass per unit area of $20 \mathrm{kg} / \mathrm{m}^{2}$.

Ahmed Kamel

Numerade Educator

Problem 17

Determine the location $\bar{y}$ of the center of mass $G$ of the assembly and then calculate the moment of inertia about an axis perpendicular to the page and passing through $G$ The block has a mass of $3 \mathrm{kg}$ and the semicylinder has a mass of $5$ $\mathrm{kg}$.

Ahmed Kamel

Numerade Educator

Problem 18

Determine the moment of inertia of the assembly about an axis perpendicular to the page and passing through point $O .$ The block has a mass of $3 \mathrm{kg}$, and the semicylinder has a mass of $5 \mathrm{kg}$.

Ahmed Kamel

Numerade Educator

Problem 19

Determine the moment of inertia of the wheel about an axis which is perpendicular to the page and passes through the center of mass $G .$ The material has a specific weight $\gamma=90 \mathrm{lb} / \mathrm{ft}^{3}$.

Ahmed Kamel

Numerade Educator

Problem 20

Determine the moment of inertia of the wheel about an axis which is perpendicular to the page and passes through point O. The material has a specific weight $\gamma=90 \mathrm{lb} / \mathrm{ft}^{3}$.

Ahmed Kamel

Numerade Educator

Problem 21

The pendulum consists of the 3 -kg slender rod and the 5 -kg thin plate. Determine the location $\bar{y}$ of the center of mass $G$ of the pendulum; then calculate the moment of inertia of the pendulum about an axis perpendicular to the page and passing through $G$.

Ahmed Kamel

Numerade Educator

Problem 22

Determine the moment of inertia of the overhung crank about the $x$ axis. The material is steel having a density of $\rho=7.85 \mathrm{Mg} / \mathrm{m}^{3}$.

Ahmed Kamel

Numerade Educator

Problem 23

Determine the moment of inertia of the overhung crank about the $x^{\prime}$ axis. The material is steel having a density of $\rho=7.85 \mathrm{Mg} / \mathrm{m}^{3}$.

Ahmed Kamel

Numerade Educator

Problem 24

The door has a weight of $200 \mathrm{lb}$ and a center of gravity at $G .$ Determine how far the door moves in $2 \mathrm{s}$ starting from rest, if a man pushes on it at $C$ with a horizontal force $F=30$ lb. Also, find the vertical reactions at the rollers $A$ and $B$.

Ahmed Kamel

Numerade Educator

Problem 25

The door has a weight of $200 \mathrm{lb}$ and a center of gravity at $G .$ Determine the constant force $F$ that must be applied to the door to push it open 12 ft to the right in 5 s, starting from rest. Also, find the vertical reactions at the rollers $A$ and $B$.

Ahmed Kamel

Numerade Educator

Problem 26

The jet aircraft has a total mass of $22 \mathrm{Mg}$ and a center of mass at $G .$ Initially at take-off the engines provide a thrust $2 T=4 \mathrm{kN}$ and $T^{\prime}=1.5 \mathrm{kN}$. Determine the acceleration of the plane and the normal reactions on the nose wheel at $A$ and each of the $t w o$ wing wheels located at $B$. Neglect the mass of the wheels and, due to low velocity, neglect any lift caused by the wings.

Ahmed Kamel

Numerade Educator

Problem 27

The sports car has a weight of 4500 lb and center of gravity at $G$. If it starts from rest it causes the rear wheels to slip as it accelerates. Determine how long it takes for it to reach a speed of $10 \mathrm{ft} / \mathrm{s}$. Also, what are the normal reactions at each of the four wheels on the road?
The coefficients of static and kinetic friction at the road are $\mu_{s}=0.5$ and $\mu_{k}=0.3,$ respectively. Neglect the mass of the wheels.

Anand Jangid

Numerade Educator

Problem 28

The assembly has a mass of $8 \mathrm{Mg}$ and is hoisted using the boom and pulley system. If the winch at $B$ draws in the cable with an acceleration of $2 \mathrm{m} / \mathrm{s}^{2},$ determine the compressive force in the hydraulic cylinder needed to support the boom. The boom has a mass of $2 \mathrm{Mg}$ and mass center at $G$.

Ahmed Kamel

Numerade Educator

Problem 29

The assembly has a mass of $4 \mathrm{Mg}$ and is hoisted using the winch at $B$. Determine the greatest acceleration of the assembly so that the compressive force in the hydraulic cylinder supporting the boom does not exceed $180 \mathrm{kN}$. What is the tension in the supporting cable? The boom has a mass of $2 \mathrm{Mg}$ and mass center at $G$.

Ahmed Kamel

Numerade Educator

Problem 30

The uniform girder $A B$ has a mass of $8 \mathrm{Mg}$ Determine the internal axial, shear, and bending-moment loadings at the center of the girder if a crane gives it an upward acceleration of $3 \mathrm{m} / \mathrm{s}^{2}$.

Ahmed Kamel

Numerade Educator

Problem 31

A car having a weight of 4000 lb begins to skid and turn with the brakes applied to all four wheels. If the coefficient of kinetic friction between the wheels and the road is $\mu_{k}=0.8,$ determine the maximum critical height $h$ of the center of gravity $G$ such that the car does not overturn. Tipping will begin to occur after the car rotates $90^{\circ}$ from its original direction of motion and, as shown in the figure, undergoes translation while skidding. Hint: Draw a free-body diagram of the car viewed from the front. When tipping occurs, the normal reactions of the wheels on the right side (or passenger side) are zero.

Ahmed Kamel

Numerade Educator

Problem 32

A force of $P=300 \mathrm{N}$ is applied to the 60 -kg cart. Determine the reactions at both the wheels at $A$ and both the wheels at $B$. Also, what is the acceleration of the cart? The mass center of the cart is at $G$.

Ahmed Kamel

Numerade Educator

Problem 33

Determine the largest force $\mathbf{P}$ that can be applied to the 60 -kg cart, without causing one of the wheel reactions, either at $A$ or at $B$, to be zero. Also, what is the acceleration of the cart? The mass center of the cart is at $G$.

Ahmed Kamel

Numerade Educator

Problem 34

The trailer with its load has a mass of 150 -kg and a center of mass at $G$. If it is subjected to a horizontal force of $P=600 \mathrm{N},$ determine the trailer's acceleration and the normal force on the pair of wheels at $A$ and at $B$. The wheels are free to roll and have negligible mass.

Ahmed Kamel

Numerade Educator

Problem 35

The desk has a weight of 75 lb and a center of gravity at $G .$ Determine its initial acceleration if a man pushes on it with a force $F=60$ lb. The coefficient of kinetic friction at $A$ and $B$ is $\mu_{k}=0.2$.

Ahmed Kamel

Numerade Educator

Problem 36

The desk has a weight of $75 \mathrm{lb}$ and a center of gravity at $G .$ Determine the initial acceleration of a desk when the man applies enough force $F$ to overcome the static friction at $A$ and $B .$ Also, find the vertical reactions on each of the two legs at $A$ and at $B$. The coefficients of static and kinetic friction at $A$ and $B$ are $\mu_{s}=0.5$ and $\mu_{k}=0.2$ respectively.

Ahmed Kamel

Numerade Educator

Problem 37

The 150 -kg uniform crate rests on the 10 -kg cart. Determine the maximum force $P$ that can be applied to the handle without causing the crate to tip on the cart. Slipping does not occur.

Ahmed Kamel

Numerade Educator

Problem 38

The 150 -kg uniform crate rests on the 10 -kg cart. Determine the maximum force $P$ that can be applied to the handle without causing the crate to slip or tip on the cart. The coefficient of static friction between the crate and cart is $\mu_{s}=0.2$.

Ahmed Kamel

Numerade Educator

Problem 39

The bar has a weight per length $w$ and is supported by the smooth collar. If it is released from rest, determine the internal normal force, shear force, and bending moment in the bar as a function of $x$.

Ahmed Kamel

Numerade Educator

Problem 40

The smooth 180 -lb pipe has a length of $20 \mathrm{ft}$ and a negligible diameter. It is carried on a truck as shown. Determine the maximum acceleration which the truck can have without causing the normal reaction at $A$ to be zero. Also determine the horizontal and vertical components of force which the truck exerts on the pipe at $B$.

Ahmed Kamel

Numerade Educator

Problem 41

The smooth 180 -lb pipe has a length of $20 \mathrm{ft}$ and a negligible diameter. It is carried on a truck as shown. If the truck accelerates at $a=5 \mathrm{ft} / \mathrm{s}^{2},$ determine the normal reaction at $A$ and the horizontal and vertical components of force which the truck exerts on the pipe at $B$.

Ahmed Kamel

Numerade Educator

Problem 42

The uniform crate has a mass of $50 \mathrm{kg}$ and rests on the cart having an inclined surface. Determine the smallest acceleration that will cause the crate either to tip or slip relative to the cart. What is the magnitude of this acceleration? The coefficient of static friction between the crate and cart is $\mu_{s}=0.5$.

Ahmed Kamel

Numerade Educator

Problem 43

Determine the acceleration of the 150 -lb cabinet and the normal reaction under the legs $A$ and $B$ if $P=35$ lb. The coefficients of static and kinetic friction between the
cabinet and the plane are $\mu_{s}=0.2$ and $\mu_{k}=0.15$ respectively. The cabinet's center of gravity is located at $G$.

Ahmed Kamel

Numerade Educator

Problem 44

The uniform bar of mass $m$ is pin connected to the collar, which slides along the smooth horizontal rod. If the collar is given a constant acceleration of a, determine the bar's inclination angle $\theta$. Neglect the collar's mass.

Ahmed Kamel

Numerade Educator

Problem 45

The drop gate at the end of the trailer has a mass of $1.25 \mathrm{Mg}$ and mass center at $G$. If it is supported by the cable $A B$ and hinge at $C,$ determine the tension in the cable when the truck begins to accelerate at $5 \mathrm{m} / \mathrm{s}^{2}$. Also, what are the horizontal and vertical components of reaction at the hinge $C ?$

Anand Jangid

Numerade Educator

Problem 46

The drop gate at the end of the trailer has a mass of $1.25 \mathrm{Mg}$ and mass center at $G$. If it is supported by the cable $A B$ and hinge at $C,$ determine the maximum deceleration of the truck so that the gate does not begin to rotate forward. What are the horizontal and vertical components of reaction at the hinge $C ?$

Ahmed Kamel

Numerade Educator

Problem 47

The snowmobile has a weight of $250 \mathrm{lb}$, centered at $G_{1},$ while the rider has a weight of $150 \mathrm{lb}$, centered at $G_{2}$ If the acceleration is $a=20 \mathrm{ft} / \mathrm{s}^{2},$ determine the maximum height $h$ of $G_{2}$ of the rider so that the snowmobile's front skid does not lift off the ground. Also, what are the traction (horizontal) force and normal reaction under the rear tracks at $A ?$

Ahmed Kamel

Numerade Educator

Problem 48

The snowmobile has a weight of $250 \mathrm{lb}$, centered at $G_{1},$ while the rider has a weight of 150 lb, centered at $G_{2}$ If $h=3 \mathrm{ft},$ determine the snowmobile's maximum permissible acceleration a so that its front skid does not lift off the ground. Also, find the traction (horizontal) force and the normal reaction under the rear tracks at $A$.

Ahmed Kamel

Numerade Educator

Problem 49

The snowmobile has a weight of $250 \mathrm{lb}$, centered at $G_{1},$ while the rider has a weight of 150 lb, centered at $G_{2}$ If $h=3 \mathrm{ft},$ determine the snowmobile's maximum permissible acceleration a so that its front skid does not lift off the ground. Also, find the traction (horizontal) force and the normal reaction under the rear tracks at $A$.

Ahmed Kamel

Numerade Educator

Problem 50

If the cart's mass is $30 \mathrm{kg}$, determine the horizontal force $P$ that should be applied to the cart so that the cord $A B$ just becomes slack. The uniform rod $B C$ has a mass of 15 kg.

Ahmed Kamel

Numerade Educator

Problem 51

The pipe has a mass of $800 \mathrm{kg}$ and is being towed behind the truck. If the acceleration of the truck is $a_{t}=0.5 \mathrm{m} / \mathrm{s}^{2},$ determine the angle $\theta$ and the tension in the cable. The coefficient of kinetic friction between the pipe and the ground is $\mu_{k}=0.1$.

Ahmed Kamel

Numerade Educator

Problem 52

The pipe has a mass of $800 \mathrm{kg}$ and is being towed behind a truck. If the angle $\theta=30^{\circ},$ determine the acceleration of the truck and the tension in the cable. The coefficient of kinetic friction between the pipe and the ground is $\mu_{k}=0.1$.

Ahmed Kamel

Numerade Educator

Problem 53

The crate $C$ has a weight of 150 lb and rests on the truck elevator for which the coefficient of static friction is $\mu_{s}=0.4 .$ Determine the largest initial angular acceleration $\alpha$ starting from rest, which the parallel links $A B$ and $D E$ can have without causing the crate to slip. No tipping occurs.

Ahmed Kamel

Numerade Educator

Problem 54

The crate $C$ has a weight of $150 \mathrm{lb}$ and rests on the truck elevator. Determine the initial friction and normal force of the elevator on the crate if the parallel links are given an angular acceleration $\alpha=2 \mathrm{rad} / \mathrm{s}^{2}$ starting from rest.

Ahmed Kamel

Numerade Educator

Problem 55

The $100-\mathrm{kg}$ uniform crate $C$ rests on the elevator floor where the coefficient of static friction is $\mu_{s}=0.4$ Determine the largest initial angular acceleration $\alpha$, starting from rest at $\theta=90^{\circ},$ without causing the crate to slip. No tipping occurs.

Ahmed Kamel

Numerade Educator

Problem 56

The two uniform $4-\mathrm{kg}$ bars $D C$ and $E F$ are fixed (welded) together at $E$. Determine the normal force $N_{E}$ shear force $V_{E},$ and moment $M_{E},$ which $D C$ exerts on $E F$ at $E$ if at the instant $\theta=60^{\circ} B C$ has an angular velocity $\omega=2 \operatorname{rad} / \mathrm{s}$ and an angular acceleration $\alpha=4 \mathrm{rad} / \mathrm{s}^{2}$ as shown.

Ahmed Kamel

Numerade Educator

Problem 57

The 10-kg wheel has a radius of gyration $k_{A}=200 \mathrm{mm}$ If the wheel is subjected to a moment $M=(5 t) \mathrm{N} \cdot \mathrm{m},$ where $t$ is in seconds, determine its angular velocity when $t=3 \mathrm{s}$ starting from rest. Also, compute the reactions which the fixed pin $A$ exerts on the wheel during the motion.

Ahmed Kamel

Numerade Educator

Problem 58

The uniform $24-\mathrm{kg}$ plate is released from rest at the position shown. Determine its initial angular acceleration and the horizontal and vertical reactions at the pin $A$.

Ahmed Kamel

Numerade Educator

Problem 59

The uniform slender rod has a mass $m$. If it is released from rest when $\theta=0^{\circ},$ determine the magnitude of the reactive force exerted on it by pin $B$ when $\theta=90^{\circ}$.

Ahmed Kamel

Numerade Educator

Problem 60

The bent rod has a mass of $2 \mathrm{kg} / \mathrm{m} .$ If it is released from rest in the position shown, determine its initial angular acceleration and the horizontal and vertical components of reaction at $A$.

Ahmed Kamel

Numerade Educator

Problem 61

If a horizontal force of $P=100 \mathrm{N}$ is applied to the $300-\mathrm{kg}$ reel of cable, determine its initial angular acceleration. The reel rests on rollers at $A$ and $B$ and has a radius of gyration of $k_{O}=0.6 \mathrm{m}$.

Ahmed Kamel

Numerade Educator

Problem 62

The $10-1 b$ bar is pinned at its center $O$ and connected to a torsional spring. The spring has a stiffness $k=5 \mathrm{lb} \cdot \mathrm{ft} / \mathrm{rad}, \quad$ so $\quad$ that $\quad$ the $\quad$ torque $\quad$ developed is $M=(5 \theta)$ Ib $\cdot \mathrm{ft},$ where $\theta$ is in radians. If the bar is released from rest when it is vertical at $\theta=90^{\circ},$ determine its angular velocity at the instant $\theta=0^{\circ}$.

Ahmed Kamel

Numerade Educator

Problem 63

The $10-1 b$ bar is pinned at its center $O$ and connected to a torsional spring. The spring has a stiffness $k=5 \mathrm{lb} \cdot \mathrm{ft} / \mathrm{rad}, \quad$ so $\quad$ that $\quad$ the $\quad$ torque $\quad$ developed $\quad$ is $M=(5 \theta)$ Ib $\cdot f t,$ where $\theta$ is in radians. If the bar is released from rest when it is vertical at $\theta=90^{\circ},$ determine its angular velocity at the instant $\theta=45^{\circ}$.

Ahmed Kamel

Numerade Educator

Problem 64

A cord is wrapped around the outer surface of the $8-\mathrm{kg}$ disk. If a force of $F=\left(1 / 4 \theta^{2}\right) \mathrm{N},$ where $\theta$ is in radians, is applied to the cord, determine the disk's angular acceleration when it has turned 5 revolutions. The disk has an initial angular velocity of $\omega_{0}=1 \mathrm{rad} / \mathrm{s}$.

Ahmed Kamel

Numerade Educator

Problem 65

Disk $A$ has a weight of 5 lb and disk $B$ has a weight of 10 lb. If no slipping occurs between them, determine the couple moment $\mathbf{M}$ which must be applied to disk $A$ to give it an angular acceleration of $4 \mathrm{rad} / \mathrm{s}^{2}$.

Ahmed Kamel

Numerade Educator

Problem 66

The kinetic diagram representing the general rotational motion of a rigid body about a fixed axis passing through $O$ is shown in the figure. Show that $I_{G} \alpha$ may be eliminated by moving the vectors $m\left(\mathbf{a}_{G}\right)_{t}$ and $m\left(\mathbf{a}_{G}\right)_{n}$ to point $P,$ located a distance $r_{G P}=k_{G}^{2} / r_{O G}$ from the center of mass $G$ of the body. Here $k_{G}$ represents the radius of gyration of the body about an axis passing through $G$. The point $P$ is called the center of percussion of the body.

Ahmed Kamel

Numerade Educator

Problem 67

If the cord at $B$ suddenly fails, determine the horizontal and vertical components of the initial reaction at the pin $A,$ and the angular acceleration of the 120 -kg beam. Treat the beam as a uniform slender rod.

Ahmed Kamel

Numerade Educator

Problem 68

The device acts as a pop-up barrier to prevent the passage of a vehicle. It consists of a 100 -kg steel plate $A C$ and a $200-\mathrm{kg}$ counterweight solid concrete block located as shown. Determine the moment of inertia of the plate and block about the hinged axis through $A$. Neglect the mass of the supporting arms $A B .$ Also, determine the initial angular acceleration of the assembly when it is released from rest at $\theta=45^{\circ}$.

Ahmed Kamel

Numerade Educator

Problem 69

The 20 -kg roll of paper has a radius of gyration $k_{A}=90 \mathrm{mm}$ about an axis passing through point $A .$ It is pin supported at both ends by two brackets $A B .$ If the roll rests against a wall for which the coefficient of kinetic friction is $\mu_{k}=0.2$ and a vertical force $F=30 \mathrm{N}$ is applied to the end of the paper, determine the angular acceleration of the roll as the paper unrolls.

Ahmed Kamel

Numerade Educator

Problem 70

The 20 -kg roll of paper has a radius of gyration $k_{A}=90 \mathrm{mm}$ about an axis passing through point $A .$ It is pin supported at both ends by two brackets $A B$. If the roll rests against a wall for which the coefficient of kinetic friction is $\mu_{k}=0.2,$ determine the constant vertical force $F$ that must be applied to the roll to pull off $1 \mathrm{m}$ of paper in $t=3 \mathrm{s}$ starting from rest. Neglect the mass of paper that is removed.

Ahmed Kamel

Numerade Educator

Problem 71

The reel of cable has a mass of $400 \mathrm{kg}$ and a radius of gyration of $k_{A}=0.75 \mathrm{m} .$ Determine its angular velocity when $t=2 \mathrm{s}$, starting from rest, if the force $\mathbf{P}=\left(20 t^{2}+80\right) \mathrm{N}$ when $t$ is in seconds. Neglect the mass of the unwound cable, and assume it is always at a radius of $0.5 \mathrm{m}$.

Ahmed Kamel

Numerade Educator

Problem 72

The 30 -kg disk is originally spinning at $\omega=125 \mathrm{rad} / \mathrm{s}$ If it is placed on the ground, for which the coefficient of kinetic friction is $\mu_{C}=0.5,$ determine the time required for the motion to stop. What are the horizontal and vertical components of force which the member $A B$ exerts on the pin at $A$ during this time? Neglect the mass of $A B$.

Ahmed Kamel

Numerade Educator

Problem 73

Cable is unwound from a spool supported on small rollers at $A$ and $B$ by exerting a force $T=300 \mathrm{N}$ on the cable. Compute the time needed to unravel $5 \mathrm{m}$ of cable from the spool if the spool and cable have a total mass of $600 \mathrm{kg}$ and a radius of gyration of $k_{O}=1.2 \mathrm{m} .$ For the calculation, neglect the mass of the cable being unwound and the mass of the rollers at $A$ and $B$. The rollers turn with no friction.

Ahmed Kamel

Numerade Educator

Problem 74

The 5-kg cylinder is initially at rest when it is placed in contact with the wall $B$ and the rotor at $A .$ If the rotor always maintains a constant clockwise angular velocity $\omega=6 \mathrm{rad} / \mathrm{s},$ determine the initial angular acceleration of the cylinder. The coefficient of kinetic friction at the contacting surfaces $B$ and $C$ is $\mu_{k}=0.2$.

Ahmed Kamel

Numerade Educator

Problem 75

The wheel has a mass of 25 kg and a radius of gyration $k_{B}=0.15 \mathrm{m} .$ It is originally spinning at $\omega=40 \mathrm{rad} / \mathrm{s} .$ If it is placed on the ground, for which the coefficient of kinetic friction is $\mu_{C}=0.5,$ determine the time required for the motion to stop. What are the horizontal and vertical components of reaction which the pin at $A$ exerts on $A B$ during this time? Neglect the mass of $A B$.

Ahmed Kamel

Numerade Educator

Problem 76

The 20 -kg roll of paper has a radius of gyration $k_{A}=120 \mathrm{mm}$ about an axis passing through point $A$. It is pin supported at both ends by two brackets $A B .$ The roll rests on the floor, for which the coefficient of kinetic friction is $\mu_{k}=0.2 .$ If a horizontal force $F=60 \mathrm{N}$ is applied to the end of the paper, determine the initial angular acceleration of the roll as the paper unrolls.

Ahmed Kamel

Numerade Educator

Problem 77

Disk $D$ turns with a constant clockwise angular velocity of 30 rad/s. Disk $E$ has a weight of 60 lb and is initially at rest when it is brought into contact with $D .$ Determine the time required for disk $E$ to attain the same angular velocity as disk $D$. The coefficient of kinetic friction between the two disks is $\mu_{k}=0.3 .$ Neglect the weight of bar $B C$.

Ahmed Kamel

Numerade Educator

Problem 78

Two cylinders $A$ and $B$, having a weight of 10 lb and 5 lb, respectively, are attached to the ends of a cord which passes over a 3 -lb pulley (disk). If the cylinders are released from rest, determine their speed in $t-0.5$ s. The cord does not slip on the pulley. Neglect the mass of the cord. Suggestion: Analyze the "system" consisting of both the
cylinders and the pulley.

Ahmed Kamel

Numerade Educator

Problem 79

The two blocks $A$ and $B$ have a mass of $5 \mathrm{kg}$ and $10 \mathrm{kg},$ respectively. If the pulley can be treated as a disk of mass $3 \mathrm{kg}$ and radius $0.15 \mathrm{m}$, determine the acceleration of block $A .$ Neglect the mass of the cord and any slipping on the pulley.

Ahmed Kamel

Numerade Educator

Problem 80

The two blocks $A$ and $B$ have a mass $m_{A}$ and $m_{B}$ respectively, where $m_{B}>m_{A} .$ If the pulley can be treated as a disk of mass $M,$ determine the acceleration of block $A$ Neglect the mass of the cord and any slipping on the pulley.

Ahmed Kamel

Numerade Educator

Problem 81

Determine the angular acceleration of the $25-\mathrm{kg}$ diving board and the horizontal and vertical components of reaction at the pin $A$ the instant the man jumps off. Assume that the board is uniform and rigid, and that at the instant he jumps off the spring is compressed a maximum amount of $200 \mathrm{mm}, \omega=0,$ and the board is horizontal. Take $k=7 \mathrm{kN} / \mathrm{m}$.

Ahmed Kamel

Numerade Educator

Problem 82

The lightweight turbine consists of a rotor which is powered from a torque applied at its center. At the instant the rotor is horizontal it has an angular velocity of 15 rad/s and a clockwise angular acceleration of $8 \mathrm{rad} / \mathrm{s}^{2} .$ Determine the internal normal force, shear force, and moment at a section through $A$. Assume the rotor is a 50 -m-long slender rod, having a mass of $3 \mathrm{kg} / \mathrm{m}$.

Ahmed Kamel

Numerade Educator

Problem 83

The two-bar assembly is released from rest in the position shown. Determine the initial bending moment at the fixed joint $B$. Each bar has a mass $m$ and length $l$.

Ahmed Kamel

Numerade Educator

Problem 84

The armature (slender rod) $A B$ has a mass of $0.2 \mathrm{kg}$ and can pivot about the pin at $A .$ Movement is controlled by the electromagnet $E,$ which exerts a horizontal attractive force on the armature at $B$ of $F_{B}=\left(0.2\left(10^{-3}\right) l^{-2}\right) \mathrm{N}$ where $l$ in meters is the gap between the armature and the magnet at any instant. If the armature lies in the horizontal plane, and is originally at rest, determine the speed of the contact at $B$ the instant $l=0.01 \mathrm{m} .$ Originally $l=0.02 \mathrm{m}$.

Ahmed Kamel

Numerade Educator

Problem 85

The bar has a weight per length of $w .$ If it is rotating in the vertical plane at a constant rate $\omega$ about point $O$ determine the internal normal force, shear force, and moment as a function of $x$ and $\theta$.

Ahmed Kamel

Numerade Educator

Problem 86

The $4-\mathrm{kg}$ slender rod is initially supported horizontally by a spring at $B$ and pin at $A .$ Determine the angular acceleration of the rod and the acceleration of the rod's mass center at the instant the 100 -N force is applied.

Ahmed Kamel

Numerade Educator

Problem 87

The $100-$ kg pendulum has a center of mass at $G$ and a radius of gyration about $G$ of $k_{G}=250 \mathrm{mm} .$ Determine the horizontal and vertical components of reaction on the beam by the pin $A$ and the normal reaction of the roller $B$ at the instant $\theta=90^{\circ}$ when the pendulum is rotating at $\omega=8$ rad/s. Neglect the weight of the beam and the support.

Anand Jangid

Numerade Educator

Problem 88

The 100 -kg pendulum has a center of mass at $G$ and a radius of gyration about $G$ of $k_{G}=250 \mathrm{mm} .$ Determine the horizontal and vertical components of reaction on the beam by the pin $A$ and the normal reaction of the roller $B$ at the instant $\theta=0^{\circ}$ when the pendulum is rotating at $\omega=4 \mathrm{rad} / \mathrm{s} .$ Neglect the weight of the beam and the support.

Ahmed Kamel

Numerade Educator

Problem 89

The "Catherine wheel" is a firework that consists of a coiled tube of powder which is pinned at its center. If the powder burns at a constant rate of $20 \mathrm{g} / \mathrm{s}$ such as that the exhaust gases always exert a force having a constant magnitude of $0.3 \mathrm{N}$, directed tangent to the wheel, determine the angular velocity of the wheel when $75 \%$ of the mass is burned off. Initially, the wheel is at rest and has a mass of $100 \mathrm{g}$ and a radius of $r=75 \mathrm{mm} .$ For the calculation, consider the wheel to always be a thin disk.

Ahmed Kamel

Numerade Educator

Problem 90

If the disk in Fig. $17-19$ rolls without slipping, show that when moments are summed about the instantaneous center of zero velocity, $I C$, it is possible to use the moment equation $\Sigma M_{I C}=I_{I C} \alpha,$ where $I_{I C}$ represents the moment of inertia of the disk calculated about the instantaneous axis of zero velocity.

Ahmed Kamel

Numerade Educator

Problem 91

The 20 -kg punching bag has a radius of gyration about its center of mass $G$ of $k_{G}=0.4 \mathrm{m}$. If it is initially at rest and is subjected to a horizontal force $F=30 \mathrm{N}$ determine the initial angular acceleration of the bag and the tension in the supporting cable $A B$.

Ahmed Kamel

Numerade Educator

Problem 92

The uniform 150 -lb beam is initially at rest when the forces are applied to the cables. Determine the magnitude of the acceleration of the mass center and the angular acceleration of the beam at this instant.

Ahmed Kamel

Numerade Educator

Problem 93

The slender $12-\mathrm{kg}$ bar has a clockwise angular velocity of $\omega=2$ rad/s when it is in the position shown. Determine its angular acceleration and the normal reactions of the smooth surface $A$ and $B$ at this instant.

Penny Riley

Numerade Educator

Problem 94

The tire has a weight of $30 \mathrm{lb}$ and a radius of gyration of $k_{G}=0.6 \mathrm{ft}$. If the coefficients of static and kinetic friction between the tire and the plane are $\mu_{s}=0.2$ and $\mu_{k}=0.15,$ determine the tire's angular acceleration as it rolls down the incline. $\operatorname{Set} \theta=12^{\circ}$.

Ahmed Kamel

Numerade Educator

Problem 95

The tire has a weight of $30 \mathrm{lb}$ and a radius of gyration of $k_{G}=0.6 \mathrm{ft}$. If the coefficients of static and kinetic friction between the tire and the plane are $\mu_{s}=0.2$ and $\mu_{k}=0.15,$ determine the maximum angle $\theta$ of the inclined plane so that the tire rolls without slipping.

Ahmed Kamel

Numerade Educator

Problem 96

The spool has a mass of $100 \mathrm{kg}$ and a radius of gyration of $k_{G}=0.3 \mathrm{m}$. If the coefficients of static and kinetic friction at $A$ are $\mu_{s}=0.2$ and $\mu_{k}=0.15,$ respectively, determine the angular acceleration of the spool if $P=50 \mathrm{N}$.

Ahmed Kamel

Numerade Educator

Problem 97

Solve Prob. $17-96$ if the cord and force $P=50 \mathrm{N}$ are directed vertically upwards.

Ahmed Kamel

Numerade Educator

Problem 98

The spool has a mass of $100 \mathrm{kg}$ and a radius of gyration $k_{G}=0.3 \mathrm{m} .$ If the coefficients of static and kinetic friction at $A$ are $\mu_{s}=0.2$ and $\mu_{k}=0.15,$ respectively, determine the angular acceleration of the spool if $P=600 \mathrm{N}$.

Ahmed Kamel

Numerade Educator

Problem 99

The 12 -kg uniform bar is supported by a roller at $A$ If a horizontal force of $F=80 \mathrm{N}$ is applied to the roller, determine the acceleration of the center of the roller at the instant the force is applied. Neglect the weight and the size of the roller.

Ahmed Kamel

Numerade Educator

Problem 100

A force of $F=10 \mathrm{N}$ is applied to the 10 -kg ring as shown. If slipping does not occur, determine the ring's initial angular acceleration, and the acceleration of its mass center, $G$ Neglect the thickness of the ring.

Ahmed Kamel

Numerade Educator

Problem 101

If the coefficient of static friction at $C$ is $\mu_{s}=0.3$ determine the largest force $\mathbf{F}$ that can be applied to the $5-\mathrm{kg}$ ring, without causing it to slip. Neglect the thickness of the ring.

Ahmed Kamel

Numerade Educator

Problem 102

The 25 -lb slender rod has a length of 6 ft. Using a collar of negligible mass, its end $A$ is confined to move along the smooth circular bar of radius $3 \sqrt{2} \mathrm{ft}$. End $B$ rests on the floor, for which the coefficient of kinetic friction is $\mu_{B}=0.4$ If the bar is released from rest when $\theta=30^{\circ},$ determine the angular acceleration of the bar at this instant.

Ahmed Kamel

Numerade Educator

Problem 103

The $15-16$ circular plate is suspended from a pin at $A .$ If the pin is connected to a track which is given an acceleration $a_{A}=5 \mathrm{ft} / \mathrm{s}^{2},$ determine the horizontal and vertical components of reaction at $A$ and the angular acceleration of the plate. The plate is originally at rest.

Ahmed Kamel

Numerade Educator

Problem 104

If $P=30$ Ib, determine the angular acceleration of the 50 -lb roller. Assume the roller to be a uniform cylinder and that no slipping occurs.

Ahmed Kamel

Numerade Educator

Problem 105

If the coefficient of static friction between the 50 -lb roller and the ground is $\mu_{s}=0.25,$ determine the maximum force $P$ that can be applied to the handle, so that roller rolls on the ground without slipping. Also, find the angular acceleration of the roller. Assume the roller to be a uniform cylinder.

Ahmed Kamel

Numerade Educator

Problem 106

The uniform bar of mass $m$ and length $L$ is balanced in the vertical position when the horizontal force $\mathbf{P}$ is applied to the roller at $A .$ Determine the bar's initial angular acceleration and the acceleration of its top point $B$.

Ahmed Kamel

Numerade Educator

Problem 107

Solve Prob. $17-106$ if the roller is removed and the coefficient of kinetic friction at the ground is $\mu_{k}$.

Ahmed Kamel

Numerade Educator

Problem 108

The semicircular disk having a mass of $10 \mathrm{kg}$ is rotating at $\omega=4 \mathrm{rad} / \mathrm{s}$ at the instant $\theta=60^{\circ} .$ If the coefficient of static friction at $A$ is $\mu_{s}=0.5,$ determine if the disk slips at this instant.

Ahmed Kamel

Numerade Educator

Problem 109

The 500 -kg concrete culvert has a mean radius of $0.5 \mathrm{m} .$ If the truck has an acceleration of $3 \mathrm{m} / \mathrm{s}^{2},$ determine the culvert's angular acceleration. Assume that the culvert does not slip on the truck bed, and neglect its thickness.

Ahmed Kamel

Numerade Educator

Problem 110

The $15-16$ disk rests on the 5 -lb plate. A cord is wrapped around the periphery of the disk and attached to the wall at $B$. If a torque $M=40 \mathrm{lb} \cdot \mathrm{ft}$ is applied to the disk, determine the angular acceleration of the disk and the time needed for the end $C$ of the plate to travel 3 ft and strike the wall. Assume the disk does not slip on the plate and the plate rests on the surface at $D$ having a coefficient of kinetic friction of $\mu_{k}=0.2 .$ Neglect the mass of the cord.

Ahmed Kamel

Numerade Educator

Problem 111

The semicircular disk having a mass of $10 \mathrm{kg}$ is rotating at $\omega=4 \mathrm{rad} / \mathrm{s}$ at the instant $\theta=60^{\circ} .$ If the coefficient of static friction at $A$ is $\mu_{s}=0.5,$ determine if the disk slips at this instant.

Ahmed Kamel

Numerade Educator

Problem 112

The circular concrete culvert rolls with an angular velocity of $\omega=0.5 \mathrm{rad} / \mathrm{s}$ when the man is at the position shown. At this instant the center of gravity of the culvert and the man is located at point $G,$ and the radius of gyration about $G$ is $k_{G}=3.5 \mathrm{ft}$. Determine the angular acceleration of the culvert. The combined weight of the culvert and the $\operatorname{man}$ is 500 lb. Assume that the culvert rolls without slipping, and the man does not move within the culvert.

Ahmed Kamel

Numerade Educator

Problem 113

The uniform disk of mass $m$ is rotating with an angular velocity of $\omega_{0}$ when it is placed on the floor. Determine the initial angular acceleration of the disk and the acceleration of its mass center. The coefficient of kinetic friction between the disk and the floor is $\mu_{k}$.

Ahmed Kamel

Numerade Educator

Problem 114

The uniform disk of mass $m$ is rotating with an angular velocity of $\omega_{0}$ when it is placed on the floor. Determine the time before it starts to roll without slipping. What is the angular velocity of the disk at this instant? The coefficient of kinetic friction between the disk and the floor is $\mu_{k}$.

Ahmed Kamel

Numerade Educator

Problem 115

A cord is wrapped around each of the two $10-\mathrm{kg}$ disks. If they are released from rest determine the angular acceleration of each disk and the tension in the cord $C$. Neglect the mass of the cord.

Ahmed Kamel

Numerade Educator

Problem 116

The disk of mass $m$ and radius $r$ rolls without slipping on the circular path. Determine the normal force which the path exerts on the disk and the disk's angular acceleration if at the instant shown the disk has an angular velocity of $\omega$.

Ahmed Kamel

Numerade Educator

Problem 117

The uniform beam has a weight $W$. If it is originally at rest while being supported at $A$ and $B$ by cables, determine the tension in cable $A$ if cable $B$ suddenly fails. Assume the beam is a slender rod.

Supratim Pal

Numerade Educator

Problem 118

The 500 -lb beam is supported at $A$ and $B$ when it is subjected to a force of 1000 lb as shown. If the pin support at $A$ suddenly fails, determine the beam's initial angular acceleration and the force of the roller support on the beam. For the calculation, assume that the beam is a slender rod so that its thickness can be neglected.

Ahmed Kamel

Numerade Educator

Problem 119

The solid ball of radius $r$ and mass $m$ rolls without slipping down the $60^{\circ}$ trough. Determine its angular acceleration.

Ahmed Kamel

Numerade Educator

Problem 120

By pressing down with the finger at $B$, a thin ring having a mass $m$ is given an initial velocity $\mathbf{v}_{0}$ and a backspin $\omega_{0}$ when the finger is released. If the coefficient of kinetic friction between the table and the ring is $\mu_{k},$ determine the distance the ring travels forward before back spinning stops.

Ahmed Kamel

Numerade Educator