Pavel Panchekha


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Problem Set 4 (Given 10/9/11)

Level I

  1. A rotating arm is attached to the floor — so it can rotate around an axis parallel to the ground. Determine the forces and the torques on the arm.
  2. At which points is the acceleration of a particle on the end of a spring maximal? The velocity? At what point (at what displacement \(x\)) is the velocity half the maximum?
  3. As we discussed in class, rubber bands are like one-way springs (they contract but do not expand). Imagine taking a rubber band of spring constant \(k\) and hanging an object of mass \(m\) from the ceiling on the rubber band. Then pull the object down a distance \(x\) and release it. Describe the motion of the object for various values of \(x\).

Level II

  1. Solve the AP Physics C 2009 Mechanics, question 2.
  2. Solve the AP Physics C 2008 Mechanics, question 3.
  3. Solve the AP Physics C 2007 Mechanics, question 1.
  4. Find the equation for the motion of a pendulum, with the small-angle approximation, for an irregularly-shaped pendulum. You're going to need the distance from the pendulum axis to the center of mass of the pendulum, and the moment of inertia of the pendulum.

Level III

  1. Consider an object on a carousel or merry-go-round, so that its reference frame is accelerating. As we've discussed, an accelerating reference frame causes "pseudo-forces" to appear – forces that are a consequence of the reference frame, not any actual, physical forces. Calculate the pseudo-forces on this rotating reference frame, assuming uniform rotational velocity.

    Hint: Imagine a straight line from the object's reference frame; easiest would be a radial line. What does that curve look like from the ground (in an inertial reference frame)? What forces would the object have the exert to move in this path? These would be the forces you have to apply to counteract the pseudo-forces.