A particle moves in a circular path of radius `0.5 m` at a speed that uniformly increases. Find the angular acceleration of particle if its speed changes from `2.0m//s` to `4.0 m//s` in `4.0s`

Explanation

To find the angular acceleration of a particle moving in a circular path with a changing speed, we can follow these steps:

### Step 1: Identify the given values
- Initial speed \( v_i = 2.0 \, \text{m/s} \)
- Final speed \( v_f = 4.0 \, \text{m/s} \)
- Time interval \( t = 4.0 \, \text{s} \)
- Radius of the circular path \( r = 0.5 \, \text{m} \)

### Step 2: Calculate the change in speed
The change in speed (\( \Delta v \)) can be calculated as:
\[ \Delta v = v_f - v_i = 4.0 \, \text{m/s} - 2.0 \, \text{m/s} = 2.0 \, \text{m/s} \]

### Step 3: Calculate the tangential acceleration
Tangential acceleration (\( a_t \)) is defined as the change in speed divided by the time interval:
\[ a_t = \frac{\Delta v}{t} = \frac{2.0 \, \text{m/s}}{4.0 \, \text{s}} = 0.5 \, \text{m/s}^2 \]

### Step 4: Calculate the angular acceleration
Angular acceleration (\( \alpha \)) can be found using the formula:
\[ \alpha = \frac{a_t}{r} \]
Substituting the values we have:
\[ \alpha = \frac{0.5 \, \text{m/s}^2}{0.5 \, \text{m}} = 1.0 \, \text{rad/s}^2 \]

### Final Answer
The angular acceleration of the particle is:
\[ \alpha = 1.0 \, \text{rad/s}^2 \]

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