Question

Two cars of masses \( m_1 \) and \( m_2 \) are moving in circles of radii \( r_1 \) and \( r_2 \) respectively. Their speeds are such that they make complete circles in the same time \( t \). The ratio of their centripetal force is

• A. \( m_1 : m_2 \)
• B. \( r_1 : r_2 \)
• C. 1 : 1
• D. \( m_1 r_1 : m_2 r_2 \)

Hint:

When objects move in circles with the same time period, their centripetal forces are proportional to the product of their mass and radius.

Answer 1

The Correct Option is D

Step 1: Understanding centripetal force.
The centripetal force \( F_c \) acting on an object moving in a circle is given by the formula: \[ F_c = \frac{mv^2}{r} \] where \( m \) is the mass, \( v \) is the velocity, and \( r \) is the radius.
Step 2: Considering the same time period.
Since the two cars complete the circles in the same time \( t \), their velocities are related by: \[ v_1 = \frac{2 \pi r_1}{t}, \quad v_2 = \frac{2 \pi r_2}{t} \] Thus, the centripetal forces for the two cars are: \[ F_{c1} = \frac{m_1 \left( \frac{2 \pi r_1}{t} \right)^2}{r_1}, \quad F_{c2} = \frac{m_2 \left( \frac{2 \pi r_2}{t} \right)^2}{r_2} \] Simplifying, we get the ratio of the centripetal forces as: \[ \frac{F_{c1}}{F_{c2}} = \frac{m_1 r_1}{m_2 r_2} \] Step 3: Conclusion.
Thus, the ratio of the centripetal forces is \( m_1 r_1 : m_2 r_2 \), which corresponds to option (D).