General Physics I Quiz Samples for Chapter 9 Center of Mass and Linear Momentum April 20, 2020 Name: Department: Student ID #: Notice +2 (-1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (✘) if the statement is correct (incorrect). !!!: corrections to an incorrect answer. Textbook: Walker, Halliday, Resnick, Principles of Physics, Tenth Edition, John Wiley & Sons (2014). 9-1 Center of Mass 1. () The center of mass of a system of n particles is defined to be the point whose coordinates are given by x com = 1 M n X i=1 m i x i , y com = 1 M n X i=1 m i y i , z com = 1 M n X i=1 m i z i , where m i and x i are the mass and the position of the ith particle and M = n X i=1 m i is the total mass of the system. If we use the vector notation, then r com = 1 M n X i=1 m i r i , where r i is the position vector of the ith particle. 2. () The center of mass of a system of particles is the point that moves as though all of the system’s mass were concentrated there. 3. () The center of mass of a system of particles is the point that moves as though all external forces were applied there. 4. () The center of mass of a two-body system is r com = m 1 r 1 + m 2 r 2 m 1 + m 2 . 5. (a) (✘) The center of mass of an object must lie within the object. !!!The center of mass of a C-shaped object does not lie within the object. (b) (✘) All the mass of an object is actually concentrated at its center of mass. !!!The center of mass of a system of particles is the point that moves as though all of the system’s mass were concentrated there and all external forces were applied there. (c) (✘) The center of mass of an object cannot move if there is zero net force on the object. !!!If there is zero net force on an object, because of the Newton’ second law, the acceleration of the object is zero, so the object has constant velocity. It does not mean that this object cannot move. 2020 KPOPE All rights reserved. Korea University Page 1 of 5
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General Physics IQuiz Samples for Chapter 9
Center of Mass and Linear MomentumApril 20, 2020
Name: Department: Student ID #:
Notice
� +2 (−1) points per correct (incorrect) answer.
� No penalty for an unanswered question.
� Fill the blank ( ) with � (8) if the statement iscorrect (incorrect).
The center of mass of a system of n particles isdefined to be the point whose coordinates are givenby
xcom =1
M
n∑i=1
mixi,
ycom =1
M
n∑i=1
miyi,
zcom =1
M
n∑i=1
mizi,
where mi and xi are the mass and the position ofthe ith particle and
M =n∑
i=1
mi
is the total mass of the system. If we use thevector notation, then
rcom =1
M
n∑i=1
miri,
where ri is the position vector of the ith particle.
2. (�) The center of mass of a system of particles isthe point that moves as though all of the system’smass were concentrated there.
3. (�) The center of mass of a system of particles isthe point that moves as though all external forceswere applied there.
4. (�) The center of mass of a two-body system is
rcom =m1r1 +m2r2m1 +m2
.
5. (a) (8) The center of mass of an object must liewithin the object.
!!!The center of mass of a C-shaped objectdoes not lie within the object.
(b) (8) All the mass of an object is actuallyconcentrated at its center of mass.
!!!The center of mass of a system ofparticles is the point that moves as though allof the system’s mass were concentrated thereand all external forces were applied there.
(c) (8) The center of mass of an object cannotmove if there is zero net force on the object.
!!!If there is zero net force on an object,because of the Newton’ second law, theacceleration of the object is zero, so the objecthas constant velocity. It does not mean thatthis object cannot move.
(d) (8) The center of mass of a cylinder must lieon its axis.
!!!The center of mass of a nonuniformcylinder may not lie on its axis.
6. (�)
Center of mass
Two boys with masses of m1 and m2 stand on ahorizontal frictionless surface holding the ends of alight road of length L. The boys pull themselvestogether along the rod. When they meet the m1
boy will have movedm2
m1 +m2L.
9-2 Newton’s Second Law for a System ofParticles
1. (�) Newton’s Second Law for a System ofParticles:The motion of the center of mass of any system ofparticles is governed by Newton’s second law for asystem of particles, which is
Fnet = Macom.
Here Fnet is the net force of all the external forcesacting on the system, M is the total mass of thesystem, and acom is the acceleration of thesystem’s center of mass.
2. (a) (�) The center of mass of a system of particleshas a constant velocity if the external forcesacting on particles of the system sum to zero.
(b) (�) The center of mass of a system of particlesremains at the same place if it is initially atrest and the external forces sum to zero.
3. (�)
Block A, with a mass of m1, is moving with avelocity of v1 while block B, with a mass of m2, ismoving with a velocity of v2. The center of mass ofthe two-block system is moving with the velocity of
vcom =m1v1 +m2v2m1 +m2
.
4. (�)
At the same instant, a ball of mass m1 is droppedfrom h above Earth, and a second ball, with a massof m2, is thrown straight upward from Earth’ssurface with an initial speed of v. They move alongnearby lines and pass without colliding. After time
A mass m1 is attached to one end of a spring witha spring constant k and a mass m2 is attached tothe other end. The masses are placed on ahorizontal surface and the spring is compressed x.The spring is then released with the masses at restand the masses oscillate. When the spring has itsequilibrium length for the first time, the m1 masshas a speed of v. The mechanical energy that hasbeen lost to this instant is
1
2kx2 − 1
2m1v
2 − 1
2
m21
m2v2 = 0.
9-3 Linear Momentum
1. (�) The linear momentum P of a system ofparticles is
P = Mvcom,
where vcom is the velocity of the system’s center ofmass and M is the total mass of the system.
2. (a) (�) Two bodies, A and B, have equal kineticenergies. The mass of A is α times that of B.The ratio of the momentum of A to that of Bis√α : 1.
(b) (�) Two objects, P and Q, have the samemomentum. Q can have more kinetic energythan P if it is moving faster than P .
(c) (�) A m1 stone is released from rest and fallstoward the Earth. After time t has elapsed,the magnitude of its momentum is mgt.
3. (a) (�) Force equals the time rate of change ofmomentum.
(b) (�) If the total momentum of a system ischanging, a net external force must be actingon the system.
(c) (�) The unit of a momentum may beexpressed in N · s.
(d) (�) The momentum of an object at a giveninstant is in the same direction as its velocity.
(e) (�) When you step on the accelerator toincrease the speed of your car, the force thataccelerates the car is the force of friction ofthe road on the tires.
9-4 Collision and Impulse
1. (�) The Impulse–Linear MomentumTheorem:The change in the body’s linear momentum due toan impulse is
∆p = pf − pi = J ,
where pi and pf are the initial and the finalmomentum, respectively, and the impulse J isdefined by
J =
∫ tf
ti
F (t)dt = Faverage∆t.
Here, Faverage is the time average of the forceduring the collision of time interval[ti, tf = ti + ∆t].
2. (�) The physical quantity “impulse” has the samedimensions as that of momentum.
3. (a) (�) A golf ball of mass m is hit by a golf clubso that the ball leaves the tee with speed v.The club is in contact with the ball for timeT . The average force on the club on the ballduring the time T is mv/T .
(b) (�) A box of books with mass m is droppedon the floor from a height of h and comes torest. The floor exerted impulse of m
√2gh on
the box.
9-5 Conservation of Linear Momentum
1. (�) If a system is isolated so that no net externalforce acts on it, the linear momentum of thesystem remains constant:
P = constant ←→ Pf = Pi.
In any two-body collision of an isolated system, thesum of two momenta is invariant:
p1f + p2f = p1i + p2i.
Here, 1 and 2 are particle IDs and f and i indicatethe initial and final states, respectively. In terms ofthe masses and the velocities, the linearmomentum conservation can be stated as
2. (�) The center of mass of a closed, isolated systemof two colliding bodies is not affected by acollision. In particular, the velocity of the center ofmass cannot be changed by the collision.
3. (�) A shell is launched from a cannon and explodesmid-air. The explosion exerts only internal forceson the shell, but gravity is an external force. Itshorizontal momentum is conserved but its verticalmomentum is not conserved.
9-6 Momentum and Kinetic Energy in Collisions
1. (a) (�)
A puck of mass m1 is traveling at speed v. Itstrikes another puck with mass m2, which isstationary. The two pucks stick together.Their common final speed is
m1
m1 +m2v.
2. (a) (�) An inelastic collision is one in whichmomentum is conserved but kinetic energy isnot conserved.
(b) (�) An elastic collision is one in which kineticenergy and momentum are both conserved.
(c) (�) A completely inelastic collision is one inwhich the bodies stick together after thecollision.
3. (�)
Gunpowder
A block moves at speed v in the positive xdirection and hits an identical block, initially atrest. A small amount of gunpowder had beenplaced on one of the blocks. The explosion does notharm the blocks but it doubles their total kineticenergy. After the explosion, the blocks move alongthe x axis and the incident block has a speed of∣∣∣∣∣1−
√3
2v
∣∣∣∣∣ .4. (�) Cart A, with a mass of m1, travels on a
horizontal air track at a speed of v and hits cart B,which has a mass of m2 and is initially at rest.After the collision, the center of mass of the twocart system has a speed of
m1
m1 +m2v.
5. (�) A bullet of mass m is fired horizontally into ablock of wood with mass M suspended by a ropefrom the ceiling. After the bullet sticks into theblock, they swing in an arc, rising h above itslowest position. The speed of the bullet was
m+M
m
√2gh.
6. (a) (�) For a completely inelastic two-bodycollision the kinetic energy retained by theobjects is the same as 1
2Mv2com where M isthe total mass and vcom is the velocity of thecenter of mass.
(b) (�)
A sack of coal with a mass of m is dropped ona railroad flatcar with mass M which wasinitially moving at v as shown. After the sackrests on the flatcar, the speed of the flatcar is
1. (�) In an elastic collision, the total kinetic energyis conserved:
K1f +K2f = K1i +K2i.
In addition, the linear momentum is alsoconserved:
p1f + p2f = p1i + p2i.
2. (�) Blocks A and B are moving toward each otheralong the x axis. A has a mass of mA and avelocity of vA, while B has a mass of mB and avelocity of vB. They suffer an elastic collision andmove off along the x axis. The velocity of B afterthe collision is
mBvB + 2mAvA −mAvBmA +mB
.
9-8 Collisions in Two Dimensions
1. (�)
A hockey puck of mass m traveling along the xaxis at v0 hits another identical hockey puck atrest. If after the collision the second puck travelsat a speed of v2 at an angle of θ2 above the x axis,the final speed of the first puck is v1, at an angle ofθ1 below the x axis. Then θ1 is
tan−1
(v2 sin θ2
v0 − v2 cos θ2
).
2. (�) A hockey puck of mass m traveling along the xaxis at 4.5 m/s hits another identical hockey puckat rest. If after the collision the second pucktravels at a speed of 3.5 m/s at an angle of 30°above the x axis, this is an inelastic collision sincekinetic energy is not conserved.
9-9 System with Varying Mass: A Rocket
1. (�) In the absence of external forces a rocketaccelerates at an instantaneous rate given by
Rvrelative = Ma,
where R is the fuel consumption rate, vrelative isthe speed of fuel ejected relative to the rocket,Rvrelative is the thrust of the rocket engine, M isthe variable mass of the rocket. The speed changesfrom vi to vf when its mass changes from Mi toMf :
vf − vi = vrelative lnMi
Mf.
2. (a) (�) A rocket exhausts fuel with a velocity of1.5 × 103 m/s, relative to the rocket. It startsfrom rest in outer space with fuel comprising80 percent of the total mass. When all the fuelhas been exhausted its speed is 2.4 × 103 m/s.
(b) (�)
A space probe of mass M is motionless inspace. To start moving, its main engine isfired for 5 s during which time it ejectsexhaust gases at a speed of v. At the end ofthis process, it is moving at V . Note that vand V are not relative speed. The mass of theejected gas is
