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1. (2005M3) A system consists of a ball of mass M2 and a uniform rod of mass M1 and length d. The rod is attached to a horizontal frictionless table by a pivot at point P and initially rotates at an angular speed ω, as shown above left. The rotational inertia of the rod about point P is ⅓M1d2 . The rod strikes the ball, which is initially at rest. As a result of this collision, the rod is stopped and the ball moves in the direction shown above right. Express all answers in terms of M1, M2, ω, d, and fundamental constants.

a) Derive an expression for the angular momentum of the rod about point P before the collision.

b) Derive an expression for the speed v of the ball after the collision.

c) Assuming that this collision is elastic, calculate the numerical value of the ratio M1/M2

d) A new ball with the same mass M1 as the rod is now placed a distance x from the pivot, as shown above. Again assuming the collision is elastic, for what value of x will the rod stop moving after hitting the ball?

2. (2004M3) A uniform rod of mass M and length L is attached to a pivot of negligible friction as shown above. The pivot is located at a distance L/3 from the left end of the rod. Express all answers in terms of the given quantities and fundamental constants.

a) Calculate the rotational inertia of the rod about the pivot.

b) The rod is then released from rest from the horizontal position shown above. Calculate the linear speed of the bottom end of the rod when the rod passes through the vertical.

3. 2000M2. An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which has mass MJ = 1.90 x 1027 kg and radius RJ = 7.14 x 107 m.

If the radius of the planned orbit is R, use Newton's laws to show each of the following.

The orbital speed of the planned satellite is given by

The period of the orbit is given by

The explorer wants the satellite's orbit to be synchronized with Jupiter's rotation. This requires an equatorial orbit whose period equals Jupiter's rotation period of 9 hr 51 min = 3.55 x 104 s. Determine the required orbital radius in meters.

Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differs from the desired value in each of the following ways, sketch the resulting orbit on the figure. (J is the center of Jupiter, the dashed circle is the desired orbit, and P is the injection point.) Also, describe the resulting orbit qualitatively but specifically.

i. When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly faster than the correct speed for a circular orbit of that radius.

ii. When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly slower than the correct speed for a circular orbit of that radius.

4. (1981M3). A thin, uniform rod of mass M1 and length L , is initially at rest on a frictionless horizontal surface. The moment of inertia of the rod about its center of mass is M1L2/12. As shown in Figure I, the rod is struck at point P by a mass m2 whose initial velocity v is perpendicular to the rod. After the collision, mass m2 has velocity –(½)v as shown in Figure II. Answer the following in terms of the symbols given.

a) Using the principle of conservation of linear momentum, determine the velocity v’ of the center of mass of this rod after the collision.

b) Using the principle of conservation of angular momentum, determine the angular velocity ( of the rod about its center of mass after the collision.

c) Determine the change in kinetic energy of the system resulting from the collision.

5. 1984M2. Two satellites, of masses m and 3m, respectively, are in the same circular orbit about the Earth's center, as shown in the diagram above. The Earth has mass Me and radius Re. In this orbit, which has a radius of 2Re, the satellites initially move with the same orbital speed vo but in opposite directions.

a) Calculate the orbital speed vo of the satellites in terms of G, Me, and Re.

b) Assume that the satellites collide head-on and stick together. In terms of vo find the speed v of the combination immediately after the collision.

c) Calculate the total mechanical energy of the system immediately after the collision in terms of G, m, Me, and Re. Assume that the gravitational potential energy of an object is defined to be zero at an infinite distance from the Earth.

6. Model the Earth by assuming it is made up of an outer portion of rock (density = ~2.5 g/cc) and a core of iron (density = ~7.9 g/cc). Earth’s mass is ~6.0 x 1024 kg and it’s radius is ~6.4 x 103 m. Assume the mass distribution is spherically symmetric.

a) Compute the radius of the iron core.

b) If gE is the magnitude of the gravitational field at the surface (gE = 9.81 m/s2), compute the value of g as a fraction of gE at the radius of the iron core.

c) Derive expressions for the gravitational field as a function of r, where r=0 at the earth’s center. Rather than numbers, use the following constants: RE = radius of Earth; RFe = radius of iron core (answer to a)); ρFe and ρE are density of iron and rock, respectively. Your answer will actually consist of three equations: one for r ................
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