Engineering Mechanics: Dynamics in SI Units

Russell 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 $m$.

Supratim Pal

Supratim Pal

Numerade Educator

Problem 2

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 3

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 4

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

Ahmed Kamel

Numerade Educator

Problem 5

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 6

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 7

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 8

Determine the mass moment of inertia $I_{z}$ of the cone formed by revolving the shaded area around the $z$ axis. The density of the material is $\rho .$ Express the result in terms of the mass $m$ of the cone.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

Determine the moment of inertia $I_{z}$ of the torus. The mass of the torus is $m$ and the density $\rho$ is constant. Suggestion: Use a shell element.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 10

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 11

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 12

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 13

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 14

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 $2 \mathrm{~kg}$. Determine the wheel's moment of inertia about an axis perpendicular to the page and passing through point $A$.

Ahmed Kamel

Numerade Educator

Problem 15

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 16

The assembly consists of a disk having a mass of $6 \mathrm{~kg}$ and slender rods $A B$ and $D C$ which have a mass per unit length of $2 \mathrm{~kg} / \mathrm{m}$. Determine the length $L$ of $D C$ so that the center of mass is at the bearing $O .$ What is the moment of inertia of the assembly about an axis perpendicular to the page and passing through $O ?$

Ahmed Kamel

Numerade Educator

Problem 17

The pendulum consists of a 4 -kg circular plate and a $2-\mathrm{kg}$ 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 18

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 19

The pendulum consists of two slender rods $A B$ and $O C$ which have a mass per unit length of $3 \mathrm{~kg} / \mathrm{m}$. The thin circular plate has a mass per unit area of $12 \mathrm{~kg} / \mathrm{m}^{2}$. 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 $G$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

The pendulum consists of two slender rods $A B$ and $O C$ which have a mass per unit length of $3 \mathrm{~kg} / \mathrm{m}$. The thin circular plate has a mass per unit area of $12 \mathrm{~kg} / \mathrm{m}^{2}$. Determine the moment of inertia of the pendulum about an axis perpendicular to the page and passing through the pin at $O$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

The pendulum consists of the $3-\mathrm{kg}$ slender rod and the $5-\mathrm{kg}$ 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 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 two 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 25

The 4-Mg uniform canister contains nuclear waste material encased in concrete. If the mass of the spreader beam $B D$ is $50 \mathrm{~kg}$, determine the force in each of the links $A B, C D, E F,$ and $G H$ when the system is lifted with an acceleration of $a=2 \mathrm{~m} / \mathrm{s}^{2}$ for a short period of time.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

The 4-Mg uniform canister contains nuclear waste material encased in concrete. If the mass of the spreader beam $B D$ is $50 \mathrm{~kg},$ determine the largest vertical acceleration a of the system so that each of the links $A B$ and $C D$ are not subjected to a force greater than $30 \mathrm{kN}$ and links $E F$ and $G H$ are not subjected to a force greater than $34 \mathrm{kN}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

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 28

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 29

Determine the shortest time possible for the rearwheel drive, 2 -Mg truck to achieve a speed of $16 \mathrm{~m} / \mathrm{s}$ with a constant acceleration starting from rest. The coefficient of static friction between the wheels and the road surface is $\mu_{s}=0.8 .$ The front wheels are free to roll. Neglect the mass of the wheels.

Simran Hiranandani

Simran Hiranandani

Numerade Educator

Problem 30

The $150-\mathrm{kg}$ uniform crate rests on the $10-\mathrm{kg}$ 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 31

The $150-\mathrm{kg}$ uniform crate rests on the $10-\mathrm{kg}$ 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 32

The pipe has a mass of $460 \mathrm{~kg}$ and is held in place on the truck bed using the two boards $A$ and $B$. Determine the greatest acceleration of the truck so that the pipe begins to lose contact at $A$ and the bed of the truck and starts to pivot about $B$. Assume board $B$ will not slip on the bed of the truck, and the pipe is smooth. Also, what force does board $B$ exert on the pipe during the acceleration?

Ahmed Kamel

Numerade Educator

Problem 33

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 34

The mountain bike has a mass of $40 \mathrm{~kg}$ with center of mass at point $G_{1}$, while the rider has a mass of $60 \mathrm{~kg}$ with center of mass at point $G_{2}$. Determine the maximum deceleration when the brake is applied to the front wheel, without causing the rear wheel $B$ to leave the road. Assume that the front wheel does not slip. Neglect the mass of all the wheels.

Supratim Pal

Numerade Educator

Problem 35

The mountain bike has a mass of $40 \mathrm{~kg}$ with center of mass at point $G_{1}$, while the rider has a mass of $60 \mathrm{~kg}$ with center of mass at point $G_{2}$. When the brake is applied to the front wheel, it causes the bike to decelerate at a constant rate of $3 \mathrm{~m} / \mathrm{s}^{2}$. Determine the normal reaction the road exerts on the front and rear wheels. Assume that the rear wheel is free to roll. Neglect the mass of all the wheels.

Supratim Pal

Numerade Educator

Problem 36

The trailer with its load has a mass of $150-\mathrm{kg}$ 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 37

A force of $P=300 \mathrm{~N}$ is applied to the $60-\mathrm{kg}$ 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 38

Determine the largest force $\mathbf{P}$ that can be applied to the $60-\mathrm{kg}$ 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 39

If the cart's mass is $30 \mathrm{~kg}$ and it is subjected to a horizontal force of $P=90 \mathrm{~N},$ determine the tension in cord $A B$ and the horizontal and vertical components of reaction on end $C$ of the uniform $15-\mathrm{kg} \operatorname{rod} B C$.

Ahmed Kamel

Numerade Educator

Problem 40

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 $15 \mathrm{~kg}$.

Ahmed Kamel

Numerade Educator

Problem 41

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 42

The dragster has a mass of $1500 \mathrm{~kg}$ and a center of mass at $G$. If the coefficient of kinetic friction between the rear wheels and the pavement is $\mu_{k}=0.6,$ determine if it is possible for the driver to lift the front wheels, $A,$ off the ground while the rear drive wheels are slipping. Neglect the mass of the wheels and assume that the front wheels are free to roll.

Dading Chen

Numerade Educator

Problem 43

The dragster has a mass of $1500 \mathrm{~kg}$ and a center of mass at $G$. If no slipping occurs, determine the frictional force $\mathbf{F}_{B}$ which must be developed at each of the rear drive wheels $B$ in order to create an acceleration of $a=6 \mathrm{~m} / \mathrm{s}^{2}$ What are the normal reactions of each wheel on the ground? Neglect the mass of the wheels and assume that the front wheels are free to roll.

Supratim Pal

Numerade Educator

Problem 44

The pipe has a length of $3 \mathrm{~m}$ and a mass of $500 \mathrm{~kg}$. It is attached to the back of the truck using a 0.6 -m-long chain $A B .$ If the coefficient of kinetic friction at $C$ is $\mu_{k}=0.4$ determine the acceleration of the truck if the angle $\theta=10^{\circ}$ with the road as shown.

Rashmi Sinha

Numerade Educator

Problem 45

The lift truck has a mass of $70 \mathrm{~kg}$ and mass center at $G$. If it lifts the 120 -kg spool with an acceleration of $3 \mathrm{~m} / \mathrm{s}^{2},$ determine the reactions of each of the four wheels. The loading is symmetric. Neglect the mass of the movable $\operatorname{arm} C D .$

Supratim Pal

Numerade Educator

Problem 46

The lift truck has a mass of $70 \mathrm{~kg}$ and mass center at $G$. Determine the largest upward acceleration of the $120-\mathrm{kg}$ spool so that no reaction of the wheels on the ground exceeds $600 \mathrm{~N}$.

Supratim Pal

Numerade Educator

Problem 47

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 48

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 $C ?$

Ahmed Kamel

Numerade Educator

Problem 49

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 50

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 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 $100-\mathrm{kg}$ 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 54

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 $\omega=2 \mathrm{rad} / \mathrm{s}$ and an angular acceleration $\alpha=4 \mathrm{rad} / \mathrm{s}^{2}$
as shown.

Ahmed Kamel

Numerade Educator

Problem 55

The arched pipe has a mass of $80 \mathrm{~kg}$ and rests on the surface of the platform for which the coefficient of static friction is $\mu_{s}=0.3 .$ Determine the greatest angular acceleration $\alpha$ of the platform, starting from rest when $\theta=45^{\circ},$ without causing the pipe to slip on the platform.

Dading Chen

Numerade Educator

Problem 56

Determine the force developed in the links and the acceleration of the bar's mass center immediately after the cord fails. Neglect the mass of links $A B$ and $C D .$ The uniform bar has a mass of $20 \mathrm{~kg}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

The $10-\mathrm{kg}$ 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 $\operatorname{pin} A$.

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 $\operatorname{pin} B$ 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 $A$.

Ajay Singhal

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 $20-\mathrm{kg}$ 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 63

The $20-\mathrm{kg}$ 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 64

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 $I_{G} \boldsymbol{\alpha}$ 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 65

Gears $A$ and $B$ have a mass of $50 \mathrm{~kg}$ and $15 \mathrm{~kg}$, respectively. Their radii of gyration about their respective centers of mass are $k_{C}=250 \mathrm{~mm}$ and $k_{D}=150 \mathrm{~mm}$. If a torque of $M=200\left(1-e^{-0.2 t}\right) \mathrm{N} \cdot \mathrm{m},$ where $t$ is in seconds, is applied to gear $A,$ determine the angular velocity of both gears when $t=3 \mathrm{~s}$, starting from rest.

Ahmed Kamel

Numerade Educator

Problem 66

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 67

The door will close automatically using torsional springs mounted on the hinges. Each spring has a stiffness $k=50 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}$ so that the torque on each hinge is $M=(50 \theta) \mathrm{N} \cdot \mathrm{m},$ where $\theta$ is measured in radians. If the door is released from rest when it is open at $\theta=90^{\circ}$, determine its angular velocity at the instant $\theta=0^{\circ} .$ For the calculation, treat the door as a thin plate having a mass of $70 \mathrm{~kg}$.

Anand Jangid

Numerade Educator

Problem 68

The door will close automatically using torsional springs mounted on the hinges. If the torque on each hinge is $M=k \theta,$ where $\theta$ is measured in radians, determine the required torsional stiffness $k$ so that the door will close $\left(\theta=0^{\circ}\right)$ with an angular velocity $\omega=2 \mathrm{rad} / \mathrm{s}$ when it is released from rest at $\theta=90^{\circ} .$ For the calculation, treat the door as a thin plate having a mass of $70 \mathrm{~kg}$.

Shoukat Ali

Other Schools

Problem 69

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 70

The device acts as a pop-up barrier to prevent the passage of a vehicle. It consists of a $100-\mathrm{kg}$ 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 71

A cord is wrapped around the outer surface of the 8 -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 72

Block $A$ has a mass $m$ and rests on a surface having a coefficient of kinetic friction $\mu_{k} .$ The cord attached to $A$ passes over a pulley at $C$ and is attached to a block $B$ having a mass $2 m$. If $B$ is released, determine the acceleration of $A$. Assume that the cord does not slip over the pulley. The pulley can be approximated as a thin disk of radius $r$ and mass $\frac{1}{4} m .$ Neglect the mass of the cord.

Keshav Singh

Numerade Educator

Problem 73

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 74

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 75

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 76

The wheel has a mass of $25 \mathrm{~kg}$ 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 77

The disk has a mass of $20 \mathrm{~kg}$ and is originally spinning at the end of the strut with an angular velocity of $\omega=60 \mathrm{rad} / \mathrm{s}$. If it is then placed against the wall, for which the coefficient of kinetic friction is $\mu_{k}=0.3$ determine the time required for the motion to stop. What is the force in strut $B C$ during this time?

Ahmed Kamel

Numerade Educator

Problem 78

The $5-\mathrm{kg}$ 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 79

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 80

The $20-\mathrm{kg}$ 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 81

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 $F_{B}=\left(0.2\left(10^{-3}\right)^{-2}\right) \mathrm{N},$ 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 82

The 4-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 $100-\mathrm{N}$ force is applied.

Ahmed Kamel

Numerade Educator

Problem 83

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 84

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 85

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 $15 \mathrm{rad} / \mathrm{s}$ 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 86

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 87

The $100-\mathrm{kg}$ 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 $\omega=8 \mathrm{rad} / \mathrm{s}$. Neglect the weight of the beam and the support.

Ahmed Kamel

Numerade Educator

Problem 88

The $100-\mathrm{kg}$ 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 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 slender $12-\mathrm{kg}$ bar has a clockwise angular velocity of $\omega=2 \mathrm{rad} / \mathrm{s}$ 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.

Ahmed Kamel

Numerade Educator

Problem 92

The $2-\mathrm{kg}$ slender bar is supported by cord $B C$ and then released from rest at $A .$ Determine the initial angular acceleration of the bar and the tension in the cord.

Ahmed Kamel

Numerade Educator

Problem 93

The spool has a mass of $500 \mathrm{~kg}$ and a radius of gyration $k_{G}=1.30 \mathrm{~m}$. It rests on the surface of a conveyor belt for which the coefficient of static friction is $\mu_{s}=0.5$ and the coefficient of kinetic friction is $\mu_{k}=0.4$. If the conveyor accelerates at $a_{C}=1 \mathrm{~m} / \mathrm{s}^{2},$ determine the initial tension in the wire and the angular acceleration of the spool. The spool is originally at rest.

Ahmed Kamel

Numerade Educator

Problem 94

The spool has a mass of $500 \mathrm{~kg}$ and a radius of gyration $k_{G}=1.30 \mathrm{~m}$. It rests on the surface of a conveyor belt for which the coefficient of static friction is $\mu_{s}=0.5$ Determine the greatest acceleration $a_{C}$ of the conveyor so that the spool will not slip. Also, what are the initial tension in the wire and the angular acceleration of the spool? The spool is originally at rest.

Ahmed Kamel

Numerade Educator

Problem 95

The $20-\mathrm{kg}$ 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 96

The assembly consists of an $8-\mathrm{kg}$ disk and a $10-\mathrm{kg}$ bar which is pin connected to the disk. If the system is released from rest, determine the angular acceleration of the disk. The coefficients of static and kinetic friction between the disk and the inclined plane are $\mu_{s}=0.6$ and $\mu_{k}=0.4,$ respectively. Neglect friction at $B .$

James Kiss

Numerade Educator

Problem 97

Solve Prob. $17-96$ if the bar is removed. The coefficients of static and kinetic friction between the disk and inclined plane are $\mu_{s}=0.15$ and $\mu_{k}=0.1,$ respectively.

Ahmed Kamel

Numerade Educator

Problem 98

A force of $F=10 \mathrm{~N}$ is applied to the $10-\mathrm{kg}$ 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 99

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 100

Wheel $C$ has a mass of $60 \mathrm{~kg}$ and a radius of gyration of $0.4 \mathrm{~m}$, whereas wheel $D$ has a mass of $40 \mathrm{~kg}$ and a radius of gyration of $0.35 \mathrm{~m}$. Determine the angular acceleration of each wheel at the instant shown. Neglect the mass of the link and assume that the assembly does not slip on the plane.

Narayan Hari

Numerade Educator

Problem 101

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 102

Solve Prob. $17-101$ if the cord and force $P=50 \mathrm{~N}$ are directed vertically upwards.

Ahmed Kamel

Numerade Educator

Problem 103

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 104

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 105

Solve Prob. $17-104$ if the roller is removed and the coefficient of kinetic friction at the ground is $\mu_{k}$.

Ahmed Kamel

Numerade Educator

Problem 106

A "lifted" truck can become a road hazard since the bumper is high enough to ride up a standard car in the event the car is rear-ended. As a model of this case consider the truck to have a mass of $2.70 \mathrm{Mg}$, a mass center $G,$ and a radius of gyration about $\mathrm{G}$ of $\mathrm{k}_{G}=1.45 \mathrm{~m} .$ Determine the horizontal and vertical components of acceleration of the mass center $G,$ and the angular acceleration of the truck, at the moment its front wheels at $C$ have just left the ground and its smooth front bumper begins to ride up the back of the stopped car so that point $B$ has a velocity of $v_{B}=8 \mathrm{~m} / \mathrm{s}$ at $20^{\circ}$ from the horizontal. Assume the wheels are free to roll, and neglect the size of the wheels and the deformation of the material.

Ummatul Choudary

Ummatul Choudary

Numerade Educator

Problem 107

The $500-\mathrm{kg}$ 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 108

The $12-\mathrm{kg}$ 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 109

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 110

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 111

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 112

The $20-\mathrm{kg}$ disk $A$ is attached to the $10-\mathrm{kg}$ block $B$ using the cable and pulley system shown. If the disk rolls without slipping, determine its angular acceleration and the acceleration of the block when they are released. Also, what is the tension in the cable? Neglect the mass of the pulleys.

Ahmed Kamel

Numerade Educator

Problem 113

The $30-\mathrm{kg}$ uniform slender $\operatorname{rod} A B$ rests in the position shown when the couple moment of $M=150 \mathrm{~N} \cdot \mathrm{m}$ is applied. Determine the initial angular acceleration of the rod. Neglect the mass of the rollers.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 114

The $30-\mathrm{kg}$ slender rod $A B$ rests in the position shown when the horizontal force $P=50 \mathrm{~N}$ is applied. Determine the initial angular acceleration of the rod. Neglect the mass of the rollers.

Ahmed Kamel

Numerade Educator

Problem 115

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 116

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 117

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 118

A long strip of paper is wrapped into two rolls, each having a mass of $8 \mathrm{~kg} .$ Roll $A$ is pin supported about its center whereas roll $B$ is not centrally supported. If $B$ is brought into contact with $A$ and released from rest, determine the initial tension in the paper between the rolls and the angular acceleration of each roll. For the calculation, assume the rolls to be approximated by cylinders.

Ahmed Kamel

Numerade Educator

Problem 119

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.

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 $\boldsymbol{\omega}_{0}$ 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 backspinning stops.

Ahmed Kamel

Numerade Educator