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Unformatted text preview: University of California at Berkeley
Department of Physics
Physics 7A, Spring 1999 Second Midterm, section 3
April 13, 1999 12:30PM You will be given 80 minutes to work this exam. No books are allowed, but you may use
a handwritten note sheet no larger than an 8 1/2 by 11 sheet of paper. Your description
of the physics involved in a problem is worth signiﬁcantly more than any numerical
answer. Show all work, and take particular care to explain what you are doing. Write
your answers directly on the exam, and if you have to use the back of a sheet make sure
to put a note on the front. Do not use a blue book or scratch paper. sin 45° = 0.707, cos 45° = 0.707, sin 30° = 0.500, cos 30° = 0.866
Rotational Inertias for radius R or length L: uniform sphere about axis: (2/5)MR2 spherical shell about axis: (2/3)MR2
disk about axis: (1/2)MR2 hoop about axis: MR2
rod about perpendicular at midpoint: MLZ/ 12 rod about perpendicular at end: ML2/3 “* 671 ‘87 2
A m —
2F zd—t =3; Vzr :[ﬂ—Jvﬂ +(4—nijV2:
m +m2 mI +m2
NANIEZ SID NUMBER: DISCUSSION SECTION NUMBER: DISCUSSION SECTION DATE/THVIE: Read the problems carefully.
Try to do all the problems.
If you get stuck, go on to the next problem.
Don't give up! Try to remain relaxed and work steadily. wt 1) (30 points) Hitting a baseball We can model a baseball bat as a uniform bar of length 1m, mass 1 kg, pivoted about one
end. The ball has mass 0.2 kg. We’re going to work our way up to a “realistic” model of
a bat hitting a ball, but each part below should be considered separately. In all cases, the
bat and ball are moving in the plane of the sheet of paper before and after the collision.
Ignore the effects of gravity, air resistance, the roar of the crowd, and the high cost of hot
dogs. Make sure you explain your reasoning — that is most of the credit a) Consider the bat and ball colliding as shown in the drawing below. Both are moving in
a straight line, and neither is rotating. The bat is moving at 10 m/s, and the ball is
moving at 50 m/sec. They collide completely elastically. The ball hits the exact center
of the bat. What are their velocities (magnitude and direction) after the collision? Bat b) Instead of the linear motion of part (a), a baseball bat is swung in a circle as Shown
below. If the batter swings the bat such that the center of the bat is moving at 10 m/s, what’s its angular velocity? What’s the linear velocity of the tip of the bat? What’s
the linear velocity of the handle? 0) To swing the bat as described in part (b), the batter has to exert a force on it, At the
instant shown in the drawing below, in what direction is this force? What’s its % (\magnitude? d) To start the bat moving from a stationary position (but still in the plane of the paper),
does the batter need to exert a torque on it? Ifso, in which direction? To keep a moving bat moving at a constant rotational velocity, does the batter need to exert a
torque on it? If so, in which direction? 5e 0——>~~
ball e) The batter swings the bat such that its center is moving at 10 m/s, and so that the ball
hits the center while moving at 50 m/sec. The collision is so fast that the batter can’t
exert any signiﬁcant torque during it, so assume that the batter acts like a perfect
pivot (see drawing). What are the motions of the ball and bat after the collision?
Make sure directions are clear. Can£2» mamj all: (0 “4/:
K perk/ff W 'C 4. f) The batter swings the bat such that its center is moving at 10 m/s, and so that the ball
hits the end while moving at 50 m/sec. The collision is so fast that the batter can’t
exert any signiﬁcant torque during it, so assume that the batter acts like a perfect
pivot (see drawing; this is just like part (e) except where the ball hits). What are the
motions of the ball and bat after the collision? Make sure directions are clear. m 2) (25 points) Rotation of the Earth In this problem, we look at the angular momentum of the Earth—Moon system. Please
consider the Earth and Moon to be uniform spheres (as we discussed it lecture, it’s a little
more complicated than that, but we ignore that here). Consider the Earth and Moon as an
isolated system. Ignore any other celestial bodies, like the Sun, other planets, etc; our
motion with respect to them is a topic for the future, not this problem. The radius of the Earth is 6x106 meters and its mass is 6x1024 kg. The Earth turns once in
24 hours, which is 9x104 seconds. The Moon has a radius of 2x106 meters and a mass of
7x1022 kg. Its orbit about the Earth has a radius of 4x108 meters. The Moon turns on its
axis and revolves around the Earth in the same time, 27 days or 2x106 seconds. (If you
decide you need any other information, please write a clear deﬁnition and assign a distinct
variable to it so we can understand your point.) a) The drawings below are a “side” view of the Earth—Moon system, and a “top View”
(from above the northern hemisphere of the Earth). Please sketch on them the
direction of the vector representing all the angular momentum of the Earth, the vector
representing all of the angular momentum of the Moon, and the vector representing
the total angular momentum of the EarthMoon system. Make sure you label which is
which. What are the magnitudes of these three quantities? Please write symbolic
expressions ﬁrst, and then plug in numbers only when you have complete expressions. r (\ big ( tr Q 0 0
Mann) ( EM' 3 ’ ( “my
5:605, View ‘ T I ‘/ b) Because it causes tides, which we’ll discuss later in the course, the Moon exerts some small forces on the Earth in the directions shown. Call each of these Ft, and consider
that both act at the surface of the Earth and along the equator (see drawings below).
These result in a torque on the Earth. In which direction is it? (Draw a vector on the
picture) Write an expression for its magnitude. Does it result in a change in the
rotation of the Earth? If so, how does the rotation change? If not, why not? 4 «
Ola; 4"? View In part (b), we looked at forces exerted on the Earth by the Moon, which resulted in a
torque on the Earth. What can you say about torques exerted on the Moon by the
Earth? (Not just “what”, but if possible “why”) Is the total angular momentum of the
EarthMoon system a conserved quantity? l’oé‘jﬂ'a“1 / ecu/(L, 93/3 “"99 3) (3 pts) An object moves in a circle at constant speed. The work done by the
centripetal force is zero because: a) there is no friction b) the displacement for each revolution is zero c) the average force for each revolution is zero (1) the magnitude of the acceleration is zero e) the centripetal force is perpendicular to the velocity 4) (3 pts) The center of mass of a system of particles has a constant velocity if:
a) the center of mass is at the geometric center of the system
b) the forces exerted by the particles on each other sum to zero
c) the external forces acting on particles of the system sum to zero
d) the velocity of the center of mass is initially zero
e) the particles are distributed symmetrically about the center of mass 5) (3 pts) A force acting on a particle is conservative if
a) its work equals the change in kinetic energy of the particle
b) it obeys Newton's second law c) its work depends on the end points of the motion, not on the path between
them (1) it obeys Newton's third law
e) it is not a frictional force 6) (3 pts) The precessional angular velocity of a spinning top increases if:
a) it leans at a larger angle to the vertical
b) it leans at a smaller angle to the vertical
0) its spin angular velocity increases
(1) its spin angular velocity decreases
e) none of the above effect the precessional angular velocity ML” 7) (3 pts) A ladder leans against a wall. If the ladder is not to slip, which one of the [auger following must be true? a) the coefﬁcient of friction between the ladder and the
wall must not be zero b) the coefﬁcient of friction between the ladder and the
floor must not be zero c) both a and b
d) either a or b
e) neither a nor b . J‘JNY 8) (3 pts) Suppose a golf ball is hurled at a heavy bowling ball initially at rest and bounces
elastically from the bowling ball. After the collision, which ball has the greater momentum? ‘Q’ a) the golf ball
b) the bowling ball
0) both must be the same 0—9 d) depends on something we haven’t been told 9) (3 pts) Suppose a golf ball is hurled at a heavy bowling ball initially at rest and bounces
elastically from the bowling ball. After the collision, which ball has the greater kinetic energy?
0 a) the golf ball V
b) the bowling ball
c) both must be the same 09 (1) depends on something we haven’t been told 10) (3 pts) An inelastic collision is one in which
a)
b)
c)
d)
8) momentum is not conserved but kinetic energy is conserved
total mass is not conserved but momentum is conserved
neither kinetic energy nor momentum is conserved
momentum is conserved, but kinetic energy is not conserved
the total impulse is equal to the change in kinetic energy 11) (3 pts) A massless rope passes over a massless pulley attached to the ceiling. A large mass is tied to one end, and a smaller mass is tied to the other. Starting from rest the large mass moves downward and the smaller mass moves upward with the same acceleration. Which of the following is true for the system consisting of the two masses? [(3).
El a) the center of mass remains ﬁxed b) the net external force is zero c) the velocity of the center of mass is constant (1) the acceleration of the center of mass is g, downward
6) none of the above 12) (3 pts) When a certain rubber band is stretched a distance x, it exerts a restoring force
of magnitude F = Ax, where A is a constant. The work done by a person stretching this
rubber band from x=0 to x=L is a) AL2 b) A+2L
c) A+2L2
d) A/L e) 1/2 AL2 13) (3 pts) An Oakland A’s baseball player catches a ball of mass m which is moving
toward him with speed v. While bringing the ball to rest, his hand moves back a distance
d. Assuming constant acceleration, the force exerted by his hand on the ball is a) mV/d
b) mvd
c) mvl/d
d) 2mv/d
e) mv2/(2d) 14) (3 pts) A large wedge rests on a horizontal frictionless surface. A block starts from
rest and slides down the including surface of the wedge, which is not frictionless. During
the motion of the block, the center of mass of the block and wedge a) does not move b) moves horizontally with constant speed c) moves horizontally with increasing speed A
d) moves vertically with increasing speed 6) moves horizontally and vertically 15) (3 pts) If a sphere is pivoted about an axis that is tangent to its surface, its rotational
1nert1a rs a) 1/5 MR2 b) 3/5 MR2 c) MR2 d) 7/5 MR2 e) 9/5 MR2 16) (3 pts) We may apply conservation of energy to a cylinder rolling down an incline 6/ ! without slipping, thus saying no work is done by friction, because
a) there is no friction present
b) the angular velocity of the center of mass about the point of contact is zero
c) the coefﬁcient of kinetic friction is zero
(1) the linear velocity of the point of contact (relative to the surface) is zero
e) the coefﬁcients of static and kinetic friction are equal in this case 17) (3 pts) When a particle collides elastically headon with another particle, initially at
rest, the greatest fraction of the kinetic energy of the incident particle is transferred to the
stationary one if: a) the incident particle is initially travelling very fast b) the incident particle is travelling very slowly c) the incident particle is much more massive than the target particle
d) the incident particle is much less massive than the target particle
e) the incident and target particles have the same mass. ...
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