Problem 41 - Master Resource Book in JEE Main Physics

Question

Here, \(\begin{aligned} n_{0} &=15 \mathrm{rps} ; n=5 \mathrm{rps} \\ \theta &=50 \text { revolution }=50 \times 2 \text { rad } \end{aligned}\) From, \(\begin{aligned} \omega^{2}-\omega_{0}^{2} &=2 \alpha \theta \\ \alpha &=\frac{\omega^{2}-\omega_{0}^{2}}{2 \theta}=\frac{4 \pi^{2}\left(n^{2}-n_{0}^{2}\right)}{2 \theta} \\ \alpha &=\frac{4 \pi^{2}\left(5^{2}-15^{2}\right)}{2 \times 5 \times 2 \pi}=-4 \pi \mathrm{rad} \mathrm{s}^{-} \end{aligned}\)

Step-by-Step Solution

Answer

The angular acceleration is \(-4\pi\,\text{radians/s}^2\).

Step 1: Write the Given Values

We are given that \( n_0 = 15 \, \text{rps} \) (initial revolutions per second) and \( n = 5 \, \text{rps} \) (final revolutions per second). The angle \( \theta \) is given as 50 revolutions, which can be converted to radians: \( \theta = 50 \times 2 \pi \).

Step 2: Convert Revolutions to Angular Velocity

Convert the given frequencies to angular velocities. Use the formula \( \omega = 2 \pi n \). Thus, \( \omega_0 = 2 \pi \times 15 = 30 \pi \) and \( \omega = 2 \pi \times 5 = 10 \pi \).

Step 3: Use the Equation of Angular Motion

The formula for angular motion is \( \omega^2 - \omega_0^2 = 2 \alpha \theta \). We will use this formula to solve for acceleration \( \alpha \).

Step 4: Plug in the Values

Substitute the known values into the formula: \( \alpha = \frac{\omega^2 - \omega_0^2}{2\theta} = \frac{(10\pi)^2 - (30\pi)^2}{2 \times 50 \times 2 \pi} \).

Step 5: Simplify and Solve for Acceleration

Calculate \( \omega^2 - \omega_0^2 \). We have \( (10\pi)^2 = 100\pi^2 \) and \( (30\pi)^2 = 900\pi^2 \), so \( 100\pi^2 - 900\pi^2 = -800\pi^2 \). Substitute into the formula for \( \alpha \): \( \alpha = \frac{-800\pi^2}{200\pi} = -4\pi \).

Step 6: Conclude the Result

Therefore, the angular acceleration \( \alpha \) is \(-4 \pi \) radians per second squared.

Key Concepts

Angular Velocity

Angular velocity is a measure of how fast something is rotating. It's like measuring how fast your car is going but instead of a speedometer reading in miles per hour, it measures how many angles or turns you are making in a given time. The formula for angular velocity is given by \( \omega = 2 \pi n \), where \( \omega \) is the angular velocity and \( n \) is the number of revolutions per second.
Angular velocity helps us understand rotations in a way that's similar to linear velocity for objects moving in straight lines. The faster the rotation, the higher the angular velocity. In application, this concept is crucial in fields such as mechanical engineering and physics, where understanding the precise rate of spin or rotation is necessary to solve problems and design functional systems.
- Initial angular velocity \( \omega_0 = 30\pi \)- Final angular velocity \( \omega = 10\pi \)

Radians per Second

Radians per second is a unit of measurement for angular velocity. It tells you how many radians an object sweeps through in one second. Radians are a fundamental way to measure angles, as there are \(2\pi\) radians in a full circle of 360 degrees.
Understanding radians per second is important because it allows us to describe rotational speeds without having to worry about the circumference or diameter of the circle involved. This standard unit simplifies calculations and is particularly helpful for consistency in scientific and engineering applications. For instance, when converting revolutions per second to radians per second, the formula \( \omega = 2\pi n \) effortlessly provides the angular velocity in radians per second.
- 1 revolution = \(2\pi\) radians
- Initial angular velocity \( \omega_0 = 30\pi \) rad/s
- Final angular velocity \( \omega = 10\pi \) rad/s

Kinematics Equations

Kinematics equations are used to describe the motion of objects. In rotational motion, they relate angular velocity, angular acceleration, and the angular displacement. One of the key kinematics equations for angular motion is \( \omega^2 - \omega_0^2 = 2 \alpha \theta \), where:

• \( \omega \) is the final angular velocity
• \( \omega_0 \) is the initial angular velocity
• \( \alpha \) is the angular acceleration
• \( \theta \) is the angular displacement

This equation helps us find unknown values related to the motion, such as angular acceleration \( \alpha \). In this situation, by rearranging the formula \( \alpha = \frac{\omega^2 - \omega_0^2}{2\theta} \), we substitute the known values and solve for \( \alpha \). It is ideal for applications where the motion is uniformly accelerating or decelerating, as seen in this problem with \( \theta = 50 \times 2\pi \) radians.
- Solve for \( \alpha \) using the known angles and velocities.
- Result shows \( \alpha = -4\pi \) rad/s².

Revolutions per Second

Revolutions per second (rps) is a straightforward measurement of rotational speed. It indicates how many complete rotations something makes in just one second. This unit is crucial when you want to understand how fast something is spinning without delving into angular measurements like radians.
Converting revolutions per second to radians per second is straightforward with the formula \( \omega = 2 \pi n \). In this exercise, converting \( n_0 = 15 \text{ rps} \) to angular velocity gives us \( \omega_0 = 30\pi \text{ rad/s} \). Similarly, \( n = 5 \text{ rps} \) results in \( \omega = 10\pi \text{ rad/s} \).
- Initial speed: 15 revolutions per second
- Final speed: 5 revolutions per second
This conversion is foundational in combining rotational speeds to kinematic problems, where angular velocities and accelerations are discussed in terms of radians per second.