A body of mass m is made to revolve along a circle of radius r in horizontal plane at a speed `Vm//s` with help of light horizontal string such that the tension in the string is 10 N. Keeping speed constant if mass is increased to 2m und radius is decreased to `r//2` then tension in the string is

Options

A. 10N
B. 20N
C. 40N
D. 5N

Answer

To solve the problem step by step, we will analyze the forces acting on the body in circular motion and apply the given conditions.

### Step 1: Understand the initial conditions
We have a body of mass \( m \) revolving in a horizontal circle of radius \( r \) with a speed \( V \). The tension in the string is given as \( T = 10 \, \text{N} \).

### Step 2: Apply the formula for tension in circular motion
The tension in the string provides the necessary centripetal force to keep the body moving in a circle. The formula for the tension \( T \) in terms of mass \( m \), speed \( V \), and radius \( r \) is given by:
\[ T = \frac{mV^2}{r} \]
From the problem, we know that \( T = 10 \, \text{N} \). Thus, we can write:
\[ 10 = \frac{mV^2}{r} \]

### Step 3: Analyze the new conditions
Now, the mass is increased to \( 2m \) and the radius is decreased to \( \frac{r}{2} \). We need to find the new tension \( T' \) in the string while keeping the speed \( V \) constant.

### Step 4: Write the new tension formula
Using the same formula for tension, we can express the new tension \( T' \) as:
\[ T' = \frac{(2m)V^2}{\frac{r}{2}} \]

### Step 5: Simplify the new tension formula
We can simplify the expression for \( T' \):
\[ T' = \frac{2mV^2}{\frac{r}{2}} = \frac{2mV^2 \cdot 2}{r} = \frac{4mV^2}{r} \]

### Step 6: Substitute the value of \( \frac{mV^2}{r} \)
From Step 2, we know that \( \frac{mV^2}{r} = 10 \). Therefore, we can substitute this into our equation for \( T' \):
\[ T' = 4 \left(\frac{mV^2}{r}\right) = 4 \times 10 = 40 \, \text{N} \]

### Final Answer
Thus, the tension in the string when the mass is increased to \( 2m \) and the radius is decreased to \( \frac{r}{2} \) is:
\[ \boxed{40 \, \text{N}} \]