Question 7

Ap net At A(mv) At Am At

Question 11

I l. A student is asked to determine the mass of Jupiter. Knowing which of the following about Jupiter and one of its moons will allow the determination to be made? I. The time it takes for u Il. The time it takes for the moon to orbit Jupiter Ill. The average distance between the moon and Jupiter

An application of Kepler's Third Law; If you know both r and T, you can find the mass M. Alternatively, use the satellite equation (for a circular orbit, which may or may not be true): net GMm 2 2 GM For a circular orbit, v — 2mr/T, so you recover Kepler's Third Law (which is true even for elliptical orbits, with a, the ellipse's semimajor axis, replacing the radius r.)

Question 19


  • When a=0, μ= tanθ

Question 21

Center of Raft Center of Mass of System d A person is standing at one end of a uniform raft of length L that is floating motionless on water, as shown above. The center of mass of the person- raft system is a distance d from the center of the raft. The person then walks to the other end of the raft. If friction between the raft and the water is negligible, how far does the raft move relative to the water?

  • After the person has walked to the other end, the situation must be the mirror image of this one; the center of mass will be shifted to a distance d to the left of the center of the raft.

  • But the center of mass cannot move as the person walks on the boat.

  • Therefore, the boat must have moved a distance 2d to the right.

Question 24

A solid cylinder of mass nt and radius R has a string wound around it. A person holding the string pulls it vertically upward, as shown above, such that the cylinder is suspended in midair for a brief time interval At and its center of mass does not move. The tension in the string is T, and the rotational 2 inertia of the cylinder about its axis is —mR

24. The linear acceleration of the person's hand during the time interval At is (E) zero T -mg 2 (C) (D)

There is an associated, famous problem about dropping a spool (with the cord held by a motionless hand). In that case, you write Newton's Second Law for the forces and for the torques, where one associates a with acm Fnet = mg - T = ma T = TR = la = I(a/R) . Then T = la/R2 and mg ma = mg — la/R2 a = m + (1/R2) The problem here is, of course, that the center of mass acceleration is zero. In fact you have to think a little more carefully. The typical relationship for rolling without slipping, a = OR, still holds, but now it is the tangential acceleration of the rim, which is equal to the upward acceleration of the cord (and the hand.) Generally in rolling without slipping (on a horizontal surface, for example), the rim's acceleration is the same as the center of mass's; that isn't true here. Here, the center of mass doesn't move at all during the time interval At. But we can still say, with a the acceleration of the rim (equal to the acceleration of the hand), T = = la = I(a/R) From the previous problem, we know T = mg. Then rngR = I(a/R) = x (a/R) = 2rnaR = 29. Unsurprisingly, 90% of those who Cancel mR on both sides, multiply both sides by 2 and obtain a took the test got this one wrong.

Question 25

A figure skater goes into a spin with arms fully extended. Which of the following describes the changes in the rotational kinetic energy and angular momentum of the skater as the skater's arms are brought toward the body?

The angular momentum cannot change, because there are no external torques. On the other hand, we have the useful formulas and K = 2m 21 As the skater's arms come in, I, which depends on the square of the average distance of mass from the axis, is getting smaller. Since the kinetic energy depends directly on the square of the angular momentum and inversely on the moment of inertia, and I here decreases, it follows K increases. Also, work has to be done to draw the arms in; this work goes into increased kinetic energy.

Question 32

Tabletop Three identical disks are initially at rest on a frictionless, horizontal table with their edges touching to form a triangle, as shown in the top view above. An explosion occurs within the triangle, propelling the disks horizontally along the surface. Which of the following diagrams shows a possible position of the disks at a later time? (In these diagrams, the triangle is shown in its original position.)

  • The center of mass cannot move

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