A small mass m is sitting on a frictionless horizontal

Question

Question 1: A merry-go-round has radius r = 4.0 m. Bubba attempts to stop the merry-go round by grabbing the edge. He immediately falls down, but still hangs on. The rotational inertia of Bubba and the merry-go-round, together, is I = 20, 000 kg m². The frictional force between Bubba and the ground is F_f = 490 N. If the merry-go-round is spinning with angular velocity ω = 0.62 rad/s immediately after Bubba grabs it, how long until it stops?

Question 2: A small mass m is sitting on a frictionless horizontal table. There is a string attached to the mass; the string goes from the mass to a hole in the center of the table so that if you hold the string under the table the mass is forced to go in a circle. (See figure) Initially, the mass is going in a circle at radius R and angular speed ω_o. You pull the string so that the mass moves in a smaller circle of radius R/2. What is the new angular speed ω of the mass?

Question 3: For question 2, calculate the initial kinetic energy of the mass and the final kinetic energy of the mass. Was energy conserved?

Question 4: In question 3, you hopefully found that there was more energy at the end than at the beginning. The extra kinetic energy comes from work done when you pull the string. Find this work done by integrating the force F_c = mrω² times the distance dr from r = R to r = R/2, and show that the "extra" energy is equal to the work done when you pull the string.

Question 5: The wheel of a stationary exercise bicycle is in the form of a uniform disk, with mass 7.2 kg and radius 25 cm. The wheel is initially at rest, but a cyclist applies to it a constant torque of 6.0 Nm for 1.5 seconds. How far does the wheel move in that time? (θ)

Question 6: A "record turntable" (an analog data-storage device used last century for reproducing recorded music) has rotational inertia of I_t = 8.4 x 10^-3 kg m², and turns at 331/3 RPM. A "record" initially at rest drops onto the turntable. The record has rotational inertia I_r = 1.1 x 10^-3 kg m². If the turntable is to bring the record up to 331/3 RPM in 1.5 seconds, what power is required?

Question 7: An Atwood's machine consists of two masses hanging on either side of a pulley, as shown in figure. Derive the equation for the acceleration of this device, assuming that the pulley is in the shape of a disk. Assume friction at the pulley axle is negligible.

Question 8: It's all well and good to be able to integrate simple shapes to find their rotational inertia: but how does one find the all-important rotational inertia of a rubber chicken? They're hard to integrate! Here's one way to measure I experimentally: First, you assemble a low-friction rotating platform, as shown in figure. It should have a string and a pulley off to one side, supporting a mass m as shown. The string should be wrapped around the center spindle, which has radius r. Place the rubber chicken on the platform. (Here shown by the grey box M since I can't draw rubber chickens.) Now release the apparatus, and measure how much time t it takes the mass m to fall a distance h. Find an equation for the rotational inertia I of the rubber chicken and platform, in terms of r, h, m, t, and anything else that comes in handy. (If you repeat the experiment without the rubber chicken, you would obtain the rotational inertia of the empty platform, which you could then subtract from your previous result to find the rotational inertia of the rubber chicken alone.)

Question 9: One of my colleagues in grad school gave his lab students the following experiment: A ball was rolled down a curving ramp from an initial height h. The ball then rolled across a horizontal section of the ramp, and shot o↵ the end of the ramp with some horizontal speed v and initial height o↵ the ground H. The ball would travel some horizontal distance x before hitting the ground.

The idea behind the lab was that the students could use h and the mass m of the ball to find the initial potential energy, equate this initial energy mgh to the kinetic energy ½mv² of the ball as it left the table, and find v. From v and H, the students could then use projectile motion methods to predict exactly how far the ball would go before hitting the ground. The problem was that the ball consistently hit at about 85% of the "correct" distance. Air resistance and other non-conservative forces were not su"cient to account for this discrepancy.

(a) What was my colleague forgetting? Why was there this systematic error?

(b) Show that a correct analysis gives an answer for x which is about 85% of the range predicted by the original analysis.

Question 10: A pair of masses (each with mass m) is arranged with a pulley (also of mass m, and radius r) on the frictionless ramp as shown in the figure below. What is the acceleration of this system?

Question 11: A Yo-Yo has a mass m, radius R, and spindle radius r. Assume the yo-yo os approximately a disk of radius R. What is the angular speed of the yo-yo after it has faller a distance L.