Question

A body of mass 0.2 kg is attached to a light string of length 1 m and revolved in a vertical circle. What is the minimum speed at the lowest point so that the body can complete the circular motion? (Take \( g = 10 \, \text{m/s}^2 \))

• A. 2 m/s
• B. 3.16 m/s
• C. 5 m/s
• D. 6.32 m/s

Hint:

Remember: When applying energy principles, the minimum speed corresponds to the condition when the tension in the string is zero at the lowest point.

Answer 1

The Correct Option is B

Minimum Speed for Circular Motion 

We are given the following information:

• Mass of the body \( m = 0.2 \, \text{kg} \)
• Length of the string \( L = 1 \, \text{m} \)
• Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)

Step 1: Recall the conditions for circular motion.

For a body to complete the circular motion, the tension in the string at the highest point should be greater than or equal to zero. At the highest point, the only forces acting on the body are the gravitational force \( mg \) and the tension \( T \), and they must provide the centripetal force to keep the body in circular motion.

At the lowest point, the body must have sufficient speed to maintain the circular motion, and the minimum speed occurs when the tension \( T \) at the highest point is zero. In other words, the gravitational force alone provides the necessary centripetal force at the highest point.

Step 2: Using the centripetal force equation at the highest point.

The centripetal force required at the highest point is:

\[ \frac{mv^2}{L} = mg \] where \( v \) is the velocity at the highest point, \( m \) is the mass, \( L \) is the length of the string (radius of the circular motion), and \( g \) is the acceleration due to gravity.

Step 3: Solve for the minimum speed.

From the equation above, we can solve for \( v \) (the minimum speed at the lowest point):

\[ \frac{mv^2}{L} = mg \quad \Rightarrow \quad v^2 = gL \quad \Rightarrow \quad v = \sqrt{gL} \]

Substitute the values:

\[ v = \sqrt{10 \times 1} = \sqrt{10} \approx 3.16 \, \text{m/s} \]

Conclusion:

The minimum speed at the lowest point for the body to complete the circular motion is approximately \( 3.16 \, \text{m/s} \).