Question

A body of mass \(m\) is moving along a circle of radius \(r\) with linear speed \(v\). Now, to change the linear speed to \(\dfrac{v}{2}\) and to move it along the circle of radius \(4r\), required change in the centripetal force of the body is

• A. decrease by \(\dfrac{15}{16}\)
• B. increase by \(\dfrac{11}{16}\)
• C. increase by \(\dfrac{9}{16}\)
• D. decrease by \(\dfrac{5}{16}\)

Hint:

Centripetal force depends on both the square of speed and inverse of radius.

Answer 1

The Correct Option is A

Step 1: Write initial centripetal force.
\[ F_1 = \frac{mv^2}{r} \]
Step 2: New speed and radius.
\[ v' = \frac{v}{2}, \quad r' = 4r \]
Step 3: Write new centripetal force.
\[ F_2 = \frac{m(v/2)^2}{4r} \]
Step 4: Simplify the expression.
\[ F_2 = \frac{mv^2}{16r} \]
Step 5: Compare the two forces.
\[ \frac{F_2}{F_1} = \frac{1}{16} \]
Step 6: Find the change in force.
Decrease in force =
\[ F_1 - F_2 = \left(1 - \frac{1}{16}\right)F_1 = \frac{15}{16}F_1 \]
Step 7: Conclusion.
The centripetal force decreases by \(\dfrac{15}{16}\) of its original value.