Q.The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds.

The problem involves angular kinematics under uniform angular acceleration. We use the rotational analogues of linear kinematic equations.

First, convert the given angular speeds from revolutions per minute (rpm) to radians per second (rad/s):

$\omega_0 = 1200 \text{ rpm} = 1200 \times \frac{2\pi \text{ rad}}{60 \text{ s}} = 40\pi \text{ rad/s}$

$\omega = 3120 \text{ rpm} = 3120 \times \frac{2\pi \text{ rad}}{60 \text{ s}} = 104\pi \text{ rad/s}$

The time interval is $t = 16 \text{ s}$.

1. To find the angular acceleration ($\alpha$), use the kinematic equation $\omega = \omega_0 + \alpha t$: $\alpha = \frac{\omega - \omega_0}{t} = \frac{104\pi \text{ rad/s} - 40\pi \text{ rad/s}}{16 \text{ s}} = \frac{64\pi \text{ rad/s}}{16 \text{ s}} = 4\pi \text{ rad/s}^2$.
2. To find the total number of revolutions, first calculate the angular displacement ($\theta$) using $\theta = \frac{(\omega_0 + \omega)}{2}t$: …

Let's understand the core concepts before diving into the calculations. This problem deals with rotational motion, which is very similar to the linear motion you might already be familiar with. Just as we have displacement, velocity, and acceleration in linear motion, we have their rotational counterparts: angular displacement ($\theta$), angular velocity ($\omega$), and angular acceleration ($\alpha$).

• Angular Velocity ($\omega$): This measures how fast an object is rotating or revolving. It's the rate of change of angular displacement. Its standard unit is radians per second (rad/s). The problem gives it in revolutions per minute (rpm), which we'll need to convert.
• Angular Acceleration ($\alpha$): This measures how quickly the angular velocity is changing. It's the rate of change of angular velocity. Its standard unit is radians per second squared (rad/s$^2$).
• Uniform Angular Acceleration: This means the angular acceleration is constant, allowing us to use a set of kinematic equations analogous to those for linear motion.

The key is to recognize the parallels between linear and rotational kinematics:

The kinematic equations for uniform angular acceleration are:

Now, let's solve the problem step-by-step.

1. Convert Initial and Final Angular Speeds to Standard Units

   The given angular speeds are in revolutions per minute (rpm). For calculations involving angular acceleration and time in seconds, it's crucial to convert these to radians per second (rad/s). Remember that one revolution is equal to $2\pi$ radians, and one minute is 60 seconds.

   • Initial angular speed ($\omega_0$):

     $\omega_0 = 1200 \text{ rpm} = 1200 \frac{\text{revolutions}}{\text{minute}} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}}$

     $\omega_0 = \frac{1200 \times 2\pi}{60} \text{ rad/s} = 40\pi \text{ rad/s}$

   • Final angular speed ($\omega$):

     $\omega = 3120 \text{ rpm} = 3120 \frac{\text{revolutions}}{\text{minute}} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}}$

     $\omega = \frac{3120 \times 2\pi}{60} \text{ rad/s} = 104\pi \text{ rad/s}$

   The time duration is given as $t = 16 \text{ s}$.

   Watch out

   Always ensure all units are consistent before performing calculations. Mixing rpm with seconds or radians with revolutions without conversion is a common source of errors.

2. Calculate Angular Acceleration ($\alpha$)

   We have the initial angular velocity ($\omega_0$), final angular velocity ($\omega$), and time ($t$). We can use the first kinematic equation: $\omega = \omega_0 + \alpha t$.

   $104\pi \text{ rad/s} = 40\pi \text{ rad/s} + \alpha (16 \text{ s})$

   $104\pi - 40\pi = 16\alpha$

   $64\pi = 16\alpha$

   $\alpha = \frac{64\pi}{16}$

   $\alpha = 4\pi \text{ rad/s}^2$

   This is the answer to part (i).

3. Calculate Total Angular Displacement ($\theta$) …

Concept: Rotational Kinematics with Uniform Angular Acceleration

Step 1: Convert the given angular speeds to rad/s

Time $t=16$ s.

Step 2: Find the angular acceleration using $\omega=\omega_0+\alpha t$

Step 3: Find the total angular displacement using the average-velocity form $\theta=\frac{(\omega_0+\omega)}{2}t$ …