Download Packet 6 - Gravitation Oscillations

Gravitation, Oscillations - Practice Packet #6
Name:
.
List vocabulary here:
List equations here (star the equations to memorize):
1.
The mass of Planet X is one-tenth that of the Earth, and its diameter is one-half that of the Earth. The acceleration due to gravity
at the surface of Planet X is most nearly
(A) 2 m/s2
(B) 4 m/s2
(C) 5 m/s2
(D) 7 m/s2
(E) 10 m/s2
2.
A satellite travels around the Sun in an elliptical orbit as shown above. As the satellite travels from point X to point Y. which of
the following is true about its speed and angular momentum?
Speed
Angular Momentum
(A) Remains constant Remains constant
(B) Increases
Increases
(C) Decreases
Decreases
(D) Increases
Remains constant
(E) Decreases
Remains constant
3.
A newly discovered planet, "Cosmo," has a mass that is 4 times the mass of the Earth. The radius of the Earth is Re. The
gravitational field strength at the surface of Cosmo is equal to that at the surface of the Earth if the radius of Cosmo is equal to
(A) ½Re
(B) Re
(C) 2Re
(D) √𝑅𝑒
(E) Re2
4.
Two artificial satellites, 1 and 2, orbit the Earth in circular orbits having radii R1 and R2, respectively, as shown above. If R2 = 2R1,
the accelerations a2 and a1 of the two satellites are related by which of the following?
(A) a2 = 4a1
(B) a2 = 2a1
(C) a2 = a1
(D) a2 = a1/2
(E) a2 = a1/4
5.
The radius of the Earth is approximately 6,000 kilometers. The acceleration of an astronaut in a perfectly circular orbit 300
kilometers above the Earth would be most nearly
(A) 0 m/s2
(B) 0.05 m/s2
(C) 5 m/s2
(D) 9 m/s2
(E) 11 m/s2
8.
A small mass is released from rest at a very great distance from a larger stationary mass. Which of the following graphs best
represents the gravitational potential energy U of the system of the two masses as a function of time t?
(A)
(B)
U
U
(C)
O
O
9.
(D)
U
O
t
t
O
(E)
U
t
U
O
t
t
A satellite S is in an elliptical orbit around a planet P, as shown above, with r1 and r2 being its closest and farthest distances,
respectively, from the center of the planet. If the satellite has a speed v 1 at its closest distance, what is its speed at its farthest
distance?
(A)
r1
v
r2 1
(B)
r2
v
r1 1


(C) r 2  r 2 v 1
(D)
r 1r 2
v1
2
(E)
r 2 r 1
v
r 1r 2 1
Questions 10-11 refer to a ball that is tossed straight up from the surface of a small, spherical asteroid with no atmosphere. The ball
rises to a height equal to the asteroid's radius and then falls straight down toward the surface of the asteroid.
10. What forces, if any, act on the ball while it is on the way up?
(A) Only a decreasing gravitational force that acts downward
(B) Only an increasing gravitational force that acts downward
(C) Only a constant gravitational force that acts downward
(D) Both a constant gravitational force that acts downward and a decreasing force that acts upward
(E) No forces act on the ball.
11. The acceleration of the ball at the top of its path is
(A) at its maximum value for the ball's flight
(B) equal to the acceleration at the surface of the asteroid
(C) equal to one-half the acceleration at the surface of the asteroid
(D) equal to one-fourth the acceleration at the surface of the asteroid
(E) zero
13. Two identical stars, a fixed distance D apart, revolve in a circle about their mutual center of mass, as shown above. Each star has
mass M and speed v. G is the universal gravitational constant. Which of the following is a correct relationship among these
quantities?
(A) v2 = GM/D
(B) v2 = GM/2D
(C) v2 = GM/D2
(D) v2 = MGD
(E) v2 = 2GM2/D
15. The graph above shows the force of gravity on a small mass as a function of its distance R from the center of the Earth of radius
Re, if the Earth is assumed to have a uniform density. The work done by the force of gravity when the small mass approaches
Earth from far away and is placed into a circular orbit of radius R2 is best represented by the area under the curve between
(A) R = 0 and R = Re,
(B) R = 0 and R = R2
(C) R = Re, and R = R2
(D) R = Re and R =∞
(E) R = R2 and R = ∞
1.
A simple pendulum of length l. whose bob has mass m, oscillates with a period T. If the bob is replaced by one of mass 4m, the
period of oscillation is
1
(A) 4 T
2.
1
(B) 2 T
(C) T
(D) 2T
(E)4T
Which of the following is true for a system consisting of a mass oscillating on the end of an ideal spring?
(A) The kinetic and potential energies are equal at all times.
(B) The kinetic and potential energies are both constant.
(C) The maximum potential energy is achieved when the mass passes through its equilibrium position.
(D) The maximum kinetic energy and maximum potential energy are equal, but occur at different times.
(E) The maximum kinetic energy occurs at maximum displacement of the mass from its equilibrium position.
Questions 3-4
A 0. l -kilogram block is attached to an initially unstretched spring of force constant k = 40 newtons per meter as
shown above. The block is released from rest at time t = 0.
3.
What is the amplitude of the resulting simple harmonic motion of the block?
1
1
1
1
(A)
(B)
(C) m
(D) m
(E) 1 m
m
m
40
20
4
2
4.
At what time after release will the block first return to its initial position?
(A)

s
40
(B)

s
20
(C)

10
s
(D)

s
5
(E)

s
4
5.
A particle moves in simple harmonic motion represented by the graph above. Which of the following represents the velocity of
the particle as a function of time?
(A) v(t) = 4 cos t
(B) v(t)=  cost
(C) v(t) = –2 cos t
(D) v(t) = –4 sin t
(E) v(t) = –4 sin t
Questions 7-8 refer to the graph below of the displacement x versus time t for a particle in simple harmonic motion.
7.
Which of the following graphs shows the kinetic energy K of the particle as a function of time t for one cycle of motion?
8.
Which of the following graphs shows the kinetic energy K of the particle as a function of its displacement x ?
17. A simple pendulum consists of a l.0-kilogram brass bob on a string about 1.0 meter long. It has a period of 2.0 seconds. The
pendulum would have a period of 1.0 second if the
(A) string were replaced by one about 0.25 meter long
(B) string were replaced by one about 2.0 meters long
(C) bob were replaced by a 0.25-kg brass sphere
(D) bob were replaced by a 4.0-kg brass sphere
(E) amplitude of the motion were increased
18. The equation of motion of a simple harmonic oscillator is d2x/dt2 = -9x, where x is displacement and t is time. The period of
oscillation is
(A) 6
(B) 9/2
(C) 3/2
(D) 2/3
(E) 2/9
19. A pendulum with a period of 1 s on Earth, where the acceleration due to gravity is g, is taken to another planet, where its period is
2 s. The acceleration due to gravity on the other planet is most nearly
(A) g/4
(B) g/2
(C) g
(D) 2g
(E) 4g
25. A 1.0 kg mass is attached to the end of a vertical ideal spring with a force constant of 400 N/m. The mass is set in simple
harmonic motion with an amplitude of 10 cm. The speed of the 1.0 kg mass at the equilibrium position is
(A) 2 m/s
(B) 4 m/s
(C) 20 m/s
(D) 40 m/s
(E) 200 m/s
2007M2. In March 1999 the Mars Global Surveyor (GS) entered its final orbit about Mars, sending data back to Earth. Assume a
circular orbit with a period of 1.18 × 102 minutes = 7.08 × 103 s and orbital speed of 3.40 × 103 m/s . The mass of the GS is 930
kg and the radius of Mars is 3.43 × 106 m.
a.
Calculate the radius of the GS orbit.
b.
c.
d.
Calculate the mass of Mars.
Calculate the total mechanical energy of the GS in this orbit.
If the GS was to be placed in a lower circular orbit (closer to the surface of Mars), would the new orbital period of the GS be
greater than or less than the given period?
_________Greater than
Justify your answer.
e.
_________ Less than
In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars at 3.71 × 105 m above the surface and its
furthest distance at 4.36 × 105 m above the surface. If the speed of the GS at closest approach is 3.40 × 103 m/s, calculate the
speed at the furthest point of the orbit.
1989M3. A 2-kilogram block is dropped from a height of 0.45 meter above an uncompressed spring, as shown above. The spring has
an elastic constant of 200 newtons per meter and negligible mass. The block strikes the end of the spring and sticks to it.
a. Determine the speed of the block at the instant it hits the end of the spring.
b. Determine the period of the simple harmonic motion that ensues.
c. Determine the distance that the spring is compressed at the instant the speed of the block is maximum.
d. Determine the maximum compression of the spring.
e. Determine the amplitude of the simple harmonic motion.
2003M2. An ideal spring is hung from the ceiling and a pan of mass M is suspended from the end of the spring, stretching it a
distance D as shown above. A piece of clay, also of mass M, is then dropped from a height H onto the pan and sticks to it.
Express all algebraic answers in terms of the given quantities and fundamental constants.
a. Determine the speed of the clay at the instant it hits the pan.
b. Determine the speed of the pan just after the clay strikes it.
c. Determine the period of the simple harmonic motion that ensues.
d. Determine the distance the spring is stretched (from its initial unstretched length) at the moment the speed of the pan is a
maximum. Justify your answer.
e. The clay is now removed from the pan and the pan is returned to equilibrium at the end of the spring. A rubber ball, also of mass
M, is dropped from the same height H onto the pan, and after the collision is caught in midair before hitting anything else.
Indicate below whether the period of the resulting simple harmonic motion of the pan is greater than, less than, or the same as it
was in part c.
_____Greater than
_____Less than
_____The same as
Justify your answer.