Deck 8: Systems of Particles and Extended Objects

Question 1

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m)and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m).The magnitude of the position vector of the center of mass of the system is

A)0.92 m.
B)1.2 m.
C)2.9 m.
D)3.6 m.
E)4.1 m.

Answer 1

1.2 m.

Question 2

A 4500-kg truck traveling due east at a speed of 101 km/h approaches a 650-kg car traveling due west at 130 km/h.The net momentum of the system consisting of the car and truck is

A)1.50 * 10⁵ kg m/s due west.
B)1.50 * 10⁵ kg m/s due east.
C)1.50 * 10⁵ kg m/s due south.
D)1.03 * 10⁵ kg m/s due east.
E)1.03 * 10⁵ kg m/s due west.

Answer 2

1.03 * 10⁵ kg m/s due east.

Question 3

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.The velocity of the center of mass of the system consisting of the two toys is

A)2.1 m/s .
B)-2.1 m/s .
C)3.1 m/s .
D)-3.1 m/s .
E)33 m/s .

Answer 3

-3.1 m/s .

Question 4

A system consists of two particles: particle 1,having the mass m₁ = 2.0 kg has the velocity, and particle having the mass m₂ = 5.0 kg has the velocity, The magnitude of the velocity of the center of mass of the system is

A)0.92 m/s.
B)1.2 m/s.
C)2.9 m/s.
D)3.7 m/s.
E)4.1 m/s.

Question 5

A large number N of small identical stones are all located on the positive x-axis.The first is located at x = 1 m,the second at x = 3 m,the third at x = 5 m,and so on.The x-coordinates of all of the stones measured in meters are odd numbers.Where is their center of mass located?

A)x = N/2 m
B)x = N m
C)x = N² m
D)x = N(N-1)/2 m

Question 6

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m)and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m).The x-coordinate of the center of mass of the system is

A)-1.0 m.
B)0.92 m.
C)-0.67 m.
D)-2.0 m.
E)-1.3 m.

Question 7

How much force is necessary to keep a 10-cm diameter fire hose stationary if it shoots water at a velocity of 12 m/s?

A)665 N
B)1130 N
C)4522 N
D)3245 N
E)2356 N

Question 8

A 4350-kg truck traveling on a bridge at a velocity of (96.2 km/h,45.3 km/h)passes over a 724-kg car traveling at a velocity of (112 km/h,-35.5 km/h).The net momentum of the system consisting of the car and truck is

A)(9.35 * 10⁴ kg m/s,4.77 * 10⁴ kg m/s).
B)(1.39 * 10⁵ kg m/s,6.19 * 10⁴ kg m/s).
C)(1.39 * 10⁵ kg m/s,4.77 * 10⁴ kg m/s).
D)(3.76 * 10⁴ kg m/s,2.45 * 10⁴ kg m/s).
E)(1.39 * 10⁵ k m/s,-4.77 * 10⁴ kg m/s).

Question 9

A thin straight wire of uniform density is oriented in the xy-plane.The ends of the wire are located at (x,y)= (-2.00 m,-1.00 m)and (x,y)= (2.00 m,3.00 m).Where is its center of mass located?

A)(x,y)= (0,0)
B)(x,y)= (0,1.00 m)
C)(x,y)= (1.00 m,0)
D)(x,y)= (1.00 m,1.00 m)

Question 10

A rocket explodes into three pieces of equal mass.Immediately after the explosion their velocities are (120 m/s,south),(240 m/s,north)and (120 m/s east).What was the velocity of the rocket's center of mass before the explosion?

A)40 m/s,northeast
B)57 m/s,northeast
C)120 m/s,northeast
D)170 m/s,northeast

Question 11

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.What is the relative momentum?

A)2.1 kg*m/s
B)-2.1 kg*m/s .
C)16 kg*m/s .
D)-16 kg*m/s .
E)33 kg*m/s .

Question 12

A toy car of mass 4.0 kg is sitting stationary when a child rolls a toy truck of mass 6.5 kg straight towards it in the negative x-direction with a speed of 5.0 m/s.The velocity of the car with respect to the center of mass of the system consisting of the two toys is

A)2.1 m/s
B)-2.1 m/s .
C)3.1 m/s .
D)-3.1 m/s .
E)33 m/s .

Question 13

A carbon monoxide molecule consists of a carbon atom bonded to an oxygen atom.The mass of the carbon atom is 75% the mass of the oxygen atom.The carbon atom is located at the origin and the oxygen atom is located at x = d.Where is the center of mass of the carbon monoxide molecule located?

A)0.50d
B)0.57d
C)0.60d
D)0.75d

Question 14

A cylindrical bottle of oil and salt water with 1/2 volume oil ( $\rho$ =0.910 g/cm³)and 1/2 volume salt water ( $\rho$ = 1.03 g/cm³)is at rest on a table.Initially the oil and salt water are separated,with the oil floating on top of the water.The bottle is shaken so the oil and water uniformly mix and the bottle is returned to the table.How has the height of the center of mass changed after mixing?

A)It is higher.
B)It is lower.
C)It is the same.
D)There is not enough information to answer this question.

Question 15

A system consists of two particles: particle 1,having the mass m₁ = 4.0 kg is located at (-4.0 m,1.0 m)and particle 2,having the mass m₂ = 5.0 kg is located at (2.0 m,1.0 m).The direction of the position vector of the center of mass of the system measured from the positive x-axis is

A)-56.3^o
B)-79.2^o
C)101^o
D)123^o
E)112^o

Question 16

A circular pizza of radius R has a circular piece of radius R/3 removed from one side,as shown in the figure.The center of mass of the pizza with the hole is at a distance of_____ from the center of mass of the original pizza.

A)R/20
B)R/2
C)R/30
D)R/3
E)R/12

Question 17

A circular pizza of radius R has a circular piece of radius removed from one side,as shown in the figure.The center of mass of the pizza with the hole is at a distance of R/6 from the center of mass of the original pizza.The radius of the circular piece removed is

A)R/20.
B)R/2.
C)R/30.
D)R/3.
E)R/12.

Question 18

A cartoon character can levitate himself from the thrust obtained by firing his two revolvers down into the ground.If the bullets have a mass of 16 g and a downward velocity of 330 m/s,find his maximum weight if he can fire 12 bullets from each revolver in 10 seconds.

A)1.3 N
B)5.3 N
C)53 N
D)64 N
E)More information is needed.

Question 19

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by ,where a = 0.100 kg/ and b = 0.200 kg/m.The mass of the rod is

A)30.1 kg.
B)31.3kg.
C)32.8 kg.
D)34.4 kg.
E)35.9 kg.

Question 20

A long,thin rod lies along the x-axis.One end of the rod is located at x = 2.00 m and the other end of the rod is located at x = 6.00 m.The linear mass density of the rod is given by ,where a = 0.100 kg/ and b = 0.200 kg/m.What is the location of the center of mass of the rod?

A)x = 4.56 m
B)x = 4.00 m
C)x = 4.82 m
D)x = 5.10 m
E)x = 3.10 m

Question 21

A 25-kg boy jumps off of a raft.If the boy jumps off at a speed of 3.90 m/s relative to the lake and the raft moves in the opposite direction with a speed of 1.0 m/s relative to the lake,find the mass of the raft.Ignore the effects of water resistance.

A)22 kg
B)78 kg
C)109 kg
D)120 kg
E)98 kg

Question 22

The linear mass density (x)for a one dimensional object is given in the graph shown.What is the location of the center of mass for this object? Express your answer in terms of x₀.

A)1.22 x₀
B)2.11 x₀
C)0.88 x₀
D)3 x₀
E)1.5 x₀

Question 23

Assume that we have a uniform,square metal plate whose sides measure 6.00 cm and whose mass is 0.325 kg.The plate is located with its lower-left corner at the origin.A 2.00 cm by 2.00 cm square is cut out at the upper-right corner.Find the x-component of the position of the center of mass for the remaining plate.

A)1.12 cm
B)2.20 cm
C)2.45 cm
D)2.50 cm
E)2.75 cm

Question 24

A 25-kg boy jumps off of a 130-kg raft.If the boy jumps off at a speed of 4.90 m/s relative to the lake,what is the speed of the raft relative to the lake? Ignore the effects of water resistance.

A)0.94 m/s
B)1.1 m/s
C)1.6 m/s
D)2.5 m/s
E)0.50 m/s

Question 25

A ping pong ball has a mass of 2.47 g,a diameter of 4.0 cm and is made of celluloid.( )Find the thickness of the celluloid.

A)0.017 cm
B)0.034 cm
C)0.051 cm
D)0.42 cm

Question 26

Two equal masses are connected by an ideal spring and supported in a vertical configuration by only one mass,with the other mass hanging freely.What is the behavior of the bottom mass immediately after the top mass is released?

A)It moves upwards relative to the ground.
B)It moved downwards relative to the ground.
C)It is stationary relative to the ground.
D)It rotates about the system center of mass.
E)There is not enough information to tell.

Question 27

Find the y-coordinate of the center of mass of a 10-cm square plate whose density (in kg/m³)is .The lower left-hand side of the plate is at the origin and the upper right-hand corner is at (10 cm,10 cm).The plate is 0.30 cm thick.

A)7.5 cm
B)6.7 cm
C)8.9 cm
D)1.0 cm
E)5.0 cm

Question 28

A 25-kg boy jumps off of a 100-kg raft.If the boy jumps off at a speed of 4.90 m/s relative to the lake,what is the speed of the raft relative to the boy? Ignore the effects of water resistance.

A)7.5 m/s
B)6.1 m/s
C)19.6 m/s
D)2.5 m/s
E)1.2 m/s

Question 29

A rocket is moving in outer space where no external forces exist.Suppose its mass is m_R and its velocity is v_R at a particular time.When a mass m_F of fuel is ejected,the mass of the rocket becomes (m_R - m_F)and its velocity increases to (v_R+ v_R).Suppose the fuel is always ejected at a constant speed v_F with respect to the rocket.Derive an expression for v_R as a function of m_R given the rocket's initial mass and speed are m₀ and v₀,respectively.

A)v_R = v₀ + v_F ln(m_R/m₀)
B)v_R = v₀ - v_F ln(m_R/m₀)
C)v_R = v₀ - v_F exp(m₀/m_R)
D)v_R = v₀ - v_F ln(m₀/m_R )
E)v_R = v₀ + v_F exp(m_R/m₀)

Question 30

A 25-kg boy jumps off of a raft (m = 92 kg).If the boy jumps off at a speed of 4.40 m/s relative to the lake and the raft moves in the opposite direction with a speed of 1.2 m/s,find the mass of his friend who remains seated on the raft.Ignore the effects of water resistance.

A)22 kg
B)38 kg
C)112 kg
D)130 kg
E)0 (He only has an imaginary friend on the raft.)

Question 31

The center of mass of a uniform,symmetric object may be located

A)at the center of the object.
B)within the body of the object.
C)outside the body of the object.
D)A,B,or C can be true.
E)None are correct.

Question 32

The center of mass of an irregular rigid object is always located

A)at the center of the object.
B)somewhere within the object.
C)outside of the object
D)Both A and B are correct.
E)None are correct.

Question 33

The coordinates for the center of mass for the extended object shown in figure are (L/4,-L/5).What are the coordinates of the 2-kg mass?

A)(0,0)
B)(-L,L)
C)(-0.375 L,0.1 L)
D)(-4 L,5 L)
E)(-5 L,4 L)

Question 34

A rod has a linear density $\lambda$ = 3x,where x is the distance along the rod,which spans 0 < x < 1 m.What is the coordinate of the center of mass of the rod?

A)0.25 m
B)0.33 m
C)0.5 m
D)0.67 m
E)0.75 m

Question 35

Find the weight of a 10-cm square plate whose density (in kg/m³)is .The lower left-hand side of the plate is at the origin and the upper right-hand corner is at (10 cm,10 cm).The plate is 0.30 cm thick.

A)1.21 * 10^-3 N
B)0.69 N
C)3.89 * 10^-2 N
D)2.07 * 10^-3 N
E)6.28 * 10^-2 N

Question 36

Find the x-coordinate of the center of mass of a 10-cm square plate whose density (in kg/m³)is .The lower left-hand side of the plate is at the origin and the upper right-hand corner is at (10 cm,10 cm).The plate is 0.30 cm thick.

A)6.7 cm
B)8.9 cm
C)1.0 cm
D)5.0 cm
E)7.5 cm

Question 37

In the water molecule,the hydrogen and oxygen atoms are separated by 10^-10 m and the bond angle between the two hydrogen-oxygen bonds is 105 ^$\circ$ .the mass of the hydrogen atom is 1 u and that of the oxygen atom is 16 u.The center of mass of the water molecule along the line of symmetry from the oxygen atom is

A)6.76 pm.
B)8.91 pm.
C)10.1 pm.
D)14.6 pm.
E)17.8 pm.

Question 38

In the water molecule,the hydrogen and oxygen atoms are separated by 10^-10 m and the bond angle between the two hydrogen-oxygen bonds is 105 ^$\circ$ .the mass of the hydrogen atom is 1 u and that of the oxygen atom is 16 u.The center of mass of the water molecule along the line of symmetry from the oxygen atom is

A)6.76 * 10^-12 m
B)8.91 * 10^-12 m
C)10.1 * 10^-12 m
D)14.6 * 10^-12 m
E)17.8 * 10^-12 m

Question 39

A 1900-kg cannon shoots a 5-kg cannon ball at a velocity of 25 m/s.What is the recoil velocity of the cannon?

A)0.033 m/s
B)0.066 m/s
C)0.132 m/s
D)0 m/s
E)1.23 m/s

Question 40

A uniform wire is bent into a semi-circle of radius R and laid out such that it is entirely in the x-y plane .The semicircle has its ends fixed on the x axis at x = -R and x = +R.With the exception of its end points,the semicircle is entirely above the x-axis (i.e.the y-coordinates of all points on its arc are positive).What is the y coordinate of its center-of-mass?

A)2R/ $\pi$
B)R/3
C)R/2
D)2R
E)R/

Question 41

A flat metal disk has a diameter of 27 cm.A smaller disk with a diameter of 11 cm is then cut out of the larger disk.The hole left behind in the larger disk has its edge located at the center of the larger disk.How far from the center of the larger disk is the center of mass of the larger disk with the circular hole?

A)0.91 cm
B)1.09 cm
C)1.83 cm
D)2.04 cm
E)2.24 cm

Question 42

Suppose we have a rocket in outer space.The rocket consists of a payload of 5280.0 kg.Assume that the rocket starts from rest,accelerates to its final velocity.After reaching this final velocity it begins its 3.82 * 10⁵ km trip to the moon.The speed at which the rocket expels propellant is 5.11 km/s.If the rocket is to make the trip from the Earth to the Moon,a distance of 3.82 * 10⁵ km,in 7.5 hours,what mass of fuel is required? (in kg)

A)121000 kg
B)130300 kg
C)84200 kg
D)74700 kg
E)200000 kg

Question 43

A 16 cm by 16 cm checkerboard has a mass of 207 g.There are three 31.0-g checkers located on the checkerboard as shown in the figure.Relative to O,the origin,located at the bottom,left-hand corner of the checkerboard,where is the center of mass of the checkerboard and checkers system?

A)(7.9 cm,6.5 cm)
B)(8.1 cm,6.7 cm)
C)(8.1 cm,7.1 cm)
D)(8.3 cm,6.5 cm)
E)(8.3 cm,6.8 cm)

Question 44

Starting at rest,two students standing on massless carts pass a 7.0-kg ball back and forth.The student on the left has a mass of 40 kg and can throw the ball with a relative speed of 9.0 m/s.The student on the right has a mass of 30 kg.If the ball is thrown horizontally from the student on the left to the student on the right,how fast,relative to the student on the left will the student on the right move after catching the pass?

A)0.9 m/s
B)1.7 m/s
C)2.8 m/s
D)3.0 m/s
E)3.9 m/s

Question 45

A long thin rod has its one end at x = 0 and the other end at x = 2 m.Its composition is such that the linear mass density changes with length according to $\lambda$ (x)= ax² + bx+c,where x is the distance from one end of the rod.Given: a = 0.75 kg/m³,b = 0.5 kg/m²,and c = .25 kg/m,the mass of the rod is

A)2.9 kg
B)2.1 kg
C)3.5 kg
D)0.75 kg
E)4.3 kg

Question 46

Mass of the Saturn V rocket is 2.5 * 10⁶ kg.It burns fuel at the rate of 2 *10⁴ kg/s and has an exhaust speed of 2.8 km/s relative to the rocket.The thrust of the rocket is

A)5.6 * 10⁴ N.
B)2.8 * 10⁷ N.
C)5.6 * 10⁷ N.
D)9.8 * 10⁸ N.

Question 47

A large truck of mass 3600 kg approaches a car of mass 1250 kg traveling in the opposite direction.If the velocity of the center of mass of the system is zero,and the truck's speed is 18 m/s,how fast is the car moving?

A)52 m/s
B)6.3 m/s
C)70 m/s
D)31 m/s

Question 48

Ignoring air resistance,a catapult on a level field tosses a 5.0-kg stone a horizontal distance of 43 m.A second 5.0-kg stone tossed in an identical fashion breaks apart in the air into 2 pieces,one with a mass of 3.2 kg and one with a mass of 1.8 kg.The breaking apart happens in such a way that both of the pieces hit the ground at the same time.If the 1.8-kg piece lands a distance of 59 m away from the catapult,how far away from the catapult does the 3.2-kg piece land?

A)15 m
B)27 m
C)34 m
D)50 m
E)52 m

Question 49

An explosive hockey puck of mass 0.127 kg slides along the ice at 32 m/s in the +x direction,and explodes,leaving a 0.075-kg chunk stationary.What is the speed of the other chunk?

A)0.0 m/s
B)32 m/s
C)54 m/s
D)19 m/s
E)78 m/s

Question 50

Suppose we have a rocket in outer space.The rocket consists of a payload of 5280.0 kg and 130300.0 kg of fuel.Assume that the rocket starts from rest,accelerates to its final velocity.After reaching this final velocity it begins its 3.82 * 10⁵ km trip to the moon.What is the speed at which we must expel the propellant to make the trip from the Earth to the Moon,a distance of 3.82 * 10⁵ km,in 7.5 hours? (in km/s)

A)3.74
B)4.37
C)5.11
D)5.98
E)7.00

Question 51

A student with a mass of 40 kg can throw a 15-kg ball with a relative speed of 5.0 m/s.The student is standing at rest on a cart with a mass of 12 kg that can move over the ground without friction.If the student throws the ball horizontally,what is the speed of the ball relative to the ground?

A)1.4 m/s
B)3.1 m/s
C)3.9 m/s
D)5.0 m/s
E)5.9 m/s

Question 52

Contrary to advice given him,a boy tries bowling on the ice.He releases his 7.3-kg bowling ball when he is sliding at 2.0 m/s,and the ball travels subsequently down the ice at 13 m/s.If,after releasing the ball,the boy is moving backward from his original direction of motion at 0.25 m/s,what is the boy's mass?

A)36 kg
B)50 kg
C)47 kg
D)64 kg

Question 53

A long thin rod has its one end at x = 0 and the other end at x =1 m.Its composition is such that the linear mass density changes with length according to $\lambda$ (x)= ax² + bx+c,where x is the distance from one end of the rod.Given: a = 0.75 kg/m³,b = 0.5 kg/m²,and c = .25 kg/m,the center of mass of the rod relative to x = 0 is

A)0.41 m.
B)0.52 m.
C)0.60 m.
D)0.64 m.

Question 54

A student with a mass of 40 kg can throw a 15-kg ball with a relative speed of 5.0 m/s.The student is standing at rest on a cart with a mass of 12 kg that can move over the ground without friction.If the student throws the ball horizontally,what is the student's speed relative to the ground?

A)1.4 m/s
B)5.0 m/s
C)3.9 m/s
D)1.9 m/s
E)1.1 m/s

Question 55

Suppose we have a rocket in outer space.The rocket consists of a payload of 5280.0 kg and 121000 kg of fuel.Assume that the rocket starts from rest,accelerates to its final velocity.After reaching this final velocity it begins its 3.82 * 10⁵ km trip to the moon.The speed at which the rocket expels propellant is 4.22 km/s.If the rocket is to make the trip from the Earth to the Moon,a distance of 3.82 * 10⁵ km,how long will the trip take to complete ?

A)15 hours
B)6.7 hours
C)23 hours
D)7.5 hours
E)7.9 hours

Question 56

A thin rectangular plate of uniform density $\sigma$ ₁ = 1.67 kg/m² has a length a = 0.740 m and a width of b = 0.330 m.The lower left hand corner is placed at the origin,(x,y)= (0,0).A circular hole is cut in the plate of radius r = 0.064 m with center at (x,y)= (0.084 m,0.084 m)and replaced with a circular disk with the same radius composed of another material of the same thickness with uniform density $\sigma$ 2 = 5.94 kg/m².What is the distance of the resulting plate's center of mass from the origin? (in m)

A)0.157
B)0.209
C)0.278
D)0.370
E)0.492

Question 57

A particle of mass 3 kg moving at 8 m/s makes an elastic collision with a particle of mass 5 kg moving at -4 m/s.The velocity of the center mass of the particles is

A)3 m/s.
B)1.8 m/s.
C)0.5 m/s.
D)0.35

Question 58

An explosive hockey puck of mass 0.127 kg slides along the ice at 32 m/s in the +x direction,and explodes,sending a 0.037-kg chunk moving in the +x direction at 83 m/s.What is the velocity of the other piece of the puck?

A)32 m/s in the +x direction
B)11 m/s in the +x direction
C)7.8 m/s in the +x direction
D)11 m/s in the -x direction
E)7.8 m/s in the -x direction

Question 59

Four particles are located at each corner of a square of length 2 cm.If a mass M is located at (x₁,y₁)= (-1 cm,1 cm),a mass 2M is located at (x₂,y₂)= (1 cm,1 cm),a mass 2 M is located at (x₃,y₃)= (1 cm,-1 cm),and a mass M is located at (x₄,y₄)= (-1 cm,-1 cm),where is the center of mass located?

A)(0 cm,0 cm)
B)(1/3 cm,0 cm)
C)(0 cm,1/3 cm)
D)(2 cm,0 cm)
E)(0 cm,2 cm)

Question 60

Three faces of a cube of side L are removed,therefore,it has three faces in xy,xz,and yz planes.The center of mass of the system relative to the origin,where the three remaining faces meet,is

A)x = y = z = L/3.
B)x = y = z = L/2.
C)x = y = z = 2L/3.
D)x = y = z = 2L/5.

Question 61

Which of the following statements is the one that is true?

A)Momentum is a scalar quantity.
B)If the kinetic energy of an object does not change,its momentum can not change.
C)A spacecraft is traveling in space with its engines off.After a blast from the engines,the center of mass of the system (space craft + exhaust gases)continues with a higher velocity.
D)The center of mass of an object must always lie within the object.
E)If two objects of different mass are subject to the same impulse,they will have the same change in momentum.

Question 62

A 1.3-kg booster attached to a 6-kg rocket is initially traveling at a speed of 175 m/s.If the booster is ejected in backward direction with speed of 60 m/s relative to the rocket/booster rest frame,how fast will the rocket then travel?

A)115 m/s
B)162 m/s
C)175 m/s
D)188 m/s
E)235 m/s