Problem 29 - Master Resource Book in JEE Main Physics

Question

A fan is making 600 revolution per minute. If after some time it makes 1200 revolution per minute, then increase in its angular velocity is (a) \(10 \pi \mathrm{rad} / \mathrm{s}\) (b) \(20 \pi \mathrm{rad} / \mathrm{s}\) (c) \(40 \pi \mathrm{rad} / \mathrm{s}\) (d) \(60 \pi \mathrm{rad} / \mathrm{s}\)

Step-by-Step Solution

Answer

The increase in angular velocity is \(20 \pi \mathrm{rad/s}\).

Step 1: Understanding Angular Velocity

Angular velocity, denoted by \( \omega \), is the rate of change of the angle through which an object rotates. It is measured in radians per second (\( \mathrm{rad/s} \)). One revolution corresponds to \( 2\pi \) radians.

Step 2: Convert to Radians Per Second

To find the angular velocity in \( \mathrm{rad/s} \), convert revolutions per minute (rpm) to radians per second. Given: \[ \omega_1 = 600 \text{ rpm} \] Convert to rad/s: \[ \omega_1 = \frac{600 \text{ rev}}{1 \text{ min}} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} \] \[ \omega_1 = 600 \times \frac{2\pi}{60} \text{ rad/s} = 20 \pi \text{ rad/s} \]

Step 3: Calculate New Angular Velocity

The new angular velocity after the increase is 1200 rpm: \[ \omega_2 = 1200 \text{ rpm} \] Convert to rad/s: \[ \omega_2 = \frac{1200 \text{ rev}}{1 \text{ min}} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} \] \[ \omega_2 = 1200 \times \frac{2\pi}{60} \text{ rad/s} = 40 \pi \text{ rad/s} \]

Step 4: Determine Increase in Angular Velocity

The increase in angular velocity is the difference between the new and the initial angular speeds: \[ \Delta \omega = \omega_2 - \omega_1 \] \[ \Delta \omega = 40\pi \text{ rad/s} - 20\pi \text{ rad/s} = 20\pi \text{ rad/s} \]

Key Concepts

Radians Per Second

When we talk about angular velocity, it is often expressed in terms of radians per second. But what does this mean? Imagine a circular path: every full revolution, a point on the path travels a distance that equates to the circle's circumference. In terms of a circle, this corresponds to an angular distance of \(2\pi\) radians. Radians are a unit of measure based on the radius of a circle. So, one full revolution is equivalent to rotating through an angle of \(2\pi\) radians. This relationship allows us to translate rotational speeds from more intuitive measures, like revolutions per minute (rpm), to this standard unit of radians per second.

Conversion of Units

To properly use angular velocity in calculations, often we need to convert from revolutions per minute to radians per second. This conversion is crucial for mathematical modeling and ensures consistency with SI units. The conversion process can be broken down into simple steps:

• Recognize that 1 revolution equals \(2\pi\) radians.
• Adjust time units from minutes to seconds by dividing by 60, since there are 60 seconds in a minute.

For instance, if a fan is rotating at 600 rpm, converting this to radians per second involves multiplying 600 by \(2\pi\), and then dividing by 60. This gives us an angular velocity of \(20\pi\) rad/s.

Revolutions Per Minute

Revolutions per minute, often abbreviated as rpm, is a common unit used to describe the speed of rotation. It's intuitive because it directly relates to how many complete turns an object makes every minute. This measure is frequently used in various fields, from mechanics to music, making it an essential concept for understanding rotational dynamics. By knowing the rpm of an object, such as our fan initially at 600 rpm, we gain insight into how quickly it’s spinning. However, to delve into more technical calculations, converting from rpm to radians per second becomes necessary.

Increase in Speed

Sometimes, machines accelerate, changing their angular velocity. Calculating the increase in angular velocity is a key part of dynamic analysis. It shows how much faster an object is spinning after a certain period. To find the increase, you calculate the difference between the new and the original angular velocities. For example, if a fan initially spins at 600 rpm and speeds up to 1200 rpm, we first convert both values to radians per second (\(20\pi\) rad/s and \(40\pi\) rad/s, respectively). The increase in speed is simply: \[\Delta\omega = 40\pi - 20\pi = 20\pi \text{ rad/s}\] This difference indicates that the fan's speed in terms of angular velocity increased by \(20\pi\) radians per second.