A student of mass 36.1 kg wants to measure the mass of a playground merry-go-rou

Question

A student of mass 36.1 kg wants to measure the mass of a playground merry-go-round, which consists of a solid metal disk of radius R = 1.50 m that is mounted in a horizontal position on a low-friction axle. She tries an experiment: She runs with speed v = 5.05 m/s toward the outer rim of the merry-go-round and jumps on to the outer rim, as shown in the figure. The merry-go-round is initially at rest before the student jumps on and rotates at 1.30 rad/s immediately after she jumps on. You may assume that the student’s mass is concentrated at a point.

a) What is the mass of the merry-go-round?

b) If it takes 51.1 s for the merry-go-round to come to a stop after the student has jumped on, what is the absolute value of the average torque due to friction in the axle?

c) How many times does the merry-go-round rotate before it stops, assuming that the torque due to friction is constant?

Explanation / Answer

given,

mass of student m = 36.1 kg

radius of the disk r = 1.5 m

speed of the student = 5.05 m/s

initial speed speed of the merry go round = 1.3 rad/sec

by conservation of angular momentum

initial angular momentum = final angular momentum

initial angular momentum = m * v * r

final angular momentum = (m * r^2 + 0.5 * M * r^2) * initial angular velocity

m * v * r = (m * r^2 + 0.5 * M * r^2) * initial angular velocity

36.1 * 5.05 * 1.5 = (36.1 * 1.5^2 + 0.5 * M * 1.5^2) * 1.3

mass of the merry-go-round = 114.779 kg

final angular velocity = 0 rad/sec

time = 51.1 sec

angular acceleration = final angular velocity - initial angular velocity / time

angular acceleration = 0 - 1.3 / 51.1

angular acceleration = -0.02544 rad/sec^2

torque = moment of inertia * angular acceleration

torque = (36.1 * 1.5^2 + 0.5 * 114.779 * 1.5^2) * (-0.02544)

average torque due to friction in the axle = -5.351 Nm

by third equation of motion

final velocity^2 = initial velocity^2 + 2 * acceleration * s

0 = 1.3^2 + 2 * (-0.02544) * s

s = 33.2154 rad

since in one rotation 2 * pi distance is moved so,

number of rotation * 2 * pi = 33.2154

number of rotation = 5.2864

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