An object moves in a circular path of radius 0.5m with a speed of 1ms$^{-1}$. What is its angular velocity?

1. Analyze the question

We are given:

• Radius of the circular path ($r$) = $0.5\text{ m}$
• Linear speed ($v$) = $1\text{ ms}^{-1}$
• We need to find the angular velocity ($\omega$).

2. Explain the concept

In circular motion, there is a direct relationship between the linear speed ($v$) of an object moving along the circumference and its angular velocity ($\omega$). This relationship is defined by the formula:

Where:

• $v$ is the linear velocity (in $\text{ms}^{-1}$)
• $r$ is the radius (in $\text{m}$)
• $\omega$ is the angular velocity (in $\text{rads}^{-1}$)

To find the angular velocity, we rearrange the formula to:

3. Show the solution

Using the values provided:

• $v = 1\text{ ms}^{-1}$
• $r = 0.5\text{ m}$

Substitute these into the formula:

Since $0.5$ is the same as $\frac{1}{2}$:

4. Identify the answer

Comparing our result to the given options: A. $8\text{ rads}^{-1}$ B. $4\text{ rads}^{-1}$ C. $1\text{ rads}^{-1}$ D. $2\text{ rads}^{-1}$

The correct option is D.

5. Tip

When dealing with circular motion problems, always check your units first. If the radius is in centimeters, convert it to meters before calculating to ensure your answer is in the standard SI unit ($\text{rads}^{-1}$). Also, remember the "V-R-Omega" ($v = r\omega$) mnemonic to quickly recall this relationship during your exam!