A sphere of radius R is surrounded by a concentric spherical shell of inner radius 2R and outer radius 3R, as shown above. The inner sphere is an insulator containing a net charge + Q distributed uniformly throughout its volume. The spherical shell is a conductor containing a net charge + q different from + Q.
Use Gauss's law to determine the electric field for the following values of r, the distance from the center of the insulator.
a. 0 < r < R
b. R < r < 2R
c. 2R < r < 3R
Determine the surface charge density (charge per unit area) on
d. the inside surface of the conducting shell,
e. the outside surface of the conducting shell
You are given a long, thin, rectangular bar of known mass M and length l with a pivot attached to one end. The bar has a nonuniform mass density, and the center of mass is located a known distance x from the end with the pivot. You are to determine the rotational inertia Ib of the bar about the pivot by suspending the bar from the pivot, as shown above, and allowing it to swing. Express all algebraic answers in terms of Ib , the given quantities, and fundamental constants.
(a)
i. By applying the appropriate equation of motion to the bar, write the differential equation for the angle θ the bar makes with the vertical.
ii. By applying the small-angle approximation to your differential equation, calculate the period of the bar’s motion.
(b) Describe the experimental procedure you would use to make the additional measurements needed to determine Ib . Include how you would use your measurements to obtain Ib and how you would minimize experimental error.
(c) Now suppose that you were not given the location of the center of mass of the bar. Describe an experimental procedure that you could use to determine it, including the equipment that you would need.
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 x 102 minutes = 7.08 x 103 s and orbital speed of 3.40 x 103 m/s . The mass of the GS is 930 kg and the radius of Mars is 3.43x106m .
(a) Calculate the radius of the GS orbit.
(b) Calculate the mass of Mars.
(c) Calculate the total mechanical energy of the GS in this orbit.
(d) 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 ______Less than
Justify your answer.
(e) In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars at 3.71 x 105 m above the surface and its furthest distance at 4.36 x 105 m above the surface. If the speed of the GS at closest approach is 3.40 x 103 m/s , calculate the speed at the furthest point of the orbit.
A block of mass m slides at velocity vo, across a horizontal frictionless surface toward a large curved movable ramp of mass 3m as shown in Figure I. The ramp, initially at rest, also can move without friction and has a smooth circular frictionless face up which the block can easily slide. When the block slides up the ramp, it momentarily reaches a maximum height as shown in Figure II and then slides back down the frictionless face to the horizontal surface as shown in Figure III.
a. Find the velocity v1, of the moving ramp at the instant the block reaches its maximum height.
b. To what maximum height h does the center of mass of the block rise above its original height? c. Determine the final speed v, of the ramp and the final speed v' of the block after the block returns to the level surface. State whether the block is moving to the right or to the left.