Question

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

• A. $5\ \text{rad/s}$
• B. $1\ \text{rad/s}$
• C. $2.5\ \text{rad/s}$
• D. $\text{zero}\ \text{rad/s}$

Hint:

Think about the physical nature of harmonic oscillations! A sine wave displacement function reaches its maximum peak value when its internal angle equals $\frac{\pi}{2}$. At $t=3$, the angle is $\frac{3\pi}{6} = \frac{\pi}{2}$, meaning the object has reached its extreme turning point. Since an object momentarily stops to change direction at any extreme turning point, its velocity must equal zero!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
An object moves along a circular path with an angular displacement ($\theta$) that varies as a sinusoidal function of time $t$. We need to calculate its instantaneous angular velocity ($\omega$) at the exact moment when $t = 3\ \text{seconds}$.

Step 2: Key Formula or Approach:
The instantaneous angular velocity ($\omega$) is defined as the first derivative of the angular displacement function with respect to time: $$\omega = \frac{d\theta}{dt}$$ Using the chain rule for trigonometric differentiation: $$\frac{d}{dt}[\sin(at)] = a\cos(at)$$

Step 3: Detailed Explanation:
Let's differentiate the given displacement equation $\theta = 5 \sin\left(\frac{\pi t}{6}\right)$ with respect to time $t$: $$\omega = \frac{d}{dt} \left[ 5 \sin\left(\frac{\pi t}{6}\right) \right]$$ $$\omega = 5 \times \frac{\pi}{6} \cos\left(\frac{\pi t}{6}\right) = \frac{5\pi}{6} \cos\left(\frac{\pi t}{6}\right)$$ Now, substitute the target timestamp value $t = 3\ \text{seconds}$ into this angular velocity equation: $$\omega = \frac{5\pi}{6} \cos\left(\frac{\pi \times 3}{6}\right)$$ Simplify the fraction inside the cosine argument: $$\omega = \frac{5\pi}{6} \cos\left(\frac{\pi}{2}\right)$$ We are given that $\cos\left(\frac{\pi}{2}\right) = 0$. Substituting this value yields: $$\omega = \frac{5\pi}{6} \times 0 = 0\ \text{rad/s}$$

Step 4: Final Answer:
The angular velocity of the body at $t = 3\ \text{seconds}$ is $\text{zero}\ \text{rad/s}$, which corresponds to option (D).