Question 287: Work, Energy and Power & Circular Motion

Question

The angular displacement of body performing circular motion is given by $$	heta=5 \sin \frac{\pi t}{6}$$. The angular velocity of the body at $$t=3$$ second will be $$\left[\sin \frac{\pi}{2}=1, \cos \frac{\pi}{2}=0ight]$$

Accepted Answer

The given angular displacement of the body performing circular motion is expressed as:
$$	heta = 5 \sin \frac{\pi t}{6}$$
To determine the angular velocity, we need to differentiate the angular displacement $$	heta$$ with respect to time $$t$$. The angular velocity $$\omega$$ is given by:
$$\omega = \frac{d	heta}{dt}$$
Let's find the derivative of $$	heta = 5 \sin \frac{\pi t}{6}$$ with respect to $$t$$.
First, apply the chain rule to differentiate the sine function:
$$\frac{d	heta}{dt} = 5 \cdot \frac{d}{dt} \left( \sin \frac{\pi t}{6} ight)$$
Next, differentiate $$\sin \frac{\pi t}{6}$$ with respect to $$t$$. Using the chain rule:
$$\frac{d}{dt} \left( \sin \frac{\pi t}{6} ight) = \cos \frac{\pi t}{6} \cdot \frac{d}{dt} \left( \frac{\pi t}{6} ight)$$
$$= \cos \frac{\pi t}{6} \cdot \frac{\pi}{6}$$
Now, substituting this back into the expression for $$\frac{d	heta}{dt}$$:
$$\frac{d	heta}{dt} = 5 \cdot \cos \frac{\pi t}{6} \cdot \frac{\pi}{6}$$
$$\frac{d	heta}{dt} = \frac{5\pi}{6} \cos \frac{\pi t}{6}$$
We need to evaluate the angular velocity at $$t = 3$$ seconds:
$$\omega(t=3) = \frac{5\pi}{6} \cos \left( \frac{\pi \cdot 3}{6} ight)$$
$$\omega(t=3) = \frac{5\pi}{6} \cos \left( \frac{\pi}{2} ight)$$
Given that $$\cos \frac{\pi}{2} = 0$$:
$$\omega(t=3) = \frac{5\pi}{6} \cdot 0$$
$$\omega(t=3) = 0$$
Therefore, the angular velocity of the body at $$t = 3$$ seconds is:
Option D: zero $$\frac{\mathrm{rad}}{\mathrm{s}}$$

Answer

$$5 \frac{\mathrm{rad}}{\mathrm{s}}$$

Answer

$$1 \frac{\mathrm{rad}}{\mathrm{s}}$$

Answer

$$2.5 \frac{\mathrm{rad}}{\mathrm{s}}$$

Answer

zero $$\frac{\mathrm{rad}}{\mathrm{s}}$$

Work, Energy and Power & Circular Motion

The angular displacement of body performing circular motion is given by $\theta=5 \sin \frac{\pi t}{6}. The angular velocity of the body at t=3 second will be \left[\sin \frac{\pi}{2}=1, \cos \frac{\pi}{2}=0\right]

Choose the correct answer: