An object is moving in a circle at constant speed `v`. The magnitude of rate of change of momentum of the object.

Options

A. is zero
B. is proportional to `v`
C. is proportional to `v^(2)`
D. is proportional to `v^(3)`

Explanation

To solve the problem of finding the magnitude of the rate of change of momentum of an object moving in a circle at constant speed \( v \), we can follow these steps:

### Step 1: Understand the motion
The object is moving in a circular path with a constant speed \( v \). Although the speed is constant, the direction of the velocity is continuously changing, which means the momentum of the object is also changing.

### Step 2: Define momentum
Momentum \( p \) is defined as the product of mass \( m \) and velocity \( v \):
\[ p = m \cdot v \]

### Step 3: Determine the rate of change of momentum
The rate of change of momentum is given by the derivative of momentum with respect to time:
\[ \frac{dp}{dt} = \frac{d(mv)}{dt} \]
Since mass \( m \) is constant, we can write:
\[ \frac{dp}{dt} = m \frac{dv}{dt} + v \frac{dm}{dt} \]
However, since mass \( m \) is constant, \( \frac{dm}{dt} = 0 \), thus:
\[ \frac{dp}{dt} = m \frac{dv}{dt} \]

### Step 4: Analyze the velocity change
In circular motion, even though the speed \( v \) is constant, the direction of the velocity vector is changing. The change in velocity \( \Delta v \) over a time interval \( \Delta t \) leads to a centripetal acceleration \( a_c \) given by:
\[ a_c = \frac{v^2}{r} \]
where \( r \) is the radius of the circular path.

### Step 5: Relate acceleration to force
The net force acting on the object, which is the centripetal force \( F_c \), is related to the mass and acceleration:
\[ F_c = m \cdot a_c = m \cdot \frac{v^2}{r} \]

### Step 6: Connect force to rate of change of momentum
According to Newton's second law, the force is also equal to the rate of change of momentum:
\[ F = \frac{dp}{dt} \]
Thus, we have:
\[ \frac{dp}{dt} = m \cdot \frac{v^2}{r} \]

### Conclusion
The magnitude of the rate of change of momentum of the object is:
\[ \frac{dp}{dt} = \frac{mv^2}{r} \]