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1977 Mechanics III
Two stars each of mass M form a binary star system such that
both stars move in the same circular orbit of radius R. The
universal gravitational constant is G.
a. Use Newton's laws of motion and gravitation to find an
expression for the speed v of either star in terms of R, G,
and M.
b. Express the total energy E of the binary star system in terms of R, G, and M.
Suppose instead, one of the stars had a mass 2M.
c. On the following diagram, show circular orbits for this star system
d. Find the ratio of the speeds, v2M/vM
1992- Mechanics III
A spacecraft of mass 1,000 kilograms is in an
elliptical orbit about the Earth, as shown
above. At point A the spacecraft is at a
distance rA= 1.2 x 107 meters from the center
of the Earth and its velocity, of magnitude vA
= 7.1 x 103 meters per second, is
perpendicular to the line connecting the
center of the Earth to the spacecraft. The
mass and radius of the Earth are ME = 6.0 x
1024 kilograms and rE = 6.4 X 106 meters,
respectively
Determine each of the following for the spacecraft when it is at point A .
a. The total mechanical energy of the spacecraft, assuming that the gravitational potential energy is
zero at an infinite distance from the Earth.
b. The magnitude of the angular momentum of the spacecraft about the center of the Earth
Later the spacecraft is at point B on the exact opposite side of the orbit at a distance rB = 3.6 x 107
meters from the center of the Earth.
c. Determine the speed vB of the spacecraft at point B.
Suppose that a different spacecraft is at point A, a distance r A=1.2 x107 meters from the center of
the Earth. Determine each of the following.
d. The speed of the spacecraft if it is in a circular orbit around the Earth
e The minimum speed of the spacecraft at point A if it is to escape completely from the Earth
1994- Mechanics III
A satellite of mass m is in an elliptical orbit
around the Earth, which has mass M e and
radius Re. The orbit varies from closest
approach of a at point A to maximum
distance of b from the center of the Earth at
point B. At point A, the speed of the satellite is
vo. Assume that the gravitational potential
energy U = 0 when masses are an infinite
distance apart. Express your answers in
terms of a, b, m, Me, Re, vo, and G.
a. Write the appropriate definite integral,
including limits, that can be evaluated to
show that the potential energy of the satellite when it is a distance r from the center of the Earth is
given by
U, = -GMem/r
b. Determine the total energy of the satellite when it is at A.
c. What is the magnitude of the angular momentum of the satellite about the center of the Earth
when it is at A?
d. Determine the velocity of the satellite as it passes point B in its orbit.
As the satellite passes point A, a rocket engine on the
satellite is fired so that its orbit is changed to a
circular orbit of radius a about the center of the Earth.
e. Determine the speed of the satellite for this circular
orbit.
f. Determine the work done by the rocket engine to
effect this change.
2005- Mechanics II
A student is given the set of orbital data for some of the moons of Saturn shown below and is asked
to use the data to determine the mass MS of Saturn. Assume the orbits of these moons are circular.
(a) Write an algebraic expression for the gravitational force between Saturn and one of its moons.
(b) Use your expression from part (a) and the assumption of circular orbits to derive an equation for
the orbital period T of a moon as a function of its orbital radius R.
(c) Which quantities should be graphed to yield a straight line whose slope could be used to
determine Saturn's mass?
(d) Complete the data table by calculating the two quantities to be graphed. Label the top of each
column, including units.
(e) Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers
to indicate the scale.
(f) Using the graph, calculate a value for the mass of Saturn.
1999- Mechanics II
A spherical, nonrotating planet has a radius R and a
uniform density ρ throughout its volume. Suppose a
narrow tunnel were drilled through the planet along one
of its diameters, as shown in the figure above, in which a
small ball of mass m could move freely under the
influence of gravity. Let r be the distance of the ball
from the center of the planet.
a. Show that the magnitude of the force on the ball at
a distance r < R from the center of
the planet is given by
F = -Cr. where C = 4/3(πGρm).
b. On the axes below, sketch the force F on the ball as a function of distance r from the center of the
planet.
The ball is dropped
into the tunnel from
rest at point P at the
planet's surface.
c. Determine the
work done by
gravity as the ball
moves from the
surface to the center
of the planet.
d. Determine the speed of the ball when it reaches the center of the planet.
e. Fully describe the subsequent motion of the ball from the time it reaches the center of the planet.
f. Write an equation that could be used to calculate the time it takes the ball to move from point P to the
center of the planet. It is not necessary to solve this equation.
Homework Thursday 4-26-2012
A particle of mass m slides down a fixed, frictionless sphere
of radius R. starting from rest
at the top.
a. In terms of m, g, R. and θ, determine each of the following for the
particle while it is sliding on the sphere.
i. The kinetic energy of the particle
ii.
The centripetal acceleration of the mass
iii.
The tangential acceleration of the mass
c. Determine the value of θ at which the particle leaves the sphere.
A ring of radius a has a total charge +Q distributed
uniformly around its circumference. As shown in
Figure I, the point P is on the axis of the ring at a
distance b from the center of the ring.
a. On Figure I above, show the direction of the electric
Figure I
field at point P.
b. Determine the magnitude of the electric field intensity at point P.
As shown in Figure II, the ring is now rotated about its
axis at a uniform angular velocity ω in a clockwise
direction as viewed from point P. The charge moves with
the ring.
c. Determine the current of this moving charge.
d. On Figure II above, draw the direction of the magnetic
field at point P.
e. Determine the magnitude B of the magnetic field at point
A system consists of a mass M 2, and a
uniform rod of mass M1, and length .
The rod is initially rotating with an angular
speed ω on a horizontal frictionless table
about a vertical axis fixed at one end
through point P. The moment of inertia of the rod about P is M 2 /3. The rod strikes the stationary
mass M2. As a result of this collision, the rod is stopped and the mass M2 moves away with speed v.
a. Using angular momentum conservation determine the speed v in terms of Ml, M2, , and ω.
b. Determine the linear momentum of this system just before the collision in terms of Ml, , and ω.
c. Determine the linear momentum of this system just after the collision in terms of M, , and ω.
d. What is responsible for the change in the linear momentum of this system during the collision?
e. Why is the angular momentum of this system about point P conserved during the collision?