An object of mass m moves with constant speed in a circular path of radius r under the action of a force of constant magnitude F. the kinetic energy of object is

Options

A. `(1)/(2)`FR
B. FR
C. 2FR
D. `(1)/(4)`FR

Answer

To find the kinetic energy of an object of mass \( m \) moving with constant speed in a circular path of radius \( r \) under the action of a constant force \( F \), we can follow these steps:

### Step 1: Understand the relationship between force and circular motion
In uniform circular motion, the net force acting on the object is the centripetal force, which is required to keep the object moving in a circle. The centripetal force \( F_c \) is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the constant speed of the object.

### Step 2: Relate the constant force to centripetal force
Since it is given that the object is moving under the action of a constant force \( F \), we can equate this force to the centripetal force:
\[ F = \frac{mv^2}{r} \]

### Step 3: Solve for \( v^2 \)
Rearranging the equation to solve for \( v^2 \):
\[ mv^2 = Fr \implies v^2 = \frac{Fr}{m} \]

### Step 4: Write the expression for kinetic energy
The kinetic energy \( KE \) of the object is given by:
\[ KE = \frac{1}{2} mv^2 \]

### Step 5: Substitute \( v^2 \) into the kinetic energy formula
Now, substitute the expression for \( v^2 \) into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\frac{Fr}{m}\right) \]

### Step 6: Simplify the expression
Simplifying the expression gives:
\[ KE = \frac{1}{2} \cdot Fr \]

### Final Answer
Thus, the kinetic energy of the object is:
\[ KE = \frac{Fr}{2} \]