Pavel Panchekha


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Problem Set 3 (Given 10/2/11)

Level I

  1. A hammock of mass 5 kg hangs from two trees attached by a rope on either side. If the rope makes an angle of 30 degrees with the horizontal at its point of contact with the tree on either side, find the tension in the rope on either side of the hammock.
  2. A block of mass \(m\) rests on a larger block of mass \(M\). The static friction coefficient between the blocks is \(\mu\), and there is no friction between \(M\) and the ground. What is the maximum force that may be applied to \(M\) such that m does not slip against it?
  3. A ball of mass 2 kg is attached to the end of a light rope. This particular type of rope is known to break if the tension in it exceeds 30 N. Neglecting gravity, what is the maximum speed with which one could spin the rope in a circle without breaking it?
  4. Two masses sit on a see-saw. A 3 kg mass sits on the left side a distance 0.5 meters from the center, and a 4 kg mass sits on the right side a distance 0.75 meters from the center. You find an additional 2 kg mass. How far from the center of the see-saw and on what side must it be placed to keep the see-saw balanced?

Level II

  1. A block of mass \(M\) sits on an inclined plane that makes an angle \(\theta\) with the horizontal. The plane moves with an acceleration \(a\). Assuming no friction between the block and the plane, find the acceleration of the mass as a function of \(a\).
  2. A car travels on a banked curve of radius \(r\) that makes an angle theta with the horizontal. The car has mass \(M\), and the coefficient of static friction between the car and the ramp is \(\mu\). Find the maximum and minimum speeds with which the car can travel around the curve without slipping.
  3. Revisiting the Atwood machine: A rope hangs over a pulley of mass \(M\). Attached to either side of the rope are two blocks of mass \(m_1\) and \(m_2\) respectively, where \(m_1 > m_2\). Unlike in the example from class, the pulley is not frictionless; instead, the friction between the rope and the pulley is sufficient for no slipping to occur. Assume that the pulley may be treated as a disk and is free to rotate about its center of mass. Find the accelerations of the blocks once the system is released. (Hint: Because there is friction between the rope and the pulley, the tension in the rope is no longer the same throughout. Instead, there will be some tension T1 on one side of the pulley and a different tension T2 on the opposite side.

Level III

  1. Find the moment of inertia of an \(a \times b\) rectangle about the center of mass. What about an equilateral triangle with side length s?
  2. Prove the parallel axis theorem: given the moment of inertia \(I_{CM}\) about the center of mass of an object along a given axis, the center of mass about some other, parallel, axis is \(I = I_{CM} + m d^2\), where d is the distance between the two axes.