Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 8

Extended Systems - all with Video Answers

Educators

Chapter Questions

Problem 1

A $4.0 \mathrm{~kg}$ particle-like object is located at $x=0, y=2.0 \mathrm{~m} ;$ a $3.0 \mathrm{~kg}$ particle-like object is located at $x=$ $3.0 \mathrm{~m}, y=1.0 \mathrm{~m}$. At what (a) $x$ and (b) $y$ coordinates must a $2.0 \mathrm{~kg}$ particle-like object be placed for the center of mass of the threeparticle system to be located at the origin?

Ivan Kochetkov

Ivan Kochetkov

Numerade Educator

Problem 2

Consider Fig. 8-14. Three masses located in the $x-y$ plane have the following coordinates; a $5 \mathrm{~kg}$ mass has coordinates given by $(2,-3) \mathrm{m} ;$ a $4 \mathrm{~kg}$ mass has coordinates $(-4,2) \mathrm{m} ; \mathrm{a}$ $2 \mathrm{~kg}$ mass has coordinates $(3,3) \mathrm{m}$. Find the coordinates of the center of mass to two significant figures.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 3

(a) How far is the center of mass of the Earth-Moon system from the center of Earth? (Appendix $\mathrm{C}$ gives the masses of Earth and the Moon and the distance between the two.) (b) Express the answer to (a) as a fraction of Earth's radius $R_{e}$.

Manish Jain

Numerade Educator

Problem 4

A distance of $1.131 \times 10^{-10} \mathrm{~m}$ lies between the centers of the carbon and oxygen atoms in a carbon monoxide (CO) gas molecule. Locate the center of mass of a CO molecule relative to the carbon atom. (Find the masses of $\mathrm{C}$ and $\mathrm{O}$ in Appendix $\mathrm{F}$.)

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 5

What are (a) the $x$ coordinate and (b) the $y$ coordinate of the center of mass of the three-particle system shown in Fig. 8$15 ?$ (c) What happens to the center of mass as the mass of the topmost particle is gradually increased?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 6

Three thin rods, each of length $L$, are arranged in an inverted $U$, as shown in Fig. 8- 16. The two rods on the arms of the $U$ each have mass $M$; the third rod has mass $3 M$. Where is the center of mass of the assembly?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 7

A uniform square plate $6 \mathrm{~m}$ on a side has had a square piece $2 \mathrm{~m}$ on a side cut out of it (Fig. 8-17). The center of that piece is at $x=2 \mathrm{~m}, y=0 .$ The center of the square plate is at $x=y=$ 0. Find (a) the $x$ coordinate and (b) the $y$ coordinate of the center of mass of the remaining piece.

Penny Riley

Numerade Educator

Problem 8

Figure $8-18$ shows the dimensions of a composite slab; half the slab is made of aluminum (density $=2.70 \mathrm{~g} / \mathrm{cm}^{3}$ ) and half is made of iron (density $=7.85 \mathrm{~g} / \mathrm{cm}^{3}$ ). Where is the center of mass of the slab?

Manish Jain

Numerade Educator

Problem 9

In the ammonia $\left(\mathrm{NH}_{3}\right)$ molecule (see Fig. 8-19), the three hydrogen (H) atoms form an equilateral triangle; the center of the triangle is $9.40 \times$ $10^{-11} \mathrm{~m}$ from each hydrogen atom. The nitrogen (N) atom is at the apex of a pyramid, with the three hydrogen atoms forming the base. The nitrogen-to-hydrogen atomic mass ratio is $13.9$. and the nitrogen-to-hydrogen distance is $10.14 \times 10^{-11} \mathrm{~m}$. Locate the center of mass of the molecule relative to the nitrogen atom.

Manish Jain

Numerade Educator

Problem 10

Figure $8-20$ shows a cubical box that has been constructed from a metal plate of uniform density and negligible thickness. The box is open at the top and has edge length 40 $\mathrm{cm}$. Find (a) the $x$ coordinate, $(\mathrm{b})$ the $y$ coordinate, and (c) the $z$ coordinate of the center of mass of the box.

Manish Jain

Numerade Educator

Problem 11

A right cylindrical can with mass $M$, height $H$, and uniform density is initially filled with soda of mass $m$ (Fig. 8-21). We punch small holes in the top and bottom to drain the soda; we then consider the height $h$ of the center of mass of the can and any soda within it. What is $h$ (a) initially and (b) when all the soda has drained? (c) What happens to $h$ during the draining of the soda? (d) If $x$ is the height of the remaining soda at any given instant, find $x$ (in terms of $M, H$, and $m$ ) when the center of mass reaches its lowest point.

Manish Jain

Numerade Educator

Problem 12

In Fig. $8-22 a$, a uniform wire forms an isosceles triangle of base $B$ and height $H .$ (a) Find the $x$ and $y$ coordinates of the figure's center of mass by assuming that each side can be replaced with a particle of the same mass as that side and positioned at the center of the side. (Be careful: Note that the base and, say, the left-hand side do not have the same mass.) (b) Use Eq. 8-12 to find the $x$ and $y$ coordinates of the center of mass of the left-hand side.

Manish Jain

Numerade Educator

Problem 13

Figure $8-22 b$ shows a uniform, solid plate in the shape of an isosceles triangle with base $B$ and height $H$. What are the $x$ and $y$ coordinates of the plate's center of mass?

Manish Jain

Numerade Educator

Problem 14

In Fig. $8-22 c$, a uniform wire forms a semicircle of radius $R$. What are the $x$ and $y$ coordinates of the figure's center of mass?

Manish Jain

Numerade Educator

Problem 15

Figure $8-22 d$ shows a uniform, solid plate in the shape of a semicircle with radius $R$. What are the $x$ and $y$ coordinates of the plate's center of mass?

Manish Jain

Numerade Educator

Problem 16

The Great Pyramid of Cheops at El Gizeh, Egypt (Fig. $8-23 a$ ), had height $H=147 \mathrm{~m}$ before its topmost stone fell. Its base is a square with edge length $L=230 \mathrm{~m}$ (see Fig. $8-23 b$ ). Its volume $V$ is equal to $L^{2} H / 3$. Assuming $\rho=1.8 \times$ $10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ is its uniform density, find the original height of its center of mass above the base.

Manish Jain

Numerade Educator

Problem 17

At a certain instant, four particles have the $x y$ coordinates and velocities given in the following table. At that instant, what are (a) the coordinates of their center of mass and (b) the velocity of their center of mass?
$$\begin{array}{cccc}
\text { Particle } & \text { Mass (kg) } & \text { Position (m) } & \text { Velocity (m/s) } \\
\hline 1 & 2.0 & 0,3.0 & -9.0 \mathrm{~m} / \mathrm{s} \hat{\mathrm{j}} \\
2 & 4.0 & 3.0,0 & 6.0 \mathrm{~m} / \mathrm{s} \hat{\mathrm{i}} \\
3 & 3.0 & 0,-2.0 & 6.0 \mathrm{~m} / \mathrm{s} \hat{\mathrm{j}} \\
4 & 12 & -1.0,0 & -2.0 \mathrm{~m} / \mathrm{s} \hat{\mathrm{i}} \\
\hline
\end{array}$$

Manish Jain

Numerade Educator

Problem 18

Show that the ratio of the distances of two particles from their center of mass is the inverse ratio of their masses.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 19

A $2.00 \mathrm{~kg}$ particle has the $x y$ coordinates $(-1.20 \mathrm{~m}, 0.500 \mathrm{~m})$ and a $4.00 \mathrm{~kg}$ particle has the $x y$ coordinates $(0.600 \mathrm{~m},-0.750 \mathrm{~m})$ Both lie on a horizontal plane. At what $x y$ coordinates must you place a $3.00 \mathrm{~kg}$ particle such that the center of mass of the three-particle system has the coordinates $\quad(-0.500$ $\mathrm{m},-0.700 \mathrm{~m}) ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 20

What are (a) the $x$ coordinate and (b) the $y$ coordinate of the center of mass for the uniform plate shown in Fig. $8-24$ ?

Manish Jain

Numerade Educator

Problem 21

At $t_{1}=0$, a $1.0 \mathrm{~kg}$ jelly jar is projected vertically upward from the base of a 50 -m-tall building with an initial velocity of $40 \mathrm{~m} / \mathrm{s}$. At the same instant and directly overhead, a $2.0 \mathrm{~kg}$ peanut butter jar is dropped from rest from the top of the building. How far above ground level is the center of mass of the two-jar system at $t_{2}=3.0 \mathrm{~s}$ ?

Ivan Kochetkov

Numerade Educator

Problem 22

Two skaters, one with mass $65 \mathrm{~kg}$ and the other with mass $40 \mathrm{~kg}$, stand on an ice rink holding a pole of length $10 \mathrm{~m}$ and negligible mass. Starting from the ends of the pole, the skaters pull themselves along the pole until they meet. How far does the $40 \mathrm{~kg}$ skater move?

Zachary Warner

Numerade Educator

Problem 23

An old Chrysler with mass $2400 \mathrm{~kg}$ is moving along a straight stretch of road at $80 \mathrm{~km} / \mathrm{h}$. It is followed by a Ford with mass $1600 \mathrm{~kg}$ moving at 60 $\mathrm{km} / \mathrm{h} .$ How fast is the center of mass of the two cars moving?

Zachary Warner

Numerade Educator

Problem 24

A man of mass $m$ clings to a rope ladder suspended below a balloon of mass $M$; see Fig. $8-25$. The balloon is stationary with respect to the ground. (a) If the man begins to climb the ladder at speed $v$ (with respect to the ladder), in what direction and with what speed (with respect to the ground) will the balloon move? (b) What is the state of the motion after the man stops climbing?

Manish Jain

Numerade Educator

Problem 25

A stone is dropped at $t_{1}=0 .$ A second stone, with twice the mass of the first, is dropped from the same point at $t_{2}=100 \mathrm{~ms}$. (a) How far below the release point is the center of mass of the two stones at $t_{3}=300 \mathrm{~ms}$ ? (Neither stone has yet reached the ground.) (b) How fast is the center of mass of the twostone system moving at that time?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 26

A $1000 \mathrm{~kg}$ automobile is at rest at a traffic signal. At the instant the light turns green, the automobile starts to move with a constant acceleration of $4.0 \mathrm{~m} / \mathrm{s}^{2}$. At the same instant a $2000 \mathrm{~kg}$ truck, traveling at a constant speed of $8.0 \mathrm{~m} / \mathrm{s}$, overtakes and passes the automobile. (a) How far is the center of mass of the automobile-truck system from the traffic light at $t_{2}=3.0 \mathrm{~s}$ ? (b) What is the speed of the center of mass of the automobiletruck system then?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 27

A shell is shot with an initial velocity $\vec{v}_{1}$ of $20 \mathrm{~m} / \mathrm{s}$, at an angle of $60^{\circ}$ with the horizontal. At the top of the trajectory, the shell explodes into two fragments of equal mass (Fig. 8-26). One fragment, whose speed immediately after the explosion is zero, falls vertically. How far from the gun does the other fragment land, assuming that the terrain is level and that air drag is negligible?

Manish Jain

Numerade Educator

Problem 28

A big olive $(m=0.50 \mathrm{~kg})$ lies at the origin and a big Brazil nut $(M=1.5 \mathrm{~kg})$ lies at the point $(1.0,2.0) \mathrm{m}$ in an $x y$ plane. At $t_{1}=0$, a force $\vec{F}_{o}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}+(3.0 \mathrm{~N}) \hat{\mathrm{j}}$ begins to act on the olive, and a force $\vec{F}_{n}=(-3.0 \mathrm{~N}) \hat{\mathrm{i}}+(-2.0 \mathrm{~N}) \hat{\mathrm{j}}$ begins to act on the nut. In unit-vector notation, what is the displacement of the center of mass of the olive-nut system at $t_{2}=4.0 \mathrm{~s}$, with respect to its position at $t_{1}=0$ ?

Manish Jain

Numerade Educator

Problem 29

Two identical containers of sugar are connected by a massless cord that passes over a massless, frictionless pulley with a diameter of $50 \mathrm{~mm}$ (Fig. 8-27). The two containers are at the same level. Each originally has a mass of 500 g. (a) What is the horizontal position of their center of mass? (b) Now $20 \mathrm{~g}$ of sugar is transferred from one container to the other, but the containers are prevented from moving. What is the new horizontal position of their center of mass, relative to the central axis through the lighter container? (c) The two containers are now released. In what direction does the center of mass move? (d) What is its acceleration?

Manish Jain

Numerade Educator

Problem 30

Ricardo, of mass $80 \mathrm{~kg}$, and Carmelita, who is lighter, are enjoying Lake Merced at dusk in a $30 \mathrm{~kg}$ canoe. When the canoe is at rest in the placid water, they exchange seats, which are $3.0 \mathrm{~m}$ apart and symmetrically located with respect to the canoe's center. Ricardo notices that the canoe moves $40 \mathrm{~cm}$ relative to a submerged log during the exchange and calculates Carmelita's mass, which she has not told him. What is it?

Manish Jain

Numerade Educator

Problem 31

In Fig. $8-28 a$, a $4.5 \mathrm{~kg}$ dog stands on an $18 \mathrm{~kg}$ flatboat and is $6.1 \mathrm{~m}$ from the shore. He walks $2.4 \mathrm{~m}$ along the boat toward shore and then stops. Assuming there is no friction between the boat and the water, find how far the $\operatorname{dog}$ is then from the shore. (Hint: See Fig. $8-28 b$. The dog moves leftward and the boat moves rightward, but does the center of mass of the boat $+\operatorname{dog}$ system move?)

Manish Jain

Numerade Educator

Problem 32

A certain nucleus, at rest, transforms into three particles. Two of them are detected; their masses and velocities are as shown in Fig. 8-29. In unit-vector notation, what is the translational momentum of the third particle, with a mass of $11.7 \times 10^{-27} \mathrm{~kg} ?$

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 33

A $40 \mathrm{~kg}$ child and her $75 \mathrm{~kg}$ father simultaneously dive from a $100 \mathrm{~kg}$ boat that is initially motionless. The child dives horizontally toward the east with a speed of $2.0$ $\mathrm{m} / \mathrm{s}$, and the father dives toward the south with a speed of $1.5 \mathrm{~m} / \mathrm{s}$ at an angle of $37^{\circ}$ above the horizontal. (Assume the boat's vertical motion due to the father's dive does not alter its horizontal motion.) Determine the magnitude and direction of the velocity of the boat along the water's surface immediately after their dives.

Manish Jain

Numerade Educator

Problem 34

A $2140 \mathrm{~kg}$ railroad flatcar, which can move with negligible friction, is motionless next to a platform. A $242 \mathrm{~kg}$ sumo wrestler runs at $5.3 \mathrm{~m} / \mathrm{s}$ along the platform (parallel to the track) and then jumps onto the flatcar. What is the speed of the flatcar if he then (a) stands on it, (b) runs at $5.3 \mathrm{~m} / \mathrm{s}$ relative to the flatcar in his original direction, and (c) turns and runs at $5.3 \mathrm{~m} / \mathrm{s}$ relative to the flatcar opposite his original direction?

Manish Jain

Numerade Educator

Problem 35

A $2.00 \mathrm{~kg}$ block is released from rest over the side of a very tall building at time $t_{1}=0 .$ At time $t_{2}=$ $1.00 \mathrm{~s}$, a $3.00 \mathrm{~kg}$ block is released from rest at the same point. The first block hits the ground at $t_{3}=5.00 \mathrm{~s}$. Plot, for the time interval $t_{1}=0$ to $t_{4}=6.00 \mathrm{~s}$, (a) the position and (b) the speed of the center of mass of the two-block system. Take $y=0$ at the release point.

Manish Jain

Numerade Educator

Problem 36

At the instant a $3.0 \mathrm{~kg}$ particle has a velocity of $6.0 \mathrm{~m} / \mathrm{s}$ in the negative $y$ direction, a $4.0 \mathrm{~kg}$ particle has a velocity of $7.0 \mathrm{~m} / \mathrm{s}$ in the positive $x$ direction. What is the speed of the center of mass of the two-particle system?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 37

A $1500 \mathrm{~kg}$ car and a $4000 \mathrm{~kg}$ truck are moving north and east, respectively, with constant velocities. The center of mass of the car-truck system has a velocity of $11 \mathrm{~m} / \mathrm{s}$ in a direction $55^{\circ}$ north of east. (a) What is the magnitude of the car's velocity?
(b) What is the magnitude of the truck's velocity?

Manish Jain

Numerade Educator

Problem 38

A cannon and a supply of cannonballs are inside a sealed railroad car of length $L$, as in Fig.8-30. The cannon fires to the right, the car recoils to the left. Fired cannonballs travel a horizontal distance $L$ and remain in the car after hitting the far wall and landing on the floor there. (a) After all the cannonballs have been fired, what is the greatest distance the car could have moved from its original position? (b) What is the speed of the car just after the last cannonball has completed its motion?

Manish Jain

Numerade Educator

Problem 39

A $1400 \mathrm{~kg}$ cannon, which fires a $70.0 \mathrm{~kg}$ shell with a speed of $556 \mathrm{~m} / \mathrm{s}$ relative to the muzzle, is set at an elevation angle of $39.0^{\circ}$ above the horizontal. The cannon is mounted on frictionless rails so that it can recoil freely. (a) At what speed relative to the ground is the shell fired? (b) At what angle with the ground is the shell fired? (Hint: The horizontal component of the momentum of the system remains unchanged as the cannon is fired.

Manish Jain

Numerade Educator

Problem 40

The following table gives the masses of three objects and, at a certain instant, the coordinates $(x, y)$ and the velocities of the objects. At that instant, what are the (a) position and (b) velocity of the center of mass of the three-particle system, and (c) what is the net translational momentum of the system?

Manish Jain

Numerade Educator

Problem 41

You are on an iceboat on frictionless, flat ice; you and the boat have a combined mass $M$. Along with you are two stones with masses $m_{A}$ and $m_{B}$ such that $M=6.00 m_{A}=12.0 m_{B}$. To get the boat moving, you throw the stones rearward, either in succession on together, but in each case with a certain speed $v^{\text {rel }}$ relative to the boat after the stone is thrown. What is the resulting speed of the boat if you throw the stones (a) simultaneously, (b) in the order $m_{A}$ and then $m_{B}$, and (c) in the order $m_{B}$ and then $m_{A}$ ?

Manish Jain

Numerade Educator

Problem 42

Two particles $P$ and $Q$ are initially at rest $1.0 \mathrm{~m}$ apart. $P$ has a mass of $0.10 \mathrm{~kg}$ and $Q$ a mass of $0.30 \mathrm{~kg} . P$ and $Q$ attract each other with a constant force of $1.0 \times 10^{-2} \mathrm{~N}$. No external forces act on the system. (a) Describe the motion of the center of mass. (b) At what distance from $P$ 's original position do the particles collide?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 43

A suspicious package is sliding on a frictionless surface when it explodes into three pieces of equal masses and with the velocities (1) $7.0 \mathrm{~m} / \mathrm{s}$, north, (2) $4.0 \mathrm{~m} / \mathrm{s}, 30^{\circ}$ south of west, and (3) $4.0 \mathrm{~m} / \mathrm{s}, 30^{\circ}$ south of east. (a) What is the velocity (magnitude and direction) of the package before it explodes? (b) What is the displacement of the center of mass of the three-piece system (with respect to the point where the explosion occurs) $3.0 \mathrm{~s}$ after the explosion?

Manish Jain

Numerade Educator

Problem 44

Figure $8-31$ shows an arrangement with an air track, in which a cart is connected by a cord to a hanging block. The cart has mass $m_{A}=0.600 \mathrm{~kg}$ and its center is initially at $x y$ coordinates $(-0.500 \mathrm{~m}, 0.000 \mathrm{~m}) ;$ the block has mass $m_{B}=0.400$ $\mathrm{kg}$ and its center is initially at $x y$ coordinates $(0,-0.100 \mathrm{~m})$. The mass of the cord and pulley are negligible. The cart is released from rest, and both cart and block move until the cart hits the pulley. The friction between the cart and the air track and between the pulley and its axle is negligible. (a) In unit-vector notation, what is the acceleration of the center of mass of the cart-block system? (b) What is the velocity of the center of mass as a function of time $t ?(\mathrm{c})$ Sketch the path taken by the system's center of mass. (d) If the path is curved, does it bulge upward to the right or downward to the left? If, instead, it is straight, give the angle between it and the $x$ axis.

Manish Jain

Numerade Educator

Problem 45

For one or more of the following situations, write a problem involving physics in this chapter, using the style of the Touchstone Examples and providing realistic data, graphs of the variables, and explained solutions: (a) determining the center of mass of a large object, (b) a system separated into parts by an internal explosion, (c) someone climbing or descending a structure, (d) track and field events.

Manish Jain

Numerade Educator

Problem 46

The script for an action movie calls for a small race car (of mass $1500 \mathrm{~kg}$ and length $3.0$ $\mathrm{m}$ ) to accelerate along a flat-top boat (of mass $4000 \mathrm{~kg}$ and length $14 \mathrm{~m}$ ), from one end to the other. The car will then jump the gap between the boat and a somewhat lower dock. You are the technical advisor for the movie. The boat will initially touch the dock as shown in Fig. 8-32. Assume the boat can slide through the water without significant resistance, and that both the car and the boat can be approximated as uniform in their mass distribution. Determine what the width of the gap will be just as the car is about to make the jump.

Manish Jain

Numerade Educator

Problem 47

Suppose you examine a digital movie of two carts with different masses that undergo a collision (for example, PASCO020 in VideoPoint). You will find there is a point between the two carts that moves at the same constant velocity both before and after the collision. We call this special point the center of mass of the two-cart system. In the PASCO020 movie, where one cart has twice the mass of the other, analysis of the video indicates that the center of mass is one-third of the distance between the two carts (measured relative to the more massive cart). A similar situation is depicted in Fig. 8 -33. The figure shows a moment in time when the cart centers just happen to be $1.000 \mathrm{~m}$ apart. For the situation in Fig. $8-33$, show that the equation
$$x_{\text {com }}=\frac{m_{A} x_{A}+m_{B} x_{B}}{m_{A}+m_{B}}$$
gives a center of mass for these two carts that is one-third of the distance between them (measured from the more massive cart).

Manish Jain

Numerade Educator