Question

If the angular velocity of a disc depends on an angle rotated as ω=(θ²+θ), then its angular acceleration at θ=1 rad is

• A. $3rad/s^2$
• B. $6rad/s^2$
• C. $2rad/s^2$
• D. $9rad/s^2$

Answer 1

Answer: (A), (B)

$6rad/s^2$

Step1: Given data

$$\begin{gathered} \omega =({\theta }^2+\theta ) \\ \mathrm{where}\mathrm{\omega }\mathrm{is}\mathrm{angluar}\mathrm{velocity},\mathrm{\theta }\mathrm{angle}\mathrm{rotated} \end{gathered}$$

Step2: Angular acceleration

1. The angular acceleration of a moving object is defined as the rate at which its angular velocity changes over time.
2. We define normal acceleration as the rate of change of velocity in the same way. However, only rotating objects are affected by angular acceleration.
3. Thus, angular velocity is the velocity of an object traveling in a circular motion.

Step3: Formula used:

Angular acceleration is defined as the rate of change of angular velocity with a time of an object. $\alpha =\frac{d\omega }{dt}$

Step4: Analyzing angular acceleration

Now we can write the above partial differentiation in terms $\theta$ so that we can substitute the given $\theta$ value.

$\alpha =\frac{d\omega }{dt}\times \frac{d\theta }{d\theta }..........\left(i\right)$

In the above equation $\frac{d\theta }{dt}$ is called the rate of change of angular displacement which is equal to the angular velocity $\omega$.

It is given to us that $\omega =({\theta }^2+\theta )$. Therefore we can substitute this in the angular acceleration formula.

$\alpha =\frac{d\omega }{d\theta }\times ({\theta }^2+\theta ).........\left(ii\right)$

Step5: Calculating $\frac{d\omega }{d\theta }$

$$\begin{gathered} \frac{d\omega }{d\theta }=\frac{d}{d\theta }({\theta }^2+\theta ) \\ \frac{d\omega }{d\theta }=2\theta +1 \end{gathered}$$

Step6: Calculating the angular velocity of a disc

Substituting the value of $\frac{d\omega }{d\theta }$ in equation (ii)

$\alpha =(2\theta +1)\times ({\theta }^2+\theta ).............\left(iv\right)$

We have to find the value of the angular acceleration of the particle $\alpha$ at $\theta =1$

Substituting $\theta =1$ in equation (4)

$$\begin{gathered} \alpha =(2\times 1+1)\times (1^2+1) \\ \alpha =3\times 2 \\ \alpha =6rad/s^2 \end{gathered}$$

Hence, option B is the correct answer.