Question

An object is moving in a circle at constant speed \(v\). The magnitude of the rate of change of momentum of the object is:

• A. zero
• B. proportional to \(v^2\)
• C. proportional to \(v^3\)
• D. proportional to \(v\)

Hint:

Uniform circular motion \(\Rightarrow\) centripetal force \( \propto v^2 \).

Answer 1

The Correct Option is B

To determine the magnitude of the rate of change of momentum of an object moving in a circle at constant speed \( v \), we must first understand the concept of centripetal force and its relation to momentum.

When an object moves in a circle of radius \( r \) with constant speed \( v \), its velocity vector continuously changes direction, even though its speed remains constant. This change in direction results in an acceleration known as centripetal acceleration, which is directed towards the center of the circle. The formula for centripetal acceleration \( a_c \) is given by:

\(a_c = \frac{v^2}{r}\)

The centripetal force \( F_c \), which is responsible for this acceleration, can be expressed as:

\(F_c = m \cdot a_c = m \cdot \frac{v^2}{r}\)

where \( m \) is the mass of the object.

According to Newton's second law, the force is the rate of change of momentum \( p \):

\(F = \frac{dp}{dt}\)

Thus, the rate of change of momentum is equal to the centripetal force acting on the object:

\(\frac{dp}{dt} = m \cdot \frac{v^2}{r}\)

This equation shows that the rate at which the momentum of the object changes is proportional to \( v^2 \). Therefore, the correct answer is:

proportional to \(v^2\)
 

Other options can be ruled out because:

• The rate of change of momentum is not zero because the direction of momentum changes continuously.
• It is not proportional to \( v^3 \) or \( v \) as explained by the centripetal acceleration and force dependency on \( v^2 \).