Shown in the figure below is a 85 kg cylinder with a 25 cm radius and two 4.5 kg

Question

Shown in the figure below is a 85 kg cylinder with a 25 cm radius and two 4.5 kg masses rotating around a central axis. Treat the two masses as point masses a distance of 80 cm from the rotational axis. The initial rate of rotation is 1 revolution every 2 seconds. The point masses are then brought into the cylinder, making a 94 kg rotating cylinder. How many rotations per second occur with the new system configuration? What is the difference in rotational kinetic energy from the initial configuration to the final? Its important to note that the total energy of the system is still conserved, in this very real situation the chemical energy from food was converted into muscle work in order to draw the weights into the cylinder resulting in a change in kinetic energy.

Explanation / Answer

Here ,

initial moment of inertia , I1 = m * r^2 * 0.5 + 2 * 4.5 * 0.80^2

I1 = 0.5 * 85 * 0.25^2 + 2 * 4.5 * 0.80^2 = 8.42 Kg.m^2

I2 = 0.5 * m * r^2

I2 = 0.5 * 94 * 0.25^2

I2 = 2.94 Kg.m^2

a) let the final angular speed is w2

Using conservation of angular momentum

I1 * w1 = I2 * w2

8.42 * (1/2) = 2.94 * w2

w2 = 1.433 rev/s

the rotations per second is 1.433 rev/s

b)

difference in rotational kinetic energy = -0.5 * I1 * w1^2 + 0.5 * I2 * w2^2

difference in rotational kinetic energy = 0.5 * 2.94 * (1.433 * 2pi)^2 - 0.5 * 8.42 * (0.5 * 6.282)^2

difference in rotational kinetic energy = 77.6 J

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