A fluid is flowing through a tube of length L. The radius of the tube is r and the velocity of the fluid is v. If the radius of the tube is increased to 2r, then what will be the new velocity?

Options

A. 4v
B. v/4
C. v/2
D. 2v

Explanation

To find the new velocity of the fluid when the radius of the tube is increased from \( r \) to \( 2r \), we can use the principle of conservation of mass, often referred to as the equation of continuity for fluids.

### Step-by-Step Solution:

1. **Understand the Equation of Continuity**:
The equation of continuity states that the mass flow rate must remain constant in a steady flow. This can be expressed as:
\[ A_1 v_1 = A_2 v_2 \]
where \( A_1 \) and \( A_2 \) are the cross-sectional areas at two points in the tube, and \( v_1 \) and \( v_2 \) are the fluid velocities at those points.

2. **Calculate the Initial Area**:
The initial radius of the tube is \( r \). The cross-sectional area \( A_1 \) can be calculated using the formula for the area of a circle:
\[ A_1 = \pi r^2 \]

3. **Calculate the New Area**:
When the radius is increased to \( 2r \), the new cross-sectional area \( A_2 \) becomes:
\[ A_2 = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2 \]

4. **Set Up the Equation**:
According to the equation of continuity:
\[ A_1 v_1 = A_2 v_2 \]
Substituting the areas and velocities:
\[ (\pi r^2) v = (4\pi r^2) v' \]

5. **Cancel Common Terms**:
We can cancel \( \pi r^2 \) from both sides of the equation:
\[ v = 4 v' \]

6. **Solve for the New Velocity**:
Rearranging the equation to solve for \( v' \):
\[ v' = \frac{v}{4} \]

### Final Answer:
The new velocity \( v' \) when the radius of the tube is increased to \( 2r \) is:
\[ v' = \frac{v}{4} \]