Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 7

Work and Energy - all with Video Answers

Educators

Chapter Questions

Problem 1

[The Problems in this Section are ranked I, II, or III according to estimated difficulty, with $( 1 )$ Problems being easiest. Level (III) Problems are meant mainly as a challenge for the best students, for "extra credit. "The Problems are arranged by Sections, meaning that the reader should have read up to and including that Section, but this Chapter also has a group of Gencral Problems that are not arranged by Section and not ranked.]

(1) How much work is done by the gravitational force when a 280 -kg pile driver falls 2.80$\mathrm { m }$ ?

Ben Nicholson

Ben Nicholson

Numerade Educator

Problem 2

(I) How high will a 1.85 -kg rock go if thrown straight up by someone who does 80.0$\mathrm { J }$ of work on it? Neglect air resistance.

Averell Hause

Carnegie Mellon University

Problem 3

(I) A 75.0 -kg firefighter climbs a flight of stairs 20.0$\mathrm { m }$ high. How much work is required?

Ben Nicholson

Numerade Educator

Problem 4

(I) A hammerhead with a mass of 2.0$\mathrm { kg }$ is allowed to fall onto a nail from a height of 0.50$\mathrm { m }$ . What is the maximum amount of work it could do on the nail? Why do people not just "let it fall" but add their own force to the hammer as it falls?

Averell Hause

Carnegie Mellon University

Problem 5

(II) Estimate the work you do to mow a lawn 10$\mathrm { m }$ by 20$\mathrm { m }$ with a 50 -cm wide mower. Assume you push with a force of about 15$\mathrm { N }$ .

Ben Nicholson

Numerade Educator

Problem 6

(II) A lever such as that shown in Fig. 20 can be used to lift objects we might not otherwise be able to lift. Show that the ratio of output force, $F _ { \mathrm { O } } ,$ to input force, $F _ { \mathrm { I } } ,$ is and $\ell _ { \mathrm { O } }$ from the pivot by $F _ { \mathrm { O } } / F _ { \mathrm { I } } = \ell _ { 1 } / \ell _ { \mathrm { O } }$ . Ignore friction and the mass of the lever, and assume the work output equals work input.

Averell Hause

Carnegie Mellon University

Problem 7

(II) What is the minimum work needed to push a $950 - \mathrm { kg }$ car 310$\mathrm { m }$ up along a $9.0 ^ { \circ }$ incline? Ignore friction.

Ben Nicholson

Numerade Educator

Problem 8

(II) Eight books, each 4.0$\mathrm { cm }$ thick with mass 1.8$\mathrm { kg }$ , lie flat on a table. How much work is required to stack them one on top of another?

Averell Hause

Carnegie Mellon University

Problem 9

(II) A box of mass 6.0$\mathrm { kg }$ is accelerated from rest by a force across a floor at a rate of 2.0$\mathrm { m } / \mathrm { s } ^ { 2 }$ for 7.0$\mathrm { s }$ . Find the net work done on the box.

Ben Nicholson

Numerade Educator

Problem 10

(II) $( a )$ What magnitude force is required to give a helicopter of mass $M$ an acceleration of 0.10$g$ upward? (b) What work is done by this force as the helicopter moves a distance $h$ upward?

Averell Hause

Carnegie Mellon University

Problem 11

(II) A 380 -kg piano slides 3.9$\mathrm { m }$ down a $27 ^ { \circ }$ incline and is kept from accelerating by a man who is pushing back on it parallel to the incline (Fig. $21 ) .$ Determine: $( a )$ the force exerted by the man, $( b )$ the work done by the man on the piano, $( c )$ the work done by the force of gravity, and $( d )$ the net work done on the piano. Ignore friction.

Ben Nicholson

Numerade Educator

Problem 12

(II) A gondola can carry 20 skiers, with a total mass of up to 2250$\mathrm { kg }$ . The gondola ascends at a constant speed from the base of a mountain, at $2150 \mathrm { m } ,$ to the summit at 3345$\mathrm { m }$ . (a) How much work does the motor do in moving a full gondola up the mountain? (b) How much work does gravity do on the gondola? (c) If the motor is capable of generating 10$\%$ more work than found in $( a ) ,$ what is the acceleration of the gondola?

Averell Hause

Carnegie Mellon University

Problem 13

(II) A $17,000 - \mathrm { kg }$ jet takes off from an aircraft carrier via a catapult (Fig, 22$\mathrm { a } ) .$ The gases thrust out from the jet's engines exert a constant force of 130$\mathrm { kN }$ on the jet; the force exerted on the jet by the catapult is plotted in Fig. 22$\mathrm { b }$ . Determine: (a) the work done on the jet by the gases expelled by its engines during launch of the jet; and $( b )$ the work done on the jet by the catapult during launch of the jet.

Ben Nicholson

Numerade Educator

Problem 14

(II) A 2200 -N crate rests on the floor. How much work is required to move it at constant speed (a) 4.0$\mathrm { m }$ along the floor against a drag force of $230 \mathrm { N } ,$ and $( b ) 4.0 \mathrm { m }$ vertically?

Averell Hause

Carnegie Mellon University

Problem 15

(II) A grocery cart with mass of 16$\mathrm { kg }$ is being pushed at constant speed up a flat $12 ^ { \circ }$ ramp by a force $F _ { \mathrm { P } }$ which acts at an angle of $17 ^ { \circ }$ below the horizontal. Find the work done by each of the forces $\left( m \vec { g } , \vec { \mathbf { F } } _ { \mathrm { N } } , \vec { \mathbf { F } } _ { \mathrm { P } } \right)$ on the cart if the ramp is 15$\mathrm { m }$ long.

Ben Nicholson

Numerade Educator

Problem 16

(1) What is the dot product of $\vec { \mathbf { A } } = 2.0 x ^ { 2 } \hat { \mathbf { i } } - 4.0 x \hat { \mathbf { j } } + 5.0 \hat { \mathbf { k } }$ and $\vec { \mathbf { B } } = 11.0 \hat { \mathbf { i } } + 2.5 x \hat { \mathbf { j } } ?$

Averell Hause

Carnegie Mellon University

Problem 17

(I) For any vector $\vec { \mathbf { v } } = V _ { x } \hat { \mathbf { i } } + V _ { y } \hat { \mathbf { j } } + V _ { z } \hat { \mathbf { k } }$ show that $V _ { x } = \hat { \mathbf { i } } \cdot \vec { \mathbf { v } } , \quad V _ { y } = \hat { \mathbf { j } } \cdot \vec { \mathbf { v } } , \quad V _ { z } = \hat { \mathbf { k } } \cdot \vec { \mathbf { v } }$

Ben Nicholson

Numerade Educator

Problem 18

(I) Calculate the angle between the vectors: $\vec { \mathbf { A } } = 6.8 \hat { \mathbf { i } } - 3.4 \hat { \mathbf { j } } - 6.2 \hat { \mathbf { k } } \quad$ and $\quad \vec { \mathbf { B } } = 8.2 \hat { \mathbf { i } } + 2.3 \hat { \mathbf { j } } - 7.0 \hat { \mathbf { k } }$

Averell Hause

Carnegie Mellon University

Problem 19

(I) Show that $$\vec { \mathbf { A } } \cdot ( - \vec { \mathbf { B } } ) = - \vec { \mathbf { A } } \cdot \vec { \mathbf { B } }$$

Ben Nicholson

Numerade Educator

Problem 20

(1) Vector $\vec { \mathbf { v } } _ { 1 }$ points along the $z$ axis and has magnitude $V _ { 1 } = 75 .$ Vector $\vec { \mathbf { V } } _ { 2 }$ lies in the $x z$ plane, has magnitude $V _ { 2 } = 58 ,$ and makes a $- 48 ^ { \circ }$ angle with the $x$ axis (points below $x$ axis). What is the scalar product $\vec { \mathbf { v } } _ { 1 } \cdot \vec { \mathbf { V } } _ { 2 }$ ?

Averell Hause

Carnegie Mellon University

Problem 21

(II) Given the vector $\vec { \mathbf { A } } = 3.0 \hat { \mathbf { i } } + 1.5 \hat { \mathbf { j } }$ , find a vector $\vec { \mathbf { B } }$ that(II) Given the vector $\vec { \mathbf { A } } = 3.0 \hat { \mathbf { i } } + 1.5 \hat { \mathbf { j } }$ , find a vector $\vec { \mathbf { B } }$ that is perpendicular to $\overline { \mathbf { A } }$ . is perpendicular to $\overline { \mathbf { A } }$ .

Ben Nicholson

Numerade Educator

Problem 22

(II) A constant force $\vec { \mathbf { F } } = ( 2.0 \hat { \mathbf { i } } + 4.0 \hat { \mathbf { j } } ) \mathrm { N }$ acts on an object as it moves along a straight-line path. If the object's displacement is $\mathbf { d } = ( 1.0 \mathbf { i } + 5.0 \mathbf { j } ) \mathrm { m } ,$ calculate the work done by $\vec { \mathbf { F } }$ using these alternate ways of writing the dot product: (a) $W = F d \cos \theta ;$ (b) $W = F _ { x } d _ { x } + F _ { y } d _ { y }$

Averell Hause

Carnegie Mellon University

Problem 23

(II) If $$\vec { \mathbf { A } } = 9.0 \hat { \mathbf { i } } - 8.5 \hat { \mathbf { j } } , \quad \vec { \mathbf { B } } = - 8.0 \hat { \mathbf { i } } + 7.1 \hat { \mathbf { j } } + 4.2 \hat { \mathbf { k } } , \quad$$ and $$\vec { \mathbf { C } } = 6.8 \hat { \mathbf { i } } - 9.2 \hat { \mathbf { j } } ,$$ determine $$( a ) \vec { \mathbf { A } } \cdot ( \vec { \mathbf { B } } + \vec { \mathbf { C } } ) ; ( b ) ( \vec { \mathbf { A } } + \vec { \mathbf { C } } ) \cdot \vec { \mathbf { B } }$$ $$( c ) ( \vec { \mathbf { B } } + \vec { \mathbf { A } } ) \cdot \vec { \mathbf { C } }$$

Ben Nicholson

Numerade Educator

Problem 24

(II) Prove that $\vec { \mathbf { A } } \cdot \vec { \mathbf { B } } = A _ { x } B _ { x } + A _ { y } B _ { y } + A _ { z } B _ { z }$ starting from Eq. 2 and using the distributive property $( \vec { \mathbf { A } } \cdot \vec { \mathbf { B } } = A B \cos \theta ,$ proved in Problem 33$)$

Averell Hause

Carnegie Mellon University

Problem 25

(II) Given vectors $\vec { \mathbf { A } } = - 4.8 \hat { \mathbf { i } } + 6.8 \hat { \mathbf { j } }$ and $\vec { \mathbf { B } } = 9.6 \hat { \mathbf { i } } + 6.7 \hat { \mathbf { j } }$ , determine the vector $\vec { \mathbf { C } }$ that lies in the $x y$ plane, is perpendicular to $\vec { \mathbf { B } } ,$ and whose dot product with $\vec { \mathbf { A } }$ is $20.0 .$

Ben Nicholson

Numerade Educator

Problem 26

(II) Show that if two nonparallel vectors have the same magnitude, their sum must be perpendicular to their difference.

Averell Hause

Carnegie Mellon University

Problem 27

(II) Let $\vec { \mathbf { v } } = 20.0 \hat { \mathbf { i } } + 22.0 \hat { \mathbf { j } } - 14.0 \hat { \mathbf { k } } .$ What angles does this vector make with the $x , y ,$ and $z$ axes?

Ben Nicholson

Numerade Educator

Problem 28

(II) Use the scalar product to prove the law of cosines for a triangle:
$$c ^ { 2 } = a ^ { 2 } + b ^ { 2 } - 2 a b \cos \theta$$
where $a , b ,$ and $c$ are the lengths of the sides of a triangle and $\theta$ is the angle opposite side $c .$

Averell Hause

Carnegie Mellon University

Problem 29

(II) Vectors $\vec { \mathbf { A } }$ and $\vec { \mathbf { B } }$ are in the $x y$ plane and their scalar product is 20.0 units. If $\vec { \mathbf { A } }$ makes a $27.4 ^ { \circ }$ angle with the $x$ axis and has magnitude $A = 12.0$ units, and $\vec { \mathbf { B } }$ has magnitude $B = 24.0$ units, what can you say about the direction of $\vec { \mathbf { B } } ?$

Ben Nicholson

Numerade Educator

Problem 30

(II) $\vec { \mathbf { A } }$ and $\vec { \mathbf { B } }$ are two vectors in the $x y$ plane that make angles $\alpha$ and $\beta$ with the $x$ axis respectively. Evaluate the scalar product of $\overline { \mathbf { A } }$ and $\vec { \mathbf { B } }$ and deduce the following trigonometric identity: $\cos ( \alpha - \beta ) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$

Averell Hause

Carnegie Mellon University

Problem 31

(1I) Suppose $\quad \vec { \mathbf { A } } = 1.0 \hat { \mathbf { i } } + 1.0 \hat { \mathbf { j } } - 2.0 \hat { \mathbf { k } } \quad$ and $\quad \vec { \mathbf { B } } =$ $- 1.0 \hat { \mathbf { i } } + 1.0 \hat { \mathbf { j } } + 2.0 \hat { \mathbf { k } } , ( a )$ what is the angle between these two vectors? $( b )$ Explain the significance of the sign in part $( a )$

Ben Nicholson

Numerade Educator

Problem 32

(II) Find a vector of unit length in the $x y$ plane that is perpendicular to $3.0 \hat { \mathrm { i } } + 4.0 \hat { \mathrm { j } }$

Averell Hause

Carnegie Mellon University

Problem 33

(III) Show that the scalar product of two vectors is distributive: $\hat { \mathbf { A } } \cdot ( \vec { \mathbf { B } } + \vec { \mathbf { C } } ) = \vec { \mathbf { A } } \cdot \vec { \mathbf { B } } + \vec { \mathbf { A } } \cdot [$Hint. Use a diagram showing all three vectors in a plane and indicate dot products on the diagram. $]$

Ben Nicholson

Numerade Educator

Problem 34

(I) In pedaling a bicycle uphill, a cyclist exerts a downward force of 450$\mathrm { N }$ during each stroke. If the diameter of the circle traced by each pedal is $36 \mathrm { cm } ,$ calculate how much work is done in each stroke.

Averell Hause

Carnegie Mellon University

Problem 35

(1I) A spring has $k = 65 \mathrm { N } / \mathrm { m }$ . Draw a graph like that in Fig. 11 and use it to determine the work needed to stretch the spring from $x = 3.0 \mathrm { cm }$ to $x = 6.5 \mathrm { cm } ,$ where $x = 0$ refers to the spring's unstretched length.

Ben Nicholson

Numerade Educator

Problem 36

(II) If the hill in Example 2 of "Work and Energy" (Fig, 4) was not an even slope but rather an irregular curve as in Fig. $23 ,$ show that the same result would be obtained as in Example $2 :$ namely, that the work done by gravity depends only on the height of the hill and not on its shape or the path taken.

Averell Hause

Carnegie Mellon University

Problem 37

(II) The net force exerted on a particle acts in the positive $x$ direction. Its magnitude increases linearly from zero at $x = 0 ,$ to 380$\mathrm { N }$ at $x = 3.0 \mathrm { m }$ . It remains constant at 380$\mathrm { N }$ from $x = 3.0 \mathrm { m }$ to $x = 7.0 \mathrm { m } ,$ and then decreases linearly to zero at $x = 12.0 \mathrm { m } .$ Determine the work done to move the particle from $x = 0$ to $x = 12.0 \mathrm { m }$ graphically, by determining the area under the $F _ { x }$ versus $x$ graph.

Ben Nicholson

Numerade Educator

Problem 38

(II) If it requires 5.0$\mathrm { J }$ of work to stretch a particular spring by 2.0$\mathrm { cm }$ from its equilibrium length, how much more work will be required to stretch it an additional 4.0$\mathrm { cm } ?$

Averell Hause

Carnegie Mellon University

Problem 39

(II) In Fig. 9 assume the distance axis is the $x$ axis and that $a = 10.0 \mathrm { m }$ and $\mathrm { b } = 30.0 \mathrm { m }$ . Estimate the work done by this force in moving a 3.50 -kg object from a to b.

Ben Nicholson

Numerade Educator

Problem 40

(II) The force on a particle, acting along the $x$ axis, varies as shown in Fig. $24 .$ Determine the work done by this force to move the particle along the $x$ axis: $( a )$ from $x = 0.0$ to $x = 10.0 \mathrm { m }$ ; (b) from $x = 0.0$ to $x = 15.0 \mathrm { m } .$

Averell Hause

Carnegie Mellon University

Problem 41

(II) A child is pulling a wagon down the sidewalk. For 9.0$\mathrm { m }$ the wagon stays on the sidewalk and the child pulls with a horizontal force of 22$\mathrm { N }$ . Then one wheel of the wagon goes off on the grass so the child has to pull with a force of 38$\mathrm { N }$ at an angle of $12 ^ { \circ }$ to the side for the next 5.0$\mathrm { m } .$ Finally the wagon gets back on the sidewalk so the child makes the rest of the trip, $13.0 \mathrm { m } ,$ with a force of 22$\mathrm { N }$ . How much total work did the child do on the wagon?

Ben Nicholson

Numerade Educator

Problem 42

(II) The resistance of a packing material to a sharp object penetrating it is a force proportional to the fourth power of the penetration depth $x ;$ that is, $\vec { \mathbf { F } } = - k x ^ { 4 } \hat { \mathbf { i } }$ . Calculate the work done to force a sharp object a distance $d$ into the material.

Averell Hause

Carnegie Mellon University

Problem 43

(II) The force needed to hold a particular spring compressed an amount $x$ from its normal length is given by $F = k x + a x ^ { 3 } + b x ^ { 4 } .$ How much work must be done to compress it by an amount $X$ , starting from $x = 0 ?$

Ben Nicholson

Numerade Educator

Problem 44

(II) At the top of a pole vault, an athlete actually can do work pushing on the pole before releasing it. Suppose the pushing force that the pole exerts back on the athlete is given by $F ( x ) = \left( 1.5 \times 10 ^ { 2 } \mathrm { N } / \mathrm { m } \right) x - \left( 1.9 \times 10 ^ { 2 } \mathrm { N } / \mathrm { m } ^ { 2 } \right) x ^ { 2 }$ acting over a distance of 0.20$\mathrm { m } .$ How much work is done on the athlete?

Averell Hause

Carnegie Mellon University

Problem 45

(II) Consider a force $F _ { 1 } = A / \sqrt { x }$ which acts on an object during its journey along the $x$ axis from $x = 0.0$ to $x = 1.0 \mathrm { m } ,$ where $A = 2.0 \mathrm { N } \cdot \mathrm { m } ^ { 1 / 2 }$ . Show that during this journey, even though $F _ { 1 }$ is infinite at $x = 0.0$ , the work done on the object by this force is finite.

Ben Nicholson

Numerade Educator

Problem 46

(II) Assume that a force acting on an object is given by $\vec { \mathbf { F } } = a x \hat { \mathbf { i } } + b y \hat { \mathbf { j } }$ . where the constants $a = 3.0 \mathrm { N } \cdot \mathrm { m } ^ { - 1 }$ and $b = 4.0 \mathrm { N } \cdot \mathrm { m } ^ { - 1 } .$ Determine the work done on the object by this force as it moves in a straight line from the origin to $\vec { \mathbf { r } } = ( 10.0 \hat { \mathbf { i } } + 20.0 \hat { \mathbf { j } } ) \mathrm { m }$

Averell Hause

Carnegie Mellon University

Problem 47

(II) An object, moving along the circumference of a circle with radius $R ,$ is acted upon by a force of constant magnitude $F$ . The force is directed at all times at a $30 ^ { \circ }$ angle with respect to the tangent to the circle as shown in Fig. 25 Determine the work done by this force along the object moves along the half circle from A to B.

Ben Nicholson

Numerade Educator

Problem 48

(1II) $\mathrm { A } 2800 - \mathrm { kg }$ space vehicle, initially at rest, falls vertically from a height of 3300$\mathrm { km }$ above the Earth's surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earth's surface.

Averell Hause

Carnegie Mellon University

Problem 49

(III) A 3.0 -m-long steel chain is stretched out along the top level of a horizontal scaffold at a construction site, in such a way that 2.0$\mathrm { m }$ of the chain remains on the top level and 1.0$\mathrm { m }$ hangs vertically, Fig. $26 .$ At this point, the force on the hanging segment is sufficient to pull the entire chain over the edge. Once the chain is moving, the kinetic friction is so small that it can be neglected. How much work is performed on the chain by the force of gravity as the chain falls from the point where 2.0$\mathrm { m }$ remains on the scaffold to the point where the entire chain has left the scaffold? (Assume that the chain has a linear weight density of 18$\mathrm { N } / \mathrm { m } . )$

Ben Nicholson

Numerade Educator

Problem 50

(I) At room temperature, an oxygen molecule, with mass of $5.31 \times 10 ^ { - 26 } \mathrm { kg }$ , typically has a kinetic energy of about $6.21 \times 10 ^ { - 21 } \mathrm { J } .$ How fast is it moving?

Averell Hause

Carnegie Mellon University

Problem 51

(I) $( a )$ If the kinetic energy of a particle is tripled, by what factor has its speed increased? (b) If the speed of a particle is halved, by what factor does its kinetic energy change?

Ben Nicholson

Numerade Educator

Problem 52

(1) How much work is required to stop an electron $\left( m = 9.11 \times 10 ^ { - 31 } \mathrm { kg } \right)$ which is moving with a speed of $1.40 \times 10 ^ { 6 } \mathrm { m } / \mathrm { s } ?$

Averell Hause

Carnegie Mellon University

Problem 53

(I) How much work must be done to stop a 1300 -kg car traveling at 95$\mathrm { km } / \mathrm { h }$ ?

Ben Nicholson

Numerade Educator

Problem 54

(II) Spiderman uses his spider webs to save a runaway train, Fig. $27 .$ His web stretches a few city blocks before the $10 ^ { 4 } - \mathrm { kg }$ train comes to a stop. Assuming the web acts like a spring estimate the spring constant.

Averell Hause

Carnegie Mellon University

Problem 55

(II) A baseball $( m = 145 \mathrm { g } )$ traveling 32$\mathrm { m } / \mathrm { s }$ moves a fielder's glove backward 25$\mathrm { cm }$ when the ball is caught. What was the average force exerted by the ball on the glove?

Ben Nicholson

Numerade Educator

Problem 56

(II) An $85 - \mathrm { g }$ arrow is fired from a bow whose string exerts an average force of 105$\mathrm { N }$ on the arrow over a distance of 75$\mathrm { cm } .$ What is the speed of the arrow as it leaves the bow?

Averell Hause

Carnegie Mellon University

Problem 57

(I) A mass $m$ is attached to a spring which is held stretched a distance $x$ by a force $F ($ Fig. 28$) ,$ and then released. The spring compresses, pulling the mass. Assuming there is no friction, determine the speed of the mass $m$ when the spring returns: $( a )$ to its normal length $( x = 0 )$ (b) to half its original extension $( x / 2 )$

Ben Nicholson

Numerade Educator

Problem 58

(II) If the speed of a car is increased by $50 \% ,$ by what factor will its minimum braking distance be increased, assuming all else is the same? Ignore the driver's reaction time.

Averell Hause

Carnegie Mellon University

Problem 59

(II) A $1200 - \mathrm { kg }$ car rolling on a horizontal surface has speed $v = 66 \mathrm { km } / \mathrm { h }$ when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2$\mathrm { m }$ . What is the spring constant of the spring?

Ben Nicholson

Numerade Educator

Problem 60

(I) One car has twice the mass of a second car, but only half as much kinetic energy. When both cars increase their speed by $7.0 \mathrm { m } / \mathrm { s } ,$ they then have the same kinetic energy. What were the original speeds of the two cars?

Averell Hause

Carnegie Mellon University

Problem 61

(II) A 4.5 -kg object moving in two dimensions initially has a on the object for 2.0$\mathrm { s }$ , after which the object's velocity is $\vec { v } _ { 2 } = ( 15.0 \hat { \mathrm { i } } + 30.0 \hat { \mathrm { j } } ) \mathrm { m } / \mathrm { s } .$ Determine the work done by $\vec { \mathrm { F } }$ on the object.

Ben Nicholson

Numerade Educator

Problem 62

(I) A 265 -kg load is lifted 23.0 m vertically with an acceler- ation $a = 0.150 g$ by a single cable. Determine $( a )$ the tension in the cable; $( b )$ the net work done on the load; assuming it started from rest.

Averell Hause

Carnegie Mellon University

Problem 63

(II) $( a )$ How much work is done by the horizontal force $F _ { \mathrm { P } } = 150 \mathrm { N }$ on the 18$\mathrm { kg }$ block of Fig. 29 when the force pushes the block 5.0$\mathrm { m }$ up along the $32 ^ { \circ }$ frictionless incline? (b) How much work is done by the gravitational force on the block during this displacement? (c) How much work is done by the normal force? (d) What is the speed of the block (assume that it is zero initially) after this displacement? [Hint. Work-energy involves network done.

Ben Nicholson

Numerade Educator

Problem 64

(II) Repeat Problem 63 assuming a coefficient of friction $\mu_{\mathrm{k}}=0.10$

Averell Hause

Carnegie Mellon University

Problem 65

(II) At an accident scene on a level road, investigators measure a car's skid mark to be 98$\mathrm { m }$ long. It was a rainy day and the coefficient of friction was estimated to be 0.38 Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes. (Why does the car's mass not matter?

Ben Nicholson

Numerade Educator

Problem 66

(II) $\mathrm { A } 46.0$ -kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 225$\mathrm { N }$ . For the first 11.0$\mathrm { m }$ the floor is frictionless, and for the next 10.0$\mathrm { m }$ the coefficient of friction is $0.20 .$ What is the final speed of the crate after being pulled these 21.0$\mathrm { m }$ ?

Averell Hause

Carnegie Mellon University

Problem 67

(II) A train is moving along a track with constant speed $v _ { 1 }$ relative to the ground. A person on the train holds a ball of mass $m$ and throws it toward the front of the train with a speed $v _ { 2 }$ relative to the train. Calculate the change in kinetic energy of the ball $( a )$ in the Earth frame of reference, and
(b) in the train frame of reference. (c) Relative to each frame of reference, how much work was done on the ball? (d) Explain why the results in part (c) are not the same for the two frames - after all, it's the same ball.

Ben Nicholson

Numerade Educator

Problem 68

(III) We usually neglect the mass of a spring if it is small compared to the mass attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length $\ell$ and mass $M _ { \text { s uniformly } }$ distributed along the length of the spring. A mass $m$ is attached to the end of the spring. One end of the spring is fixed and the mass $m$ is allowed to vibrate horizontally without friction (Fig. $30 ) .$ Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed $v _ { 0 } ,$ the midpoint of the spring moves with speed $v _ { 0 } / 2 .$ Show that the kinetic energy of the mass plus spring when the mass is moving with velocity $v$ is $K = \frac { 1 } { 2 } M v ^ { 2 }$ where $M = m + \frac { 1 } { 3 } M _ { \mathrm { S } }$ is the "effective mass" of the system. [Hint: Let $D$ be the total length of the stretched spring. Then the velocity of a mass $d m$ of a spring of length $d x$ located at $x$ is $v ( x ) = v _ { 0 } ( x / D ) .$ Note also that $d m = d x \left( M _ { \mathrm { S } } / D \right) . ]$

Averell Hause

Carnegie Mellon University

Problem 69

(III) An elevator cable breaks when a 925 -kg elevator is 22.5$\mathrm { m }$ above the top of a huge spring $( k =$ $8.00 \times 10 ^ { 4 } \mathrm { N } / \mathrm { m } )$ at the bottom of the shaft. Calculate (a) the work done by gravity on the elevator before it hits the spring; $( b )$ the speed of the elevator just betore striking the spring; $( c )$ the amount the spring compresses (note that here work is done by both the spring and gravity.

Ben Nicholson

Numerade Educator

Problem 70

(a) A 3.0 -g locust reaches a speed of 3.0$\mathrm { m } / \mathrm { s }$ during its jump. What is its kinetic energy at this speed? (b) If the locust transforms energy with 35$\%$ efficiency, how much energy is required for the jump?

Averell Hause

Carnegie Mellon University

Problem 71

In a certain library the first shelf is 12.0$\mathrm { cm }$ off the ground, and the remaining 4 shelves are each spaced 33.0$\mathrm { cm }$ above the previous one. If the average book has a mass of 1.40$\mathrm { kg }$ with a height of $22.0 \mathrm { cm } ,$ and an average shelf holds 28 with a height of $22.0 \mathrm { cm } ,$ and an average shelf holds 28 books (standing vertically), how much work is required to fill all the shelves, assuming the books are all laying flat on the floor to start?

Ben Nicholson

Numerade Educator

Problem 72

A 75 -kg meteorite buries itself 5.0$\mathrm { m }$ into soft mud. The force between the meteorite and the mud is given by $F ( x ) = \left( 640 \mathrm { N } / \mathrm { m } ^ { 3 } \right) x ^ { 3 }$ , where $x$ is the depth in the mud. What was the speed of the meteorite when it initially impacted the mud?

Averell Hause

Carnegie Mellon University

Problem 73

A $6.10 - \mathrm { kg }$ block is pushed 9.25$\mathrm { m }$ up a smooth $37.0 ^ { \circ }$ inclined plane by a horizontal force of 75.0$\mathrm { N }$ . If the initial speed of the block is 3.25$\mathrm { m } / \mathrm { s }$ up the plane, calculate $( a )$ the initial kinetic energy of the block; $( b )$ the work done by the 75.0 -N force; $( c )$ the work done by gravity; $( d )$ the work done by the normal force; $( e )$ the final kinetic energy of the block.

Ben Nicholson

Numerade Educator

Problem 74

The arrangement of atoms in zinc is an example of "hexagonal close-packed" structure. Three of the nearestneighbors are found at the following $( x , y , z )$ coordinates, given in nanometers $\left( 10 ^ { - 9 } \mathrm { m } \right) :$ atom 1 is at $( 0,0,0 ) ;$ atom 2 is at $( 0.230,0.133,0 ) ;$ atom 3 is at $( 0.077,0.133,0.247 ) .$ Find the angle between two vectors: one that connects atom 1 with atom 2 and another that connects atom 1 with atom $3 .$

Averell Hause

Carnegie Mellon University

Problem 75

Two forces, $$\quad \vec { \mathbf { F } } _ { 1 } = ( 1.50 \hat { \mathbf { i } } - 0.80 \hat { \mathbf { j } } + 0.70 \hat { \mathbf { k } } ) \mathrm { N } \quad$$ and $$\quad \vec { \mathbf { F } } _ { 2 } =$$ $( - 0.70 \mathrm { i } + 1.20 \mathrm { j } ) \mathrm { N } ,$ are applied on a moving object of mass
0.20$\mathrm { kg } .$ The displacement vector produced by the two forces is $\vec { \mathrm { d } } = ( 8.0 \hat { \mathrm { i } } + 6.0 \hat { \mathrm { j } } + 5.0 \hat { \mathrm { k } } ) \mathrm { m } .$ What is the work

Ben Nicholson

Numerade Educator

Problem 76

The barrels of the 16 in. guns (bore diameter $= 16$ in. $=$ 41$\mathrm { cm }$ ) on the World War II battleship U.S.S. Massachusetis were each 15$\mathrm { m }$ long. The shells each had a mass of 1250$\mathrm { kg }$ and were fired with sufficient explosive force to provide them with a muzzle velocity of 750$\mathrm { m } / \mathrm { s } .$ Use the work-energy principle to determine the explosive force (assumed to be a constant) that was applied to the shell within the barrel of the gun. Express your answer in both newtons and in pounds.

Averell Hause

Carnegie Mellon University

Problem 77

A varying force is given by $F = A e ^ { - k x } ,$ where $x$ is the position; $A$ and $k$ are constants that have units of $\mathrm { N }$ and $\mathrm { m } ^ { - 1 }$ , respectively. What is the work done when $x$ goes from 0.10$\mathrm { m }$ to infinity?

Ben Nicholson

Numerade Educator

Problem 78

The force required to compress an imperfect horizontal spring an amount $x$ is given by $F = 150 x + 12 x ^ { 3 } ,$ where $x$ is in meters and $F$ in newtons. If the spring is compressed $2.0 \mathrm { m } ,$ what speed will it give to a 3.0 -kg ball held against it and then released?

Averell Hause

Carnegie Mellon University

Problem 79

A force $\mathbf { F } = ( 10.0 \mathbf { i } + 9.0 \mathbf { j } + 12.0 \mathbf { k } ) \mathrm { kN }$ acts on a small object of mass 95$\mathrm { g } .$ If the displacement of the object is $\mathbf { d } = ( 5.0 \mathbf { i } + 4.0 \mathbf { j } ) \mathrm { m } ,$ find the work done by the force. What is the angle between $\vec { \mathbf { F } }$ and $\mathbf { d } ?$

Ben Nicholson

Numerade Educator

Problem 80

In the game of paintball, players use guns powered by pressurized gas to propel $33 -$ g gel capsules filled with paint at the opposing team. Game rules dictate that a paintball cannot leave the barrel of a gun with a speed greater than 85$\mathrm { m } / \mathrm { s } .$ Model the shot by assuming the pressurized gas applies a constant force $F$ to a $33 - g$ capsule over the length of the $32 - \mathrm { cm }$ barrel. Determine $F ( a )$ using the work-energy principle, and $( b )$ using the kinematic equations and Newton's second law.

Averell Hause

Carnegie Mellon University

Problem 81

A softball having a mass of 0.25$\mathrm { kg }$ is pitched horizontally at 110$\mathrm { km } / \mathrm { h } .$ By the time it reaches the plate, it may have slowed by 10$\% .$ Neglecting gravity, estimate the average force of air resistance during a pitch, if the distance between the plate and the pitcher is about 15$\mathrm { m } .$

Ben Nicholson

Numerade Educator

Problem 82

An airplane pilot fell 370$\mathrm { m }$ after jumping from an aircraft without his parachute opening. He landed in a snowbank, creating a crater 1.1$\mathrm { m }$ deep, but survived with only minor injuries. Assuming the pilot's mass was 88$\mathrm { kg }$ and his terminal velocity was 45$\mathrm { m } / \mathrm { s }$ , estimate: $( a )$ the work done by the snow in bringing him to rest; (b) the average force exerted on him by the snow to stop him; and (c) the work done on him by air resistance as he fell. Model him as a particle.

Averell Hause

Carnegie Mellon University

Problem 83

Many cars have $" 5 \mathrm { mi } / \mathrm { h } ( 8 \mathrm { km } / \mathrm { h } )$ bumpers" that are designed to compress and rebound elastically without any physical damage at speeds below 8$\mathrm { km } / \mathrm { h }$ . If the material of the bumpers permanently deforms after a compression of $1.5 \mathrm { cm } ,$ but remains like an elastic spring up to that point, what must be the effective spring constant of the bumper material, assuming the car has a mass of 1050$\mathrm { kg }$ and is tested by ramming into a solid wall?

Ben Nicholson

Numerade Educator

Problem 84

What should be the spring constant $k$ of a spring designed to bring a 1300 -kg car to rest from a speed of 90$\mathrm { km } / \mathrm { h }$ so that the occupants undergo a maximum acceleration of 5.0$\mathrm { g }$ ?

Averell Hause

Carnegie Mellon University

Problem 85

Assume a cyclist of weight $m g$ can exert a force on the pedals equal to 0.90$m g$ on the average. If the pedals rotate in a circle of radius 18$\mathrm { cm }$ , the wheels have a radius of $34 \mathrm { cm } ,$ and the front and back sprockets on which the chain runs have 42 and 19 teeth respectively (Fig. 31$)$ , determine the maximum steepness of hill the cyclist can climb at constant speed. Assume the mass of the bike is 12$\mathrm { kg }$ and that of the rider is 65$\mathrm { kg }$ . Ignore friction. Assume the cyclist's average force is always: (a) downward; ( $b$ ) tangential to pedal motion.

Ben Nicholson

Numerade Educator

Problem 86

A simple pendulum consists of a small object of mass $m$ (the "bob") suspended by a cord of length $\ell$ (Fig. 32) of negligible mass. A force $\vec { \mathbf { F } }$ is applied in the horizontal direction (so $\mathbf { F } = F \hat { \mathbf { i } } ) ,$ moving the bob very slowly so the acceleration is essentially zero. (Note that the magnitude of $\mathbf { F }$ will need to vary with the angle $\theta$ that the cord makes with the vertical at any moment.) $( a )$ Determine the work done by this force, $\vec { \mathbf { F } } ,$ to move the pendulum from $\quad \theta = 0$ to $\theta = \theta _ { 0 }$ . $( b )$ Determine the work done by the gravitational force on the bob, $\vec { \mathbf { F } } _ { \mathrm { G } } = m \vec { \mathbf { g } } , \quad$ and $\mathrm { the }$ work done by the force $\vec { \mathbf { F } } _ { \mathrm { T } }$ that the
cord exerts on the bob.

Averell Hause

Carnegie Mellon University

Problem 87

A car passenger buckles himself in with a seat belt and holds his 18 -kg toddler on his lap. Use the work-energy principle to answer the following questions. (a) While traveling 25$\mathrm { m } / \mathrm { s }$ , the driver has to make an emergency stop over a distance of 45$\mathrm { m } .$ Assuming constant deceleration, how much force will the arms of the parent need to exert on the child during this deceleration period? Is this force achievable by an average parent? (b) Now assume that the car $( v = 25 \mathrm { m } / \mathrm { s } )$ is in an accident and is brought to stop over a distance of 12$\mathrm { m } .$ Assuming constant deceleration, how much force will the parent need to exert on the child? Is this force achievable by an average parent?

Ben Nicholson

Numerade Educator

Problem 88

As an object moves along the $x$ axis from $x = 0.0 \mathrm { m }$ to $x = 20.0 \mathrm { m }$ it is acted upon by a force given by $F = \left( 100 - ( x - 10 ) ^ { 2 } \right) \mathrm { N }$ . Determine the work done by the force on the object: $( a )$ by first sketching the $F$ vs. $x$ graph and estimating the area under this curve; $( b )$ by evaluating the integral $\int _ { x = 0.0 \mathrm { m } } ^ { x = 20 \mathrm { m } } F d x$ .

Averell Hause

Carnegie Mellon University

Problem 89

A cyclist starts from rest and coasts down a 4.0^ \circ hill. The mass of the cyclist plus bicycle is 85$\mathrm { kg }$ . After the cyclist has traveled $250 \mathrm { m } ,$ (a) what was the net work done by gravity on the cyclist? (b) How fast is the cyclist going? Ignore air resistance.

Ben Nicholson

Numerade Educator

Problem 90

Stretchable ropes are used to safely arrest the fall of rock climbers. Suppose one end of a rope with unstretched length $\ell$ is anchored to a cliff and a climber of mass $m$ is attached to the other end. When the climber is a height $\ell$ above the anchor point, he slips and falls under the influence of
gravity for a distance 2$\ell$ , after which the rope becomes taut and stretches a distance $x$ as it stops the climber (see Fig. $33 ) .$ Assume a stretchy rope behaves as a spring with spring constant $k . ( a )$ Applying the work-energy principle, show that
$x = \frac { m g } { k } \left[ 1 + \sqrt { 1 + \frac { 4 k \ell } { m g } } \right]$
(b) Assuming $m = 85 \mathrm { kg } , \quad \ell = 8.0 \mathrm { m }$ and $k = 850 \mathrm { N } / \mathrm { m } ,$ determine $x / \ell$ (the fractional stretch of the rope) and $k x / m g$ (the force that the rope exerts on the climber compared to his own weight) at climber's fall has been stopped.

Averell Hause

Carnegie Mellon University

Problem 91

A small mass $m$ hangs at rest from a vertical rope of length $\ell$ that is fixed to the ceiling. A force $\vec { \mathbf { F } }$ then pushes on the mass, perpendicular to the taut rope at all times, until the rope is oriented at an angle $\theta = \theta _ { 0 }$ and the mass has been raised by a vertical distance $h$ (Fig. $34 ) .$ Assume the force's magnitude $F$ is adjusted so that the mass moves at constant speed along its curved trajectory. Show that the work done by $\vec { \mathbf { F } }$ during this process equals $m g h ,$ which is equivalent to the amount of work it takes to slowly lift a mass $m$ straight up by a height $h .$ [Hint: When the angle is increased by $d \theta$ (in radians), the mass moves along an arc length $d s = \ell d \theta . ]$

Ben Nicholson

Numerade Educator

Problem 92

(II) The net force along the linear path of a particle of mass 480 g has been measured at 10.0 -cm intervals, starting at $x = 0.0 ,$ to be $26.0,28.5,28.8,29.6,32.8,40.1,46.6,42.2 ,$ $48.8,52.6,55.8,60.2,60.6,58.2,53.7,50.3,45.6,45.2,43.2$ $38.9,35.1,30.8,27.2,21.0,22.2 ,$ and $18.6 ,$ all in newtons. Determine the total work done on the particle over this entire range.

Averell Hause

Carnegie Mellon University

Problem 93

(II) When different masses are suspended from a spring, the spring stretches by different amounts as shown in the Table below. Masses are $\pm 1.0$ gram.
$\begin{array} { l } { \text { Mass (g) } 0 } & { 50 \quad 100 \quad 150 \quad 200 \quad 250 \quad 300 \quad 350 \quad 400 } \\ { \text { Stretch } ( \mathrm { cm } ) 0 \quad 5.0 \quad 9.8 \quad 14.8 \quad 19.4 \quad 24.5 \quad 29.6 \quad 34.1 \quad 39.2 } \\ \hline \end{array}$ (a) Graph the applied force (in Newtons) versus the stretch (in meters) of the spring, and determine the best-fit straight line. (b) Determine the spring constant $( \mathrm { N } / \mathrm { m } )$ of the spring from the slope of the best-fit line. (c) If the spring is stretched by $20.0 \mathrm { cm } ,$ estimate the force acting on the spring using the best-fit line.

Ben Nicholson

Numerade Educator