Problem 4 - Master Resource Book in JEE Main Physics

Question

As, \(\omega_{2}=\omega_{1}+\alpha t\) \(\therefore \quad 40 \pi=20 \pi+\alpha \times 10\) or \(\alpha=2 \pi \mathrm{rad} \mathrm{s}^{-2}\) \(\begin{aligned}&\text { From, } \quad \omega_{2}^{2}-\omega_{1}^{2} & =2 \alpha \theta \\\& & (40 \pi)^{2}-(20 \pi)^{2} & =2 \times 2 \pi \theta \\\&\Rightarrow \quad \theta & =\frac{1200 \pi^{2}}{4 \pi}=300 \pi\end{aligned}\) Number of rotations completed \(=\frac{\theta}{2 \pi}=\frac{300}{2 \pi}=150\)

Step-by-Step Solution

Answer

150 rotations

Step 1: Start with the Given Equation

We are given the equation \( \omega_{2} = \omega_{1} + \alpha t \). Substitute the given values into this equation: \( 40 \pi = 20 \pi + \alpha \times 10 \).

Step 2: Solve for Angular Acceleration (\(\alpha\))

Rearrange the equation to solve for \( \alpha \): \( \alpha \times 10 = 40 \pi - 20 \pi \). This simplifies to \( \alpha \times 10 = 20 \pi \). Divide both sides by 10 to find \( \alpha = 2 \pi \text{ rad/s}^2 \).

Step 3: Use the Second Angular Motion Equation

Apply the equation \( \omega_{2}^{2} - \omega_{1}^{2} = 2 \alpha \theta \) with known values: \((40 \pi)^{2} - (20 \pi)^{2} = 2 \times 2 \pi \times \theta \).

Step 4: Simplify and Solve for \(\theta\)

Expand both sides: \(1600\pi^2 - 400\pi^2 = 4 \pi \theta\). This is \(1200\pi^2 = 4 \pi \theta\). Solve for \(\theta\) by dividing both sides by \(4\pi\): \(\theta = \frac{1200\pi^2}{4\pi} = 300\pi\).

Step 5: Calculate Number of Rotations

Rotations completed are given by \(\frac{\theta}{2\pi}\). Fill in the \(\theta\) value: \(\frac{300\pi}{2\pi} = 150\).

Step 6: Conclusion

The number of rotations completed is 150.

Key Concepts

Angular Acceleration

Angular acceleration (\( \alpha \)) is the rate at which angular velocity changes with time.
This concept is central to analyzing rotational motion.
Just like linear acceleration in everyday motion, angular acceleration helps us understand how quickly an object speeds up or slows down its rotation.To calculate angular acceleration, we use the formula:

• \[ \omega_{2} = \omega_{1} + \alpha t \]

Angular velocity is measured in radians per second, and time is in seconds.
By rearranging the formula, angular acceleration can be isolated as:

• \[ \alpha = \frac{\omega_{2} - \omega_{1}}{t} \]

In our exercise, the initial angular velocity is given as \(20 \pi\) radians per second and the final as \(40 \pi\) radians per second over a period of 10 seconds.
Substituting these values, we obtained an angular acceleration of \(2 \pi \text{ rad/s}^2\).The positive value here indicates that the angular velocity is increasing.

Angular Displacement

Angular displacement (\( \theta \)) represents the angle in radians through which a point or line has been rotated in a specified sense about a specified axis.
This is crucial for determining how far an object has turned during its motion.The equation used here is:

• \[ \omega_{2}^{2} - \omega_{1}^{2} = 2 \alpha \theta \]

This equation links the change in angular velocity to the angular displacement and acceleration.
Solving, we find:

• \(\theta = \frac{1200 \pi^2}{4 \pi} = 300 \pi\text{ radians}\)

Given that \(2\pi\) radians is one complete rotation, the displacement tells us how far the object rotates overall.

Number of Rotations

The total number of rotations (\( n \)) is how many times the object completes a full circle during its motion.
It's directly linked to angular displacement, as each full rotation equals \(2\pi\) radians.We calculate the number of rotations using:

• \[ n = \frac{\theta}{2\pi} \]

This equation measures how many full circles are covered by the angle \(\theta\).
In our solution, \(\theta = 300 \pi\), leading to:

• \[ n = \frac{300 \pi}{2 \pi} = 150 \]

This tells us that the object rotates 150 times.
Understanding the number of rotations helps in visualizing the extent of circular movement.