A solid cylinder is mounted above the ground with its axis of rotation oriented

Question

A solid cylinder is mounted above the ground with its axis of rotation oriented horizontally. A rope is wound around the cylinder and its free end is attached to a block of mass 74.0 kg that rests on a platform. The cylinder has a mass of 215 kg and a radius of 0.430 m. Assume that the cylinder can rotate about its axis without any friction and the rope is of negligible mass. The platform is suddenly removed from under the block. The block falls down toward the ground and as it does so, it causes the rope to unwind and the cylinder to rotate.

(a) What is the angular acceleration of the cylinder?

(b) How many revolutions does the cylinder make in 5.00 s?

(c) How much of the rope unwinds in this time interval?

Explanation / Answer

Since the bucket is falling, the net force is equal to the weight of the bucket minus the tension.

m * g – T = m * a
74 * 9.8 – T = 74 * a
T = 725.2 – 74 * a
Since we need to the determine the angular acceleration of the cylinder, let’s use the following equation to convert a to . a = * r = * 0.43

T = 725.2 – 74 * * 0.43
Eq#1: T = 725.2 – 31.82 *

Let’s use the following equations to determine the torque on the cylinder.

Torque = Force * distance, and Torque = I *
F * d = I *

If you look at d, you will see that the tension is the force that is producing the torque in the cylinder. The distance is the radius of the cylinder.

F * d = T * 0.43
For a solid cylinder, I = ½ * m * r^2 = ½ * 215 * 0.43^2 = 19.88
Torque = 19.88 *

T * 0.43 = 19.88 *
Solve for T
Eq#2: T = 46.23 *

To determine the angular acceleration of the cylinder, set Eq#2 equal to Eq#1.

46.23 * = 725.2 – 31.82 *
= 725.2 ÷ 78.05 = 9.29 rad/s^2


T = 725.2 – 31.82 *9.29 = 429.5
T = 46.23 * 9.29 = 429.5

b)
Let’s use the angular acceleration in the following equation to determine the angle the cylinder rotates in 5 seconds.

f = i + ½ * * t^2, i = 0
f = ½ * 9.29 * 5^2 = 116.125 radians

During one revolution, the cylinder rotates an angle of 2. To determine the number of revolutions, divide the angle by 2.

Number = 116.125 ÷ 2 = 18.48

c)
To determine this distance, use the following equation.

d = vi * t + ½ * a * t^2, vi = 0, = * r = (9.29) * 0.43 = 3.99
d = ½ * 3.99 * 5^2 = 49.93 meters

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