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 concept of momentum
Momentum (\( p \)) of an object is defined as the product of its mass (\( m \)) and its velocity (\( v \)):
\[ p = mv \]

### Step 2: Identify the nature of motion
The object is moving in a circle at a constant speed. Although the speed is constant, the direction of the velocity is continuously changing. This means that the momentum of the object is also changing due to the change in direction.

### Step 3: Determine the rate of change of momentum
The rate of change of momentum (\( \frac{dp}{dt} \)) is equal to the net force acting on the object, according to Newton's second law:
\[ \frac{dp}{dt} = F \]

### Step 4: Identify the type of force involved
For an object moving in a circle, the force that keeps it in circular motion is called the centripetal force. This force acts towards the center of the circle and is responsible for changing the direction of the velocity.

### Step 5: Calculate centripetal force
The centripetal force (\( F_c \)) required to keep an object of mass \( m \) moving at a speed \( v \) in a circle of radius \( r \) is given by:
\[ F_c = \frac{mv^2}{r} \]

### Step 6: Relate centripetal force to rate of change of momentum
Since the rate of change of momentum is equal to the centripetal force, we can write:
\[ \frac{dp}{dt} = \frac{mv^2}{r} \]

### Step 7: Conclusion
Thus, the magnitude of the rate of change of momentum of the object is:
\[ \frac{dp}{dt} = \frac{mv^2}{r} \]

### Final Answer
The magnitude of the rate of change of momentum of the object is \( \frac{mv^2}{r} \).

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