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Problem 1: (4 pts.) Two spherical planets, each of mass M and Radius R, start out at
rest with a distance from center to center of 4R. What is the speed of one of the planets at
the moment that their surfaces touch? Give your answer in terms of G, M and R.
Problem 2: (4 pts.) A meter stick goes by me with a speed v = 0.8c. How much time
elapses on my watch between when the front of the meter stick passes me and when the
back passes me? Chose the closest to your answer.
a. 2.0×10−9 s
b. 2.5×10−9 s
c. 3.0×10−9 s
d. 2.0×10−8 s
e. 2.5×10−8 s
f. 3.0×10−8 s
Problem 3: An astronaut lands on a round asteroid whose radius is Ra = 3 km. The
escape speed off the surface of the asteroid is Vesc = 5 km/hr. The astronaut then jumps
straight up from the surface of the asteroid and reaches a height of 1 km above the surface of
the asteroid before she falls back toward the surface. That is, her maximum distance away
from the center of the asteroid is a total of 4 km.
a. (4 pts.) Give the formula for the escape velocity Vesc as a function of the mass M ,
radius Ra and Newton’s gravitational constant G. Don’t put numbers into the formula!
b. (6 pts.) How fast was the astronaut moving when she first left the surface of the
asteroid? Hint: Use the conservation of energy and don’t substitute in the “numbers”
for this problem until the very last step. You do not need to use a calculator or to
know the values of G, the mass of the asteroid or the mass of the astronaut.
Problem 4: A small block of mass m is attached to the end of a spring whose natural
length is ` and whose spring constant is k. The other end of the spring is attached to a
table, and initially the the block is moving in a circle whose radius is ` + d.
a. What is the angular frequency ω of the block in terms of k, `, and m?
Problem 5: Two particles each have rest mass m and are moving directly towards one
another, each has a speed v = 4c/5. The particles collide in a completely inelastic collision,
i.e., the particles stick together to form a single composite particle. Give all of your answers
in terms of m and c and, specifically, not in terms of v.
a. (2 pts) What is the magnitude of the momentum of one of the particles before the
collision?
b. (2 pts) What is the total energy of one of the particles before the collision?
c. (2 pts) What is the kinetic energy of one of the particles before the collision?
d. (4 pts) What is the rest mass of the composite particle after the collision?
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Problem 6: A small block of mass m is constrained to move along the positive part of the
x-axis under the influence of a potential energy function
U (x) =
B
x2
where B is a positive constant. Initially m is at a very large value of x and is moving inwards
with a speed v0 .
Give your answers in terms of m, v0 and B.
a. What are the dimensions or units associated with the constant B?
b. What is the smallest x value, xmin , that m can reach?
c. What is the speed of m when it is at 2xmin ?
Problem 7: A mass m on the end of a particular spring oscillates horizontally with an
angular frequency ω = 6 radians/s and with an amplitude of 0.25 m. At a moment when
the spring is stretched to its maximum, an additional mass ∆m = 3 kg is quickly placed on
top of the original mass. With the additional mass in place, the angular frequency changes
to 3 radians/s.
a. What was the initial mass on the spring?
b. What is the value of the spring constant?
c. What is the amplitude of the oscillation after the extra mass has been added?
Problem 8: A yo-yo of mass m, on a string of length ` is held horizontally and then
released, as shown in the diagram. Assume that there is no friction and that the yo-yo
swings freely in a vertical plane.
Give your answers in terms of m, `, θ and g.
a. (2 pts.) What is the magnitude of the acceleration |~a| at the instant after it is released?
b. (3 pts.) What is the angular speed ω = θ̇ as a function of θ?
c. (2 pts.) What is the maximum angular speed ωmax , while the yo-yo swings back and
forth?
d. (3 pts.) What is the tension in the string when the yo-yo moves through the bottom
of its arc?
Problem 9: A block of mass m is on an inclined plane (inclined at angle θ) and attached
to a string. The string loops over a massless, frictionless pulley and is then attached to a
second block also of mass m, as shown in the diagram. The blocks are initially held at rest
and level with each other with y1 = y2 = 0.
Give your answers in terms of m, g, θ, h and the length of the string L.
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a. (2 pts.) When the blocks are released they start to move. After the block on the right
has moved a distance h, what is the value of y for the block on the left?
b. (4 pts.) What is the speed of the block on the right after it has moved this distance
h?
c. (4 pts.) What is the magnitude of the acceleration of the block on the right? Is the
acceleration up or down?
Problem 10: A small block of mass m is constrained to move along the positive part of
the x-axis under the influence of a potential energy function
4a x
+
,
U (x) = B
x
a
where a and B are positive constants.
Give your answers in terms of m, a and B.
a. (1 pt.) Provide a rough sketch of U (x) on the axes provided above. No numbers or
scale is required. Mark with an “×” the location of each equilibrium.
b. (1 pt.) What are the dimensions or units associated with the constant a, and with the
constant B?
c. (2 pts.) For each equilibrium position what is the value of x and is the equilibrium
stable or unstable?
d. (3 pts.) What are the values of x for the turning points if the total energy of m is
E = 5B?
e. (3 pts.) If the block is held at x = a and then released, what is the speed of the block
when it reaches the position x = 2a?
Problem 11: A block of mass m is attached to a spring of spring-constant k and there is
damping which gives a force proportional to the velocity whose size is −bẋ. It is convenient
to define β ≡ b/2m and ω02 ≡ k/m. In this problem the damping is strong and β 2 = 4ω02 /3.
Thus the differential equation that describes the motion is either
ẍ + 2β ẋ +
4ω0
3β 2
x = 0 or ẍ + √ ẋ + ω02 x = 0
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(1)
Initially, the mass is displaced a positive distance xt=0 = A away from the equilibrium
position.
Give your answers in terms of either ω0 or β, not both!
a. (3 pts.) What value for the initial velocity ẋt=0 = v0 would make x(t) decrease in
an exponential fashion as rapidly as possible? Hint: Your answer should probably be
negative.
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b. (2 pts.) In this case, how much time would elapse for the mass to move to the position
x = A/e?
c. (3 pts.) What value for the initial velocity ẋt=0 = v0 would make x(t) decrease in an
exponential fashion as slowly as possible?
d. (2 pts.) In this case, how much time would elapse for the mass to move to the position
x = A/e?
Problem 12: An object of mass m is on a frictionless table and attached to the end of a
horizontal spring of spring-constant k and length L; the other end of the spring is attached
to the table. Let x = 0 be the position of m when the spring is not stretched. A constant
force F acts in the positive-x direction, but you are initially holding the mass in place at
x = 0. At time t = 0 you release the mass.
Give your answers in terms of m, k, L and F .
a. (2 pts.) What is the angular frequency ω of the ensuing oscillation?
b. (4 pts.) Give an expression for x(t) in terms of m, k, L and F .
c. (2 pts.) What is the amplitude of the oscillation?
d. (2 pts.) Give two different values of x where ẋ = 0.
Problem 13: A small block of mass m is attached to the end of a horizontal spring of
spring-constant k along the x-axis and is being acted upon by a force F0 cos(Ωt) in the x
direction. The friction between the block and the table is very small, but the mass has
been oscillating long enough that all of the “transient” motion has died away; otherwise the
friction is negligible.
a. (2 pt.) What is the amplitude of the oscillation as a function of F0 , k, m and Ω? Hint:
Note that the “amplitude” might be either positive or negative.
b. (2 pt.) What is the magnitude of the speed ẋ of the mass when it moves through the
equilibrium position? Give your answer in terms of F0 , k, m and Ω.
c. (4 pt.) What value of Ω2 should I choose, if I want the absolute value of the amplitude
of the oscillations to be F0 /2k? Give your answer in terms of F0 , k and m. Hint:
Watch the sign!
d. (2 pt.) For this same value for Ω as in the previous part, what is the value of x when
Ωt = 12π? Give your answer in terms of F0 , k, and m.
Problem 14: Oort is an alien from a planet near the star Xaxes, which is 10 lt-yrs from
the Earth, as measured in our frame of reference. After a physical exam, he is diagnosed to
have the dreaded Xaxian disease, flemel, and will live for only a little more than one year.
Oort’s last wishes are to visit Disney World on the planet Earth and, if he has time, to
abduct an Earthling.
pSo he gets in a Xaxian rocket and flies toward the Earth with a speed
v = 0.995c, so that 1 − v 2 /c2 ≈ 0.1.
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a. While flying toward the Earth, what does Oort (in his frame of reference) measure the
distance between Xaxes and the Earth to be?
b. Approximately how many years, as measured in the Earth’s frame of reference, would
we measure Oort to be traveling from Xaxes to the Earth?
c. Is Oort able to reach the Earth while still alive?
Problem 15: A star of mass m and a star of mass 3m orbit their common center of mass
in circular orbits. The separation between the two stars is L. Give all of your answers in
terms of m, L, constants of nature such as G or c, and mathematical constants such as
π or e. Simplify your answers!
a. How far is the center of mass from the 3m star?
b. What is the orbital frequency of the 3m star?
c. What is the speed of the 3m star in its orbit?
d. What is the speed of the 1m star in its orbit?
Problem 16: Consider a large spherical object of constant density. A satellite takes just
90 minutes for one complete orbit just above the surface of the the object. A small off-center
tunnel goes from one side of the object to the other as shown in the figure on the blackboard.
The triangle with vertices at the end points of the tunnel and the center of the object is a
45◦ -45◦ -90◦ triangle. Assume that the walls of the tunnel are frictionless and there is no air
resistance. The radius of the object is ??? If a small package is dropped into the tunnel at
one end, how long does it take for the object to arrive at the other end?
Problem 17: A commuting scheme involves boring a straight tunnel through the crust
of the Earth between New York City and San Francisco. If you were to drop a ball down
into the tunnel in NYC then how long does it take for the ball to pop up in SF? Assume
that there is no friction or air resistance in the tunnel. Assume that the earth is round and
has constant density, and that at the surface of the Earth g = 10 m/s2 . Finally assume that
the triangle connecting NYC, SF and the center of the Earth is an equilateral triangle (60◦
for each angle). Assume that the radius of the Earth is 6,400 km. Hint: Ignore the Earth’s
rotation and centrifugal force affects in this problem!
Short Answer
Problem 18:
A horizontal force of 12 N pushes a 0.5 kg book against a vertical wall.
The book is initially at rest. If the coefficients of friction are µs = 0.6 and µk = 0.8 which
of the following is true?
a.
b.
c.
d.
e.
The magnitude of the frictional force is 4.9 N
The magnitude of the frictional force is 7.2 N
The normal force is 4.9 N
The book will start moving and accelerate
If started moving downward, the book will decelerate.
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Problem 19: An ideal spring is used to fire a 15.0 g pellet horizontally. The spring has a
spring constant of 20 N/ m and is initially compressed by 7.0 cm. The kinetic energy of the
pellet as it leaves the spring is:
a.
b.
c.
d.
e.
zero
2.5×10−2 J
4.9×10−2 J
9.8×10−2 J
1.4 J
Problem 20: A kaon K o is an uncharged elementary particle with rest energy of about
mK c2 = 500 MeV. A pion π o is an uncharged elementary particle with rest energy of about
mπ c2 = 125 MeV. A K o decays into two π o particles in an average time of 9 × 10−17 s.
a. (2 pts.) After the decay of a K o , at rest in the laboratory, into two π o particles what
is the total energy of one π o , measured in the laboratory?
b. (3 pts.) What is the kinetic energy of this π?
c. (5 pts.) What is the speed of this π o ?
Problem 21: Two particles each have rest mass m and are moving directly towards one
another, each has a speed v = 3c/5. The particles collide in a completely inelastic collision,
i.e., the particles stick together to form a single composite particle. Give all of your answers
in terms of m and c and, specifically, not in terms of v.
a. (2 pts) What is the magnitude of the momentum of one of the particles before the
collision?
b. (2 pts) What is the total energy of one of the particles before the collision?
c. (2 pts) What is the kinetic energy of one of the particles before the collision?
d. (4 pts) What is the rest mass of the composite particle after the collision?
Problem 22: A yo-yo of mass m, on a string of length ` is held horizontally and then
released, as shown in the diagram on the blackboard. Assume that there is no friction and
that the yo-yo swings freely in a vertical plane.
Give your answers only in terms of m, ` and g.
a. What is the speed of the yo-yo at the bottom of the arc just before the string hits the
post?
b. What is the tension in the string just before the string hits the post?
c. What is the tension in the string just after the string hits the post?
d. What is the tension in the string when the yo-yo is at its maximum height after hitting
the post?
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Problem 23: A bullet of mass m is fired into a vat of jello with an initial speed v0 . Assume
that the the viscous force on the bullet is −b~v , for some constant b. Let t = 0 when the
bullet first hits the jello. Ignore gravity. Give your answers only in terms of m, b and v0 .
a. What is the speed of the bullet as a function of time t?
b. What is the distance into the jello as a function of time t?
c. How far into the jello does the bullet go?