UIUC Physics 211 James Scholars Problem Set 5
Consider a hollow tube of mass M = 1.2 kg and length L = 1.6 m that rotates about an axle through its center and perpendicular to its length. Inside the tube are two masses, m_1 = 0.4 kg each. These masses are initially held a distance d = 0.8 m apart by a string and centered in tube. The maximum tension the string can sustain is 100 N. You may consider that the radius of the tube is negligible (i.e. its moment of inertia is that of a 'stick') and that the masses held by the string are point-like.
1.1 Starting from rest, the cylinder starts to rotate as a result of a constant driving torque applied to it. What is the work done by this torque up to the point at which the string breaks?
1.2 Immediately after the string breaks the driving torque is removed. For the period after the string breaks and the masses are still inside the tube, find an expression for omega(x), where x is the distance of each of the masses m_1 from the center (axle) of the tube. Your solution for omega(x) should apply to the region 0.4 m < x < 0.8 m.
1.3 Now we want to think about the total angular momentum of the system as the masses m_1 fly out of the tube. What is the total angular momentum of the cylinder and masses just before the masses exit? What is the angular momentum of each mass just after it has left the tube? What is the angular momentum of the tube just after the masses have exited? What do you conclude about the external torque acting on the system in this period?
In this problem we will investigate a particular example of damped harmonic motion. A block of mass m rests on a horizontal table and is attached to a spring of force constant k. The coefficient of friction between the block and the table is mu. For this problem we will assume that the coefficients of kinetic and static friction are equal. Let the equilibrium position of the mass be x = 0. The mass is moved to the position x = +A, stretching the spring, and then released.
2.1 Apply Newton's 2nd law to the block to obtain an equation for its acceleration for the first half cycle of its motion, i.e. the part of its motion where it moves from x = +A to x < 0 and (momentarily) stops. Show that the resulting equation can be written d2x'/dt2 = -omega2 * x', where x' = x - x0 and x0 = mu*m*g/k. Write the expression for position of the block, x(t), for the first half cycle (be sure to express omega, the angular frequency, in terms of the constants given in the problem statement). What is the smallest value of x that the mass reaches at the end of this first half cycle?
2.2 Repeat the above for the second half cycle, i.e. wherein the block moves from its maximum negative position to its (new) maximum positive position. First show that the differential equation for the block's acceleration can be written d2x''/dt2 = -omega2*x'' where this time x'' = x + x0. Next, match the amplitude for the beginning of this half cycle with the amplitude at the end of the last one. Write the expression for the position of the block, x(t), for the second half cycle.
2.3 Make a graph of the motion of the block for the first 5 half cycles of the motion in the case where A = 10.5*x0. Plot the position of the block normalized to x0 as a function of the fractional period, T = 2*Pi/omega (i.e. plot x(t)/x0 vs t/T). Attach the graph to your email.
2.4 Something interesting happens at the end of the 5th half cycle - what changes in the physical situation and what is the motion after this half cycle?
A heavy rope 3 m long is attached to the ceiling and allowed to hang freely. Let y = 0 denote the bottom end of the rope. To get started on this problem, imagine cutting the rope at an arbitrary value of y. Draw a free body diagram of each of the two pieces of rope to determine the tension at the point where the rope was cut.
3.1 Determine the propagation speed of transverse waves on the rope and show that this speed is independent of the rope's mass and overall length.
3.2 How long would the rope have to be in order for the maximum propagation speed to be equal to the speed of sound in air (which we will take to be 330 m/s)?
3.3 Calculate the time it takes for a transverse wave to travel from the bottom of the 3 m long rope to the top and then back to the bottom.
3.4 Compare this round-trip time to that for a horizontal rope with the same tension as the average tension of the vertical rope.