Question

QUESTION NO. 5 &6 on
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Answer 1

Answer on Question #56595, Physics Mechanics Relativity

The flywheel of a large motor in a factory has mass $30\,\mathrm{kg}$ and moment of inertia $67.5\,\mathrm{kg}\cdot \mathrm{m}^2$ about rotation axis. The motor develops a constant torque of $600\,\mathrm{Nt}\cdot \mathrm{m}$, and the flywheel starts from rest.

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

(b) What is its angular velocity after making 4 revolutions?

(c) How much work is done by the motor during the first 4 revolutions?

(d) What is the average power output of the motor during the first 4 revolutions?

Solution

(a)

$$M = J \varepsilon$$

where $M = 600\,\mathrm{N}\cdot \mathrm{m}$ is a constant torque; $\varepsilon$ is the angular acceleration of the flywheel; $J = 67.5\,\mathrm{kg}\cdot \mathrm{m}^2$ is moment of inertia about rotation axis.

Then

$$\varepsilon = M / J = 600\,\mathrm{N}\cdot \mathrm{m} / 67.5\,\mathrm{kg}\cdot \mathrm{m}^2 \approx 8.89\,\mathrm{rad} / \mathrm{s}^2$$

(b)

Angular velocity:

$$\omega = \omega_0 + \varepsilon t$$ $$\omega_ {0} = 0 \text{ (flywheel starts from rest)}.$$

The angle of rotation

$$\varphi = \varphi_ {0} + \omega_ {0} t + \frac {1}{2} \varepsilon t ^ {2}$$

where $\omega_0 = 0$ and $\varphi_0 = 0$ (flywheel starts from rest).

4 revolutions $\varphi = 4\cdot 2\pi$, then from (3)

$$t = \sqrt {16 \pi / \varepsilon} = 4 \sqrt {\pi / \varepsilon} = 2.38s$$

Angular velocity after making 4 revolutions:

$$\omega = \varepsilon t = \varepsilon 4 \sqrt {\pi / \varepsilon} = 4 \sqrt {\pi \varepsilon} = 4 \sqrt {\pi \cdot 8.89} \approx 21.14rad / s$$

(c)

The work is done by the motor during the first 4 revolutions:

$$A = \frac {J \omega^ {2}}{2} - \frac {J \omega_ {0} ^ {2}}{2} = \frac {J \omega^ {2}}{2} - 0 = \frac {J \omega^ {2}}{2} = \frac {67.5 \cdot 8.89 ^ {2}}{2} = 2667.33J = 2.67kJ$$

(d)

The average power output of the motor during the first 4 revolutions:

$$\left\langle P \right\rangle = \frac {A}{t} = \frac {2667.33}{2.38} = 1120\text{Watt}$$

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