A particle in a circular motion covers a distance of 0.5 `pi r` where r is the radius of the circular path. What is the angular displacement ?

Answer

To solve the problem of finding the angular displacement of a particle in circular motion that covers a distance of \(0.5 \pi r\), we can follow these steps:

### Step 1: Understand the relationship between linear distance and angular displacement
The relationship between linear distance \(s\) traveled by a particle in circular motion and angular displacement \(\theta\) (in radians) is given by the formula:
\[ s = r \theta \]
where \(s\) is the linear distance, \(r\) is the radius of the circular path, and \(\theta\) is the angular displacement in radians.

### Step 2: Substitute the given values into the formula
In this problem, we are given that the linear distance \(s\) is \(0.5 \pi r\). We can substitute this value into the formula:
\[ 0.5 \pi r = r \theta \]

### Step 3: Simplify the equation
Now, we can simplify the equation by dividing both sides by \(r\) (assuming \(r \neq 0\)):
\[ 0.5 \pi = \theta \]

### Step 4: Write the final answer
Thus, the angular displacement \(\theta\) is:
\[ \theta = 0.5 \pi \text{ radians} \]

### Summary
The angular displacement of the particle is \(0.5 \pi\) radians.

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