Engineering Mechanics: Dynamics in SI Units

Russell Hibbeler

Chapter 14

Kinetics of a Particle: Work and Energy - all with Video Answers

Educators

Chapter Questions

Problem 1

The 20 -kg crate is subjected to a force having a constant direction and a magnitude $F=100 \mathrm{~N}$. When $s=15 \mathrm{~m}$, the crate is moving to the right with a speed of $8 \mathrm{~m} / \mathrm{s}$. Determine its speed when $s=25 \mathrm{~m}$. The coefficient of kinetic friction between the crate and the ground is $\mu_{k}=0.25$.

Prabhat Tyagi

Prabhat Tyagi

Numerade Educator

Problem 2

The crate, which has a mass of $100 \mathrm{~kg}$, is subjected to the action of the two forces. If it is originally at rest, determine the distance it slides in order to attain a speed of $6 \mathrm{~m} / \mathrm{s}$. The coefficient of kinetic friction between the crate and the surface is $\mu_{k}=0.2$.

Luis Rios

Numerade Educator

Problem 3

The 100 -kg crate is subjected to the forces shown. If it is originally at rest, determine the distance it slides in order to attain a speed of $v=8 \mathrm{~m} / \mathrm{s}$. The coefficient of kinetic friction between the crate and the surface is $\mu_{k}=0.2$

Prabhu Ramji

Numerade Educator

Problem 4

Determine the required height $h$ of the roller coaster so that when it is essentially at rest at the crest of the hill $A$, it will reach a speed of $100 \mathrm{~km} / \mathrm{h}$ when it comes to the bottom $B$. Also, what should be the minimum radius of curvature $\rho$ for the track at $B$ so that the passengers do not experience a normal force greater than $4 m g=(39.24 m) \mathrm{N} ?$ Neglect the size of the car and passenger.

Luis Rios

Numerade Educator

Problem 5

For protection, the barrel barrier is placed in front of the bridge pier. If the relation between the force and deflection of the barrier is $F=\left[800\left(10^{3}\right) x^{1 / 2}\right] \mathrm{N},$ where $x$ is in $\mathrm{m}$, determine the car's maximum penetration in the barrier. The car has a mass of $2 \mathrm{Mg}$ and it is traveling with a speed of $20 \mathrm{~m} / \mathrm{s}$ just before it hits the barrier.

Luis Rios

Numerade Educator

Problem 6

The force of $F=50 \mathrm{~N}$ is applied to the cord when $s=2 \mathrm{~m}$. If the $6-\mathrm{kg}$ collar is orginally at rest, determine its velocity at $s=0$. Neglect friction.

Luis Rios

Numerade Educator

Problem 7

A force of $F=250 \mathrm{~N}$ is applied to the end at $B$. Determine the speed of the $10-\mathrm{kg}$ block when it has moved $1.5 \mathrm{~m}$, starting from rest.

Luis Rios

Numerade Educator

Problem 8

As indicated by the derivation, the principle of work and energy is valid for observers in any inertial reference frame. Show that this is so, by considering the $10-\mathrm{kg}$ block which rests on the smooth surface and is subjected to a horizontal force of $6 \mathrm{~N}$. If observer $A$ is in a fixed frame $x$, determine the final speed of the block if it has an initial speed of $5 \mathrm{~m} / \mathrm{s}$ and travels $10 \mathrm{~m}$, both directed to the right and measured from the fixed frame. Compare the result with that obtained by an observer $B,$ attached to the $x^{\prime}$ axis and moving at a constant velocity of $2 \mathrm{~m} / \mathrm{s}$ relative to $A$. Hint: The distance the block travels will first have to be computed for observer $B$ before applying the principle of work and energy.

Suzanne W.

Numerade Educator

Problem 9

When the driver applies the brakes of a light truck traveling $40 \mathrm{~km} / \mathrm{h},$ it skids $3 \mathrm{~m}$ before stopping. How far will the truck skid if it is traveling at $80 \mathrm{~km} / \mathrm{h}$ when the brakes are applied?

Jacob Adamczyk

Numerade Educator

Problem 10

The "air spring" $A$ is used to protect the support $B$ and prevent damage to the conveyor-belt tensioning weight $C$ in the event of a belt failure $D .$ The force developed by the air spring as a function of its deflection is shown by the graph. If the block has a mass of $20 \mathrm{~kg}$ and is suspended a height $d=0.4 \mathrm{~m}$ above the top of the spring, determine the maximum deformation of the spring in the event the conveyor belt fails. Neglect the mass of the pulley and belt.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 11

The force $\mathbf{F}$, acting in a constant direction on the 20 -kg block, has a magnitude which varies with the position $s$ of the block. Determine how far the block must slide before its velocity becomes $15 \mathrm{~m} / \mathrm{s}$. When $s=0$ the block is moving to the right at $v=6 \mathrm{~m} / \mathrm{s}$. The coefficient of kinetic friction between the block and surface is $\mu_{k}=0.3$.

Luis Rios

Numerade Educator

Problem 12

The skier starts from rest at $A$ and travels down the ramp. If friction and air resistance can be neglected, determine his speed $v_{B}$ when he reaches $B$. Also, find the distance $s$ to where he strikes the ground at $C,$ if he makes the jump traveling horizontally at $B$. Neglect the skier's size. He has a mass of $70 \mathrm{~kg}$.

Rory Naguib

Numerade Educator

Problem 13

Design considerations for the bumper $B$ on the 5-Mg train car require use of a nonlinear spring having the load-deflection characteristics shown in the graph. Select the proper value of $k$ so that the maximum deflection of the spring is limited to $0.2 \mathrm{~m}$ when the car, traveling at $4 \mathrm{~m} / \mathrm{s}$, strikes the rigid stop. Neglect the mass of the car wheels.

Luis Rios

Numerade Educator

Problem 14

The $8-\mathrm{kg}$ cylinder $A$ and $3-\mathrm{kg}$ cylinder $B$ are released from rest. Determine the speed of $A$ after it has moved $2 \mathrm{~m}$ starting from rest. Neglect the mass of the cord and pulleys.

Keshav Singh

Numerade Educator

Problem 15

Cylinder $A$ has a mass of $3 \mathrm{~kg}$ and cylinder $B$ has a mass of $8 \mathrm{~kg}$. Determine the speed of $A$ after it has moved $2 \mathrm{~m}$ starting from rest. Neglect the mass of the cord and pulleys.

Keshav Singh

Numerade Educator

Problem 16

The collar has a mass of $20 \mathrm{~kg}$ and is supported on the smooth rod. The attached springs are undeformed when $d=0.5 \mathrm{~m} .$ Determine the speed of the collar after the applied force $F=100 \mathrm{~N}$ causes it to be displaced so that $d=0.3 \mathrm{~m}$. When $d=0.5 \mathrm{~m}$ the collar is at rest.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 17

A small box of mass $m$ is given a speed of $v=\sqrt{\frac{1}{4} g r}$ at the top of the smooth half cylinder. Determine the angle $\theta$ at which the box leaves the cylinder.

Prashant Bana

Numerade Educator

Problem 18

If the cord is subjected to a constant force of $F=300 \mathrm{~N}$ and the $15-\mathrm{kg}$ smooth collar starts from rest at $A$ determine the velocity of the collar when it reaches point $B$. Neglect the size of the pulley.

Luis Rios

Numerade Educator

Problem 19

If the force exerted by the motor $M$ on the cable is $250 \mathrm{~N},$ determine the speed of the $100-\mathrm{kg}$ crate when it is hoisted to $s=3 \mathrm{~m}$.The crate is at rest when $s=0$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 20

When a $7-\mathrm{kg}$ projectile is fired from a cannon barrel that has a length of $2 \mathrm{~m}$, the explosive force exerted on the projectile, while it is in the barrel, varies in the manner shown. Determine the approximate muzzle velocity of the projectile at the instant it leaves the barrel. Neglect the effects of friction inside the barrel and assume the barrel is horizontal.

Manish Jain

Numerade Educator

Problem 21

The steel ingot has a mass of $1800 \mathrm{~kg}$. It travels along the conveyor at a speed $v=0.5 \mathrm{~m} / \mathrm{s}$ when it collides with the "nested" spring assembly. If the stiffness of the outer spring is $k_{A}=5 \mathrm{kN} / \mathrm{m}$ determine the required stiffness $k_{B}$ of the inner spring so that the motion of the ingot is stopped at the moment the front, $C,$ of the ingot is $0.3 \mathrm{~m}$ from the wall.

Penny Riley

Numerade Educator

Problem 22

The $1.5-\mathrm{kg}$ block slides along a smooth plane and strikes a nonlinear spring with a speed of $v=4 \mathrm{~m} / \mathrm{s}$. The spring is termed "nonlinear" because it has a resistance of $F_{s}=k s^{2},$ where $k=900 \mathrm{~N} / \mathrm{m}^{2} .$ Determine the speed of the block after it has compressed the spring $s=0.2 \mathrm{~m}$.

Keshav Singh

Numerade Educator

Problem 23

A car is equipped with a bumper $B$ designed to absorb collisions. The bumper is mounted to the car using pieces of flexible tubing $T$. Upon collision with a rigid barrier at $A,$ a constant horizontal force $\mathbf{F}$ is developed which causes a car deceleration of $3 g=29.43 \mathrm{~m} / \mathrm{s}^{2}$ (the highest safe deceleration for a passenger without a seatbelt). If the car and passenger have a total mass of $1.5 \mathrm{Mg}$ and the car is initially coasting with a speed of $1.5 \mathrm{~m} / \mathrm{s}$, determine the magnitude of $\mathbf{F}$ needed to stop the car and the deformation $x$ of the bumper tubing.

Dading Chen

Numerade Educator

Problem 24

The catapulting mechanism is used to propel the $10-\mathrm{kg}$ slider $A$ to the right along the smooth track. The propelling action is obtained by drawing the pulley attached to $\operatorname{rod} B C$ rapidly to the left by means of a piston $P$. If the piston applies a constant force $F=20 \mathrm{kN}$ to $\operatorname{rod} B C$ such that it moves it $0.2 \mathrm{~m},$ determine the speed attained by the slider if it was originally at rest. Neglect the mass of the pulleys, cable, piston, and $\operatorname{rod} B C$.

Narayan Hari

Numerade Educator

Problem 25

The $12-\mathrm{kg}$ block has an initial speed of $v_{0}=4 \mathrm{~m}$ when it is midway between springs $A$ and $B$. After striking spring $B$, it rebounds and slides across the horizontal plane toward spring $A,$ etc. If the coefficient of kinetic friction between the plane and the block is $\mu_{k}=0.4,$ determine the total distance traveled by the block before it comes to rest.

Luis Rios

Numerade Educator

Problem 26

The $8-\mathrm{kg}$ block is moving with an initial speed of $5 \mathrm{~m} / \mathrm{s}$. If the coefficient of kinetic friction between the block and plane is $\mu_{k}=0.25,$ determine the compression in the spring when the block momentarily stops.

Luis Rios

Numerade Educator

Problem 27

Marbles having a mass of $5 \mathrm{~g}$ are dropped from rest at $A$ through the smooth glass tube and accumulate in the can at $C$. Determine the placement $R$ of the can from the end of the tube and the speed at which the marbles fall into the can. Neglect the size of the can.

Luis Rios

Numerade Educator

Problem 28

The collar has a mass of $20 \mathrm{~kg}$ and slides along the smooth rod. Two springs are attached to it and the ends of the rod as shown. If each spring has an uncompressed length of $1 \mathrm{~m}$ and the collar has a speed of $2 \mathrm{~m} / \mathrm{s}$ when $s=0$, determine the maximum compression of each spring due to the back-and-forth (oscillating) motion of the collar.

Keshav Singh

Numerade Educator

Problem 29

The train car has a mass of $10 \mathrm{Mg}$ and is traveling at $5 \mathrm{~m} / \mathrm{s}$ when it reaches $A .$ If the rolling resistance is $1 / 100$ of the weight of the car, determine the compression of each spring when the car is momentarily brought to rest.

Paul Gabriel

Numerade Educator

Problem 30

The $0.5-\mathrm{kg}$ ball is fired up the smooth vertical circular track using the spring plunger. The plunger keeps the spring compressed $0.08 \mathrm{~m}$ when $s=0 .$ Determine how far $s$ it must be pulled back and released so that the ball will begin to leave the track when $\theta=135^{\circ}$.

Anand Jangid

Numerade Educator

Problem 31

The "flying car" is a ride at an amusement park which consists of a car having wheels that roll along a track mounted inside a rotating drum. By design the car cannot fall off the track, however motion of the car is developed by applying the car's brake, thereby gripping the car to the track and allowing it to move with a constant speed of the track, $v_{t}=3 \mathrm{~m} / \mathrm{s}$. If the rider applies the brake when going from $B$ to $A$ and then releases it at the top of the drum, $A,$ so that the car coasts freely down along the track to $B(\theta=\pi$ rad), determine the speed of the car at $B$ and the normal reaction which the drum exerts on the car at $B$. Neglect friction during the motion from $A$ to $B$. The rider and car have a total mass of $250 \mathrm{~kg}$ and the center of mass of the car and rider moves along a circular path having a radius of $8 \mathrm{~m}$.

Luis Rios

Numerade Educator

Problem 32

The man at the window $A$ wishes to throw the 30 -kg sack on the ground. To do this he allows it to swing from rest at $B$ to point $C,$ when he releases the cord at $\theta=30^{\circ} .$ Determine the speed at which it strikes the ground and the distance $R$.

Donald Albin

Numerade Educator

Problem 33

The cyclist travels to point $A,$ pedaling until he reaches a speed $v_{A}=4 \mathrm{~m} / \mathrm{s}$. He then coasts freely up the curved surface. Determine how high he reaches up the surface before he comes to a stop. Also, what are the resultant normal force on the surface at this point and his acceleration? The total mass of the bike and man is $75 \mathrm{~kg}$. Neglect friction, the mass of the wheels, and the size of the bicycle.

Prabhat Tyagi

Numerade Educator

Problem 34

The conveyor belt delivers each $12-\mathrm{kg}$ crate to the ramp at $A$ such that the crate's velocity is $v_{A}=2.5 \mathrm{~m} / \mathrm{s}$ directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp is $\mu_{k}=0.3,$ determine the speed at which each crate slides off the ramp at $B$. Assume that no tipping occurs.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 35

The block has a mass of $0.8 \mathrm{~kg}$ and moves within the smooth vertical slot. If it starts from rest when the attached spring is in the unstretched position at $A,$ determine the constant vertical force $F$ which must be applied to the cord so that the block attains a speed $v_{B}=2.5 \mathrm{~m} / \mathrm{s}$ when it reaches $B ; s_{B}=0.15 \mathrm{~m} .$ Neglect the size and mass of the pulley. Hint: The work of $\mathbf{F}$ can be determined by finding the difference $\Delta l$ in cord lengths $A C$ and $B C$ and using $U_{F}=F \Delta l$

Keshav Singh

Numerade Educator

Problem 36

If the 60 -kg skier passes point $A$ with a speed of $5 \mathrm{~m} / \mathrm{s}$, determine his speed when he reaches point $B$. Also find the normal force exerted on him by the slope at this point. Neglect friction.

Supratim Pal

Numerade Educator

Problem 37

The spring in the toy gun has an unstretched length of $100 \mathrm{~mm}$. It is compressed and locked in the position shown. When the trigger is pulled, the spring unstretches $12.5 \mathrm{~mm}$, and the 20 -g ball moves along the barrel. Determine the speed of the ball when it leaves the gun. Neglect friction.

Vishal Gupta

Numerade Educator

Problem 38

If the track is to be designed so that the passengers of the roller coaster do not experience a normal force equal to zero or more than 4 times their weight, determine the limiting heights $h_{A}$ and $h_{C}$ so that this does not occur. The roller coaster starts from rest at position $A .$ Neglect friction.

Bret Rosen

Numerade Educator

Problem 39

The skier starts from rest at $A$ and travels down the ramp. If friction and air resistance can be neglected, determine his speed $v_{B}$ when he reaches $B$. Also, find the distance $s$ to where he strikes the ground at $C$, if he makes the jump traveling horizontally at $B$. Neglect the skier's size. He has a mass $75 \mathrm{~kg}$.

Rory Naguib

Numerade Educator

Problem 40

If the $75-\mathrm{kg}$ crate starts from rest at $A,$ determine its speed when it reaches point $B$. The cable is subjected to a constant force of $F=300 \mathrm{~N}$. Neglect friction and the size of the pulley.

Averell Hause

Carnegie Mellon University

Problem 41

If the $75-\mathrm{kg}$ crate starts from rest at $A,$ and its speed is $6 \mathrm{~m} / \mathrm{s}$ when it passes point $B,$ determine the constant force $\mathbf{F}$ exerted on the cable. Neglect friction and the size of the pulley.

Averell Hause

Carnegie Mellon University

Problem 42

A spring having a stiffness of $5 \mathrm{kN} / \mathrm{m}$ is compressed $400 \mathrm{~mm}$. The stored energy in the spring is used to drive a machine which requires $90 \mathrm{~W}$ of power. Determine how long the spring can supply energy at the required rate.

Gregory Devenport

Gregory Devenport

Numerade Educator

Problem 43

To dramatize the loss of energy in an automobile, consider a car having a weight of $25000 \mathrm{~N}$ that is traveling at $56 \mathrm{~km} / \mathrm{h}$. If the car is brought to a stop, determine how long a $100-\mathrm{W}$ light bulb must burn to expend the same amount of energy.

Luis Rios

Numerade Educator

Problem 44

If the engine of a 1.5-Mg car generates a constant power of $15 \mathrm{~kW},$ determine the speed of the car after it has traveled a distance of $200 \mathrm{~m}$ on a level road starting from rest. Neglect friction.

Sachin Rao

Numerade Educator

Problem 45

If the engine of a 1.5-Mg car generates a constant power of $15 \mathrm{~kW},$ determine the speed of the car after it has traveled a distance of $200 \mathrm{~m}$ on a level road starting from rest. Neglect friction.

Sachin Rao

Numerade Educator

Problem 46

The 2 -Mg car increases its speed uniformly from rest to $25 \mathrm{~m} / \mathrm{s}$ in $30 \mathrm{~s}$ up the inclined road. Determine the maximum power that must be supplied by the engine, which operates with an efficiency of $\varepsilon=0.8 .$ Also, find the average power supplied by the engine.

Luis Rios

Numerade Educator

Problem 47

A car has a mass $m$ and accelerates along a horizontal straight road from rest such that the power is always a constant amount $P .$ Determine how far it must travel to reach a speed of $v$

Dheeraj Sharma

Numerade Educator

Problem 48

An automobile having a mass of $2 \mathrm{Mg}$ travels up a $7^{\circ}$ slope at a constant speed of $v=100 \mathrm{~km} / \mathrm{h}$. If mechanical friction and wind resistance are neglected, determine the power developed by the engine if the automobile has an efficiency $\varepsilon=0.65$.

Luis Rios

Numerade Educator

Problem 49

A rocket having a total mass of $8 \mathrm{Mg}$ is fired vertically from rest. If the engines provide a constant thrust of $T=300 \mathrm{kN},$ determine the power output of the engines as a function of time. Neglect the effect of drag resistance and the loss of fuel mass and weight.

Aaron Shoolroy

Numerade Educator

Problem 50

The sports car has a mass of $2.3 \mathrm{Mg}$, and while it is traveling at $28 \mathrm{~m} / \mathrm{s}$ the driver causes it to accelerate at $5 \mathrm{~m} / \mathrm{s}^{2}$. If the drag resistance on the car due to the wind is $F_{D}=\left(0.3 v^{2}\right) \mathrm{N},$ where $v$ is the velocity in $\mathrm{m} / \mathrm{s}$, determine the power supplied to the engine at this instant. The engine has a running efficiency of $\varepsilon=0.68$.

Jacob Adamczyk

Numerade Educator

Problem 51

The sports car has a mass of $2.3 \mathrm{Mg}$ and accelerates at $6 \mathrm{~m} / \mathrm{s}^{2}$, starting from rest. If the drag resistance on the car due to the wind is $F_{D}=(10 v) \mathrm{N},$ where $v$ is the velocity in $\mathrm{m} / \mathrm{s},$ determine the power supplied to the engine when $t=5 \mathrm{~s}$. The engine has a running efficiency of $\varepsilon=0.68$.

Dominador Tan

Numerade Educator

Problem 52

A motor hoists a $60-\mathrm{kg}$ crate at a constant velocity to a height of $h=5$ in $2 \mathrm{~s}$. If the indicated power of the motor is $3.2 \mathrm{~kW},$ determine the motor's efficiency.

Emily Anderson

Numerade Educator

Problem 53

The 50 -kg crate is hoisted up the $30^{\circ}$ incline by the pulley system and motor $M$. If the crate starts from rest and, by constant acceleration, attains a speed of $4 \mathrm{~m} / \mathrm{s}$ after traveling $8 \mathrm{~m}$ along the plane, determine the power that must be supplied to the motor at the instant. Neglect friction along the plane. The motor has an efficiency of $\varepsilon=0.74$.

Ren Jie Tuieng

Numerade Educator

Problem 54

The $500-\mathrm{kg}$ elevator starts from rest and travels upward with a constant acceleration $a_{c}=2 \mathrm{~m} / \mathrm{s}^{2}$. Determine the power output of the motor $M$ when $t=3 \mathrm{~s}$. Neglect the mass of the pulleys and cable.

Narayan Hari

Numerade Educator

Problem 55

The escalator steps move with a constant speed of $0.6 \mathrm{~m} / \mathrm{s}$. If the steps are $125 \mathrm{~mm}$ high and $250 \mathrm{~mm}$ in length, determine the power of a motor needed to lift an average mass of $150 \mathrm{~kg}$ per step. There are 32 steps.

Luis Rios

Numerade Educator

Problem 56

If the escalator in Prob. 14-55 is not moving, determine the constant speed at which a man having a mass of $80 \mathrm{~kg}$ must walk up the steps to generate $100 \mathrm{~W}$ of power-the same amount that is needed to power a standard light bulb.

Luis Rios

Numerade Educator

Problem 57

The elevator $E$ and its freight have a total mass of $400 \mathrm{~kg}$. Hoisting is provided by the motor $M$ and the 60 -kg block $C$. If the motor has an efficiency of $\varepsilon=0.6$, determine the power that must be supplied to the motor when the elevator is hoisted upward at a constant speed of $v_{E}=4 \mathrm{~m} / \mathrm{s}$.

Sophie S

Numerade Educator

Problem 58

The crate has a mass of $150 \mathrm{~kg}$ and rests on a surface for which the coefficients of static and kinetic friction are $\mu_{s}=0.3$ and $\mu_{k}=0.2,$ respectively. If the motor $M$ supplies a cable force of $F=\left(8 t^{2}+20\right) \mathrm{N},$ where $t$ is in seconds, determine the power output developed by the motor when $t=5 \mathrm{~s}$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 59

The material hoist and the load have a total mass of $800 \mathrm{~kg}$ and the counterweight $C$ has a mass of $150 \mathrm{~kg}$. At a given instant, the hoist has an upward velocity of $2 \mathrm{~m} / \mathrm{s}$ and an acceleration of $1.5 \mathrm{~m} / \mathrm{s}^{2}$. Determine the power generated by the motor $M$ at this instant if it operates with an efficiency of $\varepsilon=0.8$.

Anand Jangid

Numerade Educator

Problem 60

The material hoist and the load have a total mass of $800 \mathrm{~kg}$ and the counterweight $C$ has a mass of $150 \mathrm{~kg}$. If the upward speed of the hoist increases uniformly from $0.5 \mathrm{~m} / \mathrm{s}$ to $1.5 \mathrm{~m} / \mathrm{s}$ in $1.5 \mathrm{~s}$, determine the average power generated by the motor $M$ during this time. The motor operates with an efficiency of $\varepsilon=0.8$.

Anand Jangid

Numerade Educator

Problem 61

An athlete pushes against an exercise machine with a force that varies with time as shown in the first graph. Also, the velocity of the athlete's arm acting in the same direction as the force varies with time as shown in the second graph. Determine the power applied as a function of time and the work done in $t=0.3 \mathrm{~s}$.

Victor Salazar

Numerade Educator

Problem 62

An athlete pushes against an exercise machine with a force that varies with time as shown in the first graph. Also, the velocity of the athlete's arm acting in the same direction as the force varies with time as shown in the second graph. Determine the maximum power developed during the 0.3 -second time period.

Rashmi Sinha

Numerade Educator

Problem 63

If the jet on the dragster supplies a constant thrust of $T=20 \mathrm{kN},$ determine the power generated by the jet as a function of time. Neglect drag and rolling resistance, and the loss of fuel. The dragster has a mass of $1 \mathrm{Mg}$ and starts from rest.

Luis Rios

Numerade Educator

Problem 64

The rocket sled has a mass of $4 \mathrm{Mg}$ and travels from rest along the horizontal track for which the coefficient of kinetic friction is $\mu_{k}=0.20 .$ If the engine provides a constant thrust $T=150 \mathrm{kN},$ determine the power output of the engine as a function of time. Neglect the loss of fuel mass and air resistance.

Chai Santi

Numerade Educator

Problem 65

The block has a mass of $150 \mathrm{~kg}$ and rests on a surface for which the coefficients of static and kinetic friction are $\mu_{s}=0.5$ and $\mu_{k}=0.4,$ respectively. If a force $F=\left(60 t^{2}\right) \mathrm{N},$ where $t$ is in seconds, is applied to the cable, determine the power developed by the force when $t=5 \mathrm{~s}$. Hint: First determine the time needed for the force to cause motion.

Luis Rios

Numerade Educator

Problem 66

The assembly consists of two blocks $A$ and $B$, which have a mass of $20 \mathrm{~kg}$ and $30 \mathrm{~kg}$, respectively. Determine the distance $B$ must descend in order for $A$ to achieve a speed of $3 \mathrm{~m} / \mathrm{s}$ starting from rest.

Luis Rios

Numerade Educator

Problem 67

The assembly consists of two blocks $A$ and $B$ which have a mass of $20 \mathrm{~kg}$ and $30 \mathrm{~kg}$, respectively. Determine the speed of each block when $B$ descends $1.5 \mathrm{~m}$. The blocks are released from rest. Neglect the mass of the pulleys and cords.

Luis Rios

Numerade Educator

Problem 68

The girl has a mass of $40 \mathrm{~kg}$ and center of mass at $G$. If she is swinging to a maximum height defined by $\theta=60^{\circ},$ determine the force developed along each of the four supporting posts such as $A B$ at the instant $\theta=0^{\circ} .$ The swing is centrally located between the posts.

Rory Naguib

Numerade Educator

Problem 69

Each of the two elastic rubber bands of the slingshot has an unstretched length of $180 \mathrm{~mm}$. If they are pulled back to the position shown and released from rest, determine the maximum height the 30 -g pellet will reach if it is fired vertically upward. Neglect the mass of the rubber bands and the change in elevation of the pellet while it is constrained by the rubber bands. Each rubber band has a stiffness $k=80 \mathrm{~N} / \mathrm{m}$

Nick Johnson

Numerade Educator

Problem 70

Two equal-length springs are "nested" together in order to form a shock absorber. If it is designed to arrest the motion of a $2-\mathrm{kg}$ mass that is dropped $s=0.5 \mathrm{~m}$ above the top of the springs from an at-rest position, and the maximum compression of the springs is to be $0.2 \mathrm{~m}$, determine the required stiffness of the inner spring, $k_{B},$ if the outer spring has a stiffness $k_{A}=400 \mathrm{~N} / \mathrm{m}$

Narayan Hari

Numerade Educator

Problem 71

The $5-\mathrm{kg}$ collar has a velocity of $5 \mathrm{~m} / \mathrm{s}$ to the right when it is at $A$. It then travels down along the smooth guide. Determine the speed of the collar when it reaches point $B$, which is located just before the end of the curved portion of the rod. The spring has an unstretched length of $100 \mathrm{~mm}$ and $B$ is located just before the end of the curved portion of the rod.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 72

The $5-\mathrm{kg}$ collar has a velocity of $5 \mathrm{~m} / \mathrm{s}$ to the right when it is at $A .$ It then travels along the smooth guide. Determine its speed when its center reaches point $B$ and the normal force it exerts on the rod at this point. The spring has an unstretched length of $100 \mathrm{~mm}$ and $B$ is located just before the end of the curved portion of the rod.

Rory Naguib

Numerade Educator

Problem 73

The roller coaster car has a mass of $700 \mathrm{~kg}$, including its passenger. If it starts from the top of the hill $A$ with a speed $v_{A}=3 \mathrm{~m} / \mathrm{s}$, determine the minimum height $h$ of the hill crest so that the car travels around the inside loops without leaving the track. Neglect friction, the mass of the wheels, and the size of the car. What is the normal reaction on the car when the car is at $B$ and when it is at $C ?$ Take $\rho_{B}=7.5 \mathrm{~m}$ and $\rho_{C}=5 \mathrm{~m}$

Luis Rios

Numerade Educator

Problem 74

The roller coaster car has a mass of $700 \mathrm{~kg}$, including its passenger. If it is released from rest at the top of the hill $A$, determine the minimum height $h$ of the hill crest so that the car travels around both inside the loops without leaving the track. Neglect friction, the mass of the wheels, and the size of the car. What is the normal reaction on the car when the car is at $B$ and when it is at $C ?$ Take $\rho_{B}=7.5 \mathrm{~m}$ and $\rho_{C}=5 \mathrm{~m}$.

Luis Rios

Numerade Educator

Problem 75

The 2 -kg ball of negligible size is fired from point $A$ with an initial velocity of $10 \mathrm{~m} / \mathrm{s}$ up the smooth inclined plane. Determine the distance from point $C$ to where it hits the horizontal surface at $D$. Also, what is its velocity when it strikes the surface?

Narayan Hari

Numerade Educator

Problem 76

The 4 -kg smooth collar has a speed of $3 \mathrm{~m} / \mathrm{s}$ when it is at $s=0 .$ Determine the maximum distance $s$ it travels before it stops momentarily. The spring has an unstretched length of $1 \mathrm{~m}$

Rory Naguib

Numerade Educator

Problem 77

The spring has a stiffness $k=200 \mathrm{~N} / \mathrm{m}$ and an unstretched length of $0.5 \mathrm{~m}$. If it is attached to the $3-\mathrm{kg}$ smooth collar and the collar is released from rest at $A$ determine the speed of the collar when it reaches $B$. Neglect the size of the collar.

Luis Rios

Numerade Educator

Problem 78

The roller coaster car having a mass $m$ is released from rest at point $A$. If the track is to be designed so that the car does not leave it at $B,$ determine the required height $h$. Also, find the speed of the car when it reaches point $C$. Neglect friction.

Luis Rios

Numerade Educator

Problem 79

A 750 -mm-long spring is compressed and confined by the plate $P$, which can slide freely along the vertical 600 -mm-long rods. The $40-\mathrm{kg}$ block is given a speed of $v=5 \mathrm{~m} / \mathrm{s}$ when it is $h=2 \mathrm{~m}$ above the plate. Determine how far the plate moves downwards when the block momentarily stops after striking it. Neglect the mass of the plate.

Averell Hause

Carnegie Mellon University

Problem 80

The spring has a stiffness $k=50 \mathrm{~N} / \mathrm{m}$ and an unstretched length of $0.3 \mathrm{~m}$. If it is attached to the $2-\mathrm{kg}$ smooth collar and the collar is released from rest at $A$ $\left(\theta=0^{\circ}\right)$, determine the speed of the collar when $\theta=60^{\circ}$ The motion occurs in the horizontal plane. Neglect the size of the collar.

Luis Rios

Numerade Educator

Problem 81

If the mass of the earth is $M_{e}$, show that the gravitational potential energy of a body of mass $m$ located a distance $r$ from the center of the earth is $V_{g}=-G M_{e} m / r$. Recall that the gravitational force acting between the earth and the body is $F=G\left(M_{e} m / r^{2}\right),$ Eq. 13-1. For the calculation, locate the datum at $r \rightarrow \infty$. Also, prove that $F$ is a conservative force.

Rory Naguib

Numerade Educator

Problem 82

A rocket of mass $m$ is fired vertically from the surface of the earth, i.e., at $r=r_{1}$. Assuming that no mass is lost as it travels upward, determine the work it must do against gravity to reach a distance $r_{2}$. The force of gravity is $F=G M_{e} m / r^{2}$ (Eq.13-1), where $M_{e}$ is the mass of the earth and $r$ the distance between the rocket and the center of the earth.

Rory Naguib

Numerade Educator

Problem 83

When $s=0$, the spring on the firing mechanism is unstretched. If the arm is pulled back such that $s=100 \mathrm{~mm}$ and released, determine the speed of the $0.3-\mathrm{kg}$ ball and the normal reaction of the circular track on the ball when $\theta=60^{\circ} .$ Assume all surfaces of contact to be smooth. Neglect the mass of the spring and the size of the ball.

Rory Naguib

Numerade Educator

Problem 84

When $s=0,$ the spring on the firing mechanism is unstretched. If the arm is pulled back such that $s=100 \mathrm{~mm}$ and released, determine the maximum angle $\theta$ the ball will travel without leaving the circular track. Assume all surfaces of contact to be smooth. Neglect the mass of the spring and the size of the ball.

Rory Naguib

Numerade Educator

Problem 85

When the $5-\mathrm{kg}$ box reaches point $A$ it has a speed $v_{A}=10 \mathrm{~m} / \mathrm{s}$. Determine the normal force the box exerts on the surface when it reaches point $B$. Neglect friction and the size of the box.

Narayan Hari

Numerade Educator

Problem 86

When the $5-\mathrm{kg}$ box reaches point $A$ it has a speed $v_{A}=10 \mathrm{~m} / \mathrm{s}$. Determine how high the box reaches up the surface before it comes to a stop. Also, what is the resultant normal force on the surface at this point and the acceleration? Neglect friction and the size of the box.

Narayan Hari

Numerade Educator

Problem 87

When the 6 -kg box reaches point $A$ it has a speed of $v_{A}=2 \mathrm{~m} / \mathrm{s}$. Determine the angle $\theta$ at which it leaves the smooth circular ramp and the distance $s$ to where it falls into the cart. Neglect friction.

Luis Rios

Numerade Educator

Problem 88

The skier starts from rest at $A$ and travels down the ramp. If friction and air resistance can be neglected, determine his speed $v_{B}$ when he reaches $B$. Also, compute the distance $s$ to where he strikes the ground at $C,$ if he makes the jump traveling horizontally at $B .$ Neglect the skier's size. He has a mass of $70 \mathrm{~kg}$.

Rory Naguib

Numerade Educator

Problem 89

A $60-\mathrm{kg}$ satellite travels in free flight along an elliptical orbit such that at $A,$ where $r_{A}=20 \mathrm{Mm},$ it has a speed $v_{A}=40 \mathrm{Mm} / \mathrm{h} .$ What is the speed of the satellite when it reaches point $B$, where $r_{B}=80 \mathrm{Mm}$ ? Hint: See Prob. $\quad$ 14-81, where $\quad M_{e}=5.976\left(10^{24}\right) \mathrm{kg}$ and $G=66.73\left(10^{-12}\right) \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)$

Rory Naguib

Numerade Educator

Problem 90

The block has a mass of $20 \mathrm{~kg}$ and is released from rest when $s=0.5 \mathrm{~m}$. If the mass of the bumpers $A$ and $B$ can be neglected, determine the maximum deformation of each spring due to the collision.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 91

The $0.75-\mathrm{kg}$ bob of a pendulum is fired from rest at position $A$ by a spring which has a stiffness $k=6 \mathrm{kN} / \mathrm{m}$ and is compressed $125 \mathrm{~mm}$. Determine the speed of the bob and the tension in the cord when the bob is at positions $B$ and $C$. Point $B$ is located on the path where the radius of curvature is still $0.6 \mathrm{~m}$, i.e., just before the cord becomes horizontal.

Maria Gabriela Cota Moreira

Maria Gabriela Cota Moreira

Numerade Educator

Problem 92

The Raptor is an outside loop roller coaster in which riders are belted into seats resembling ski-lift chairs. If the cars travel at $v_{0}=4 \mathrm{~m} / \mathrm{s}$ when they are at the top of the hill, determine their speed when they are at the top of the loop and the reaction of the 70 -kg passenger on his seat at this instant. The car has a mass of $50 \mathrm{~kg} .$ Take $h=12 \mathrm{~m}$, $\rho=5 \mathrm{~m} .$ Neglect friction and the size of the car and passenger.

Luis Rios

Numerade Educator

Problem 93

If the 20 -kg cylinder is released from rest at $h=0$, determine the required stiffness $k$ of each spring so that its motion is arrested or stops when $h=0.5 \mathrm{~m}$. Each spring has an unstretched length of $1 \mathrm{~m}$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 94

The cylinder has a mass of $20 \mathrm{~kg}$ and is released from rest when $h=0 .$ Determine its speed when $h=3 \mathrm{~m}$. Each spring has a stiffness $k=40 \mathrm{~N} / \mathrm{m}$ and an unstretched length of $2 \mathrm{~m}$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 95

A quarter-circular tube $A B$ of mean radius $r$ contains a smooth chain that has a mass per unit length of $m_{0}$. If the chain is released from rest from the position shown, determine its speed when it emerges completely from the tube.

Suzanne W.

Numerade Educator

Problem 96

The $10-\mathrm{kg}$ sphere $C$ is released from rest when $\theta=0^{\circ}$ and the tension in the spring is $100 \mathrm{~N}$. Determine the speed of the sphere at the instant $\theta=90^{\circ} .$ Neglect the mass of $\operatorname{rod} A B$ and the size of the sphere.

Anand Jangid

Numerade Educator

Problem 97

A pan of negligible mass is attached to two identical springs of stiffness $k=250 \mathrm{~N} / \mathrm{m}$. If a $10-\mathrm{kg}$ box is dropped from a height of $0.5 \mathrm{~m}$ above the pan, determine the maximum vertical displacement $d$. Initially each spring has a tension of $50 \mathrm{~N}$.

Abid Hussain

Numerade Educator