The angular velocity of a particle moving in a circle of radius `50 cm` is increased in 5 min from `100` revolutions per minute to `400` revolutions per minute. Find the tangential acceleration of the particle.

Options

A. `60 m s^-2`
B. `pi//30 m s^-2`
C. `pi//15 m s^-2`
D. `pi//60 m s^-2`

Explanation

To solve the problem step by step, we will follow the given information and apply the necessary formulas.

### Step 1: Convert initial and final angular velocities to radians per second
The initial angular velocity (\( \omega_i \)) is given as 100 revolutions per minute (rpm). To convert this to radians per second, we use the conversion factor \( 2\pi \) radians per revolution and divide by 60 seconds per minute.

\[ \omega_i = 100 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = \frac{100 \times 2\pi}{60} = \frac{100\pi}{30} = \frac{10\pi}{3} \, \text{rad/s} \]

The final angular velocity (\( \omega_f \)) is given as 400 revolutions per minute. We convert this similarly:

\[ \omega_f = 400 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = \frac{400 \times 2\pi}{60} = \frac{400\pi}{30} = \frac{40\pi}{3} \, \text{rad/s} \]

### Step 2: Calculate the angular acceleration (\( \alpha \))
The angular acceleration can be found using the formula:

\[ \omega_f = \omega_i + \alpha t \]

We know that the time \( t \) is 5 minutes, which we convert to seconds:

\[ t = 5 \, \text{min} \times 60 \, \text{s/min} = 300 \, \text{s} \]

Now substituting the values into the equation:

\[ \frac{40\pi}{3} = \frac{10\pi}{3} + \alpha \times 300 \]

Rearranging gives:

\[ \alpha \times 300 = \frac{40\pi}{3} - \frac{10\pi}{3} = \frac{30\pi}{3} = 10\pi \]

Now, solving for \( \alpha \):

\[ \alpha = \frac{10\pi}{300} = \frac{\pi}{30} \, \text{rad/s}^2 \]

### Step 3: Calculate the tangential acceleration (\( a_t \))
The tangential acceleration can be calculated using the formula:

\[ a_t = r \alpha \]

Where \( r \) is the radius. The radius is given as 50 cm, which we convert to meters:

\[ r = 50 \, \text{cm} = 0.5 \, \text{m} \]

Now substituting the values:

\[ a_t = 0.5 \, \text{m} \times \frac{\pi}{30} \, \text{rad/s}^2 = \frac{0.5\pi}{30} = \frac{\pi}{60} \, \text{m/s}^2 \]

### Final Answer
The tangential acceleration of the particle is:

\[ \boxed{\frac{\pi}{60} \, \text{m/s}^2} \]