Problem 666 - JEE Physics

Question

A wheel is subjected to uniform angular acceleration about its axis. Initially its angular velocity is zero. In the first two second it rotate through an angle \(\theta_{1}\), in the next \(2 \mathrm{sec}\). it rotates through an angle \(\theta_{2}\), find the ratio \(\left(\theta_{2} / \theta_{1}\right)=\) \(\\{\mathrm{A}\\} 1\) \(\\{B\\} 2\) \(\\{\mathrm{C}\\} 3\) \(\\{\mathrm{D}\\} 4\)

Step-by-Step Solution

Answer

The ratio \(\left(\theta_{2} / \theta_{1}\right) = 3\), which corresponds to option C.

Step 1: Find the angle θ₁

To find the angle \(\theta_1\) rotated within the first 2 seconds, we use the equation of motion mentioned above. Since the initial angular velocity \(\omega_0\) is zero, we can simplify the equation to: $$\theta_1 = \frac{1}{2} \alpha t^2$$ Where \(\alpha\) is the angular acceleration, and \(t = 2\,\text{sec}\).

Step 2: Find the angle θ₂

To find the angle \(\theta_2\) rotated within the next 2 seconds, first calculate the angular velocity \(\omega\) at the end of the first 2 seconds using the formula: $$\omega = \omega_0 + \alpha t$$ Since \(\omega_0 = 0\), the formula becomes \(\omega = \alpha t\). Now, we have to calculate the angle rotated in the next 2 seconds with the given angular velocity \(\omega\) and angular acceleration \(\alpha\). Again, we apply the equation of motion mentioned above: $$\theta_2 = \omega t + \frac{1}{2}\alpha t^2$$ Replacing \(\omega\) in the above equation, we get: $$\theta_2 = \alpha t \cdot t + \frac{1}{2}\alpha t^2$$

Step 3: Find the ratio of angles θ₂ and θ₁

To find the ratio \(\left(\frac{\theta_2}{\theta_1}\right)\), simply divide the equations of Step 2 by Step 1: $$\frac{\theta_2}{\theta_1} = \frac{\alpha t \cdot t + \frac{1}{2}\alpha t^2}{\frac{1}{2} \alpha t^2}$$ By simplification, we get: $$\frac{\theta_2}{\theta_1} = \frac{2 \alpha t \cdot t + \alpha t^2}{\alpha t^2}$$ The term \(\alpha\) can be canceled out from both the numerator and denominator. Further simplification provides: $$\frac{\theta_2}{\theta_1} = \frac{2t \cdot t + t^2}{t^2} = \frac{3t^2}{t^2} = 3$$ Thus, the ratio \(\left(\theta_{2} / \theta_{1}\right) = 3\), which corresponds to option C.

Key Concepts

Angular Velocity

Angular velocity refers to the speed at which an object rotates around an axis. It is a vital concept in understanding rotational motion. Imagine a spinning wheel—angular velocity measures how fast it spins.
Typically denoted by \( \omega \), angular velocity is expressed in radians per second. If a wheel completes a rotation faster, it has a higher angular velocity. If it takes longer, its angular velocity is lower.
In our problem, the initial angular velocity \( \omega_0 \) is zero, as the wheel starts from rest. Its angular velocity at the end of 2 seconds can be calculated using \( \omega = \alpha t \). This demonstrates how angular velocity changes over time when an object is accelerating.

Equation of Motion

The equation of motion for rotational systems is somewhat analogous to linear motion but involves terms specific to rotation. It allows us to calculate aspects like rotational displacement or angle, using variables such as angular acceleration \( \alpha \), angular velocity \( \omega \), and time \( t \).
The basic form used in rotational motion is \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \). This calculates the angle rotated over a time \( t \) based on the initial angular velocity and angular acceleration.
For the problem at hand, we used this equation to derive \( \theta_1 \) and \( \theta_2 \) for different time intervals, aiding in understanding how much the wheel rotates as time progresses.

Rotational Motion

Rotational motion occurs when an object spins or rotates around a point or axis. It's a form of motion observable in many everyday phenomena, like a spinning top or the rotation of Earth.
This type of motion is described by a similar set of physical laws and equations as linear motion, but they are adapted for circular paths. Key variables include:

• Angular velocity \( \omega \)
• Angular acceleration \( \alpha \)
• Rotational displacement or angle \( \theta \)

Understanding how these factors interplay is crucial in analyzing problems involving rotational motion, like our exercise with the wheel. The systematic change due to uniform angular acceleration exemplifies how rotational dynamics work.

Ratio of Angles

In rotational motion problems, comparing angles or other aspects between different time frames helps understand the dynamics. This is evident in finding the ratio \( \frac{\theta_2}{\theta_1} \).
The ratio provides insight into how rotation scales with time under uniform acceleration. In the exercise, calculating \( \frac{\theta_2}{\theta_1} \) required breaking down the motion using the rotational equations. Simplifying this ratio shows how angular displacement varies due to the acceleration.\br>Applied correctly, this method reveals details like the constant factor change between different intervals, helping predict and understand future movements. Here, obtaining the ratio 3 means in the next section, the wheel covers three times the angle compared to the first section.