A ball rises to the surface of a liquid with constant velocity. The density of the liquid is four lime the density of the material of the ball. The frictional force of the liquid on the rising ball is greater than the weight of the ball by a factor of

Options

A. `2`
B. `3`
C. `4`
D. `6`

Explanation

To solve the problem step by step, we will analyze the forces acting on the ball as it rises through the liquid.

### Step 1: Identify the forces acting on the ball
When the ball is rising in the liquid, there are three main forces acting on it:
1. **Weight of the ball (W)**: This acts downward and is given by the formula \( W = m \cdot g \), where \( m \) is the mass of the ball and \( g \) is the acceleration due to gravity.
2. **Buoyant force (F_b)**: This acts upward and is given by Archimedes' principle. The buoyant force is equal to the weight of the liquid displaced by the ball. Since the density of the liquid (\( \rho_l \)) is four times that of the ball (\( \rho_b \)), we can express the buoyant force as \( F_b = \rho_l \cdot V \cdot g \), where \( V \) is the volume of the ball.
3. **Viscous force (F_v)**: This acts downward and opposes the motion of the ball.

### Step 2: Relate the densities
Given that the density of the liquid is four times the density of the ball, we can write:
\[ \rho_l = 4 \cdot \rho_b \]

### Step 3: Express the weight of the ball
The weight of the ball can be expressed in terms of its density and volume:
\[ W = \rho_b \cdot V \cdot g \]

### Step 4: Calculate the buoyant force
Substituting the expression for the density of the liquid into the buoyant force equation:
\[ F_b = \rho_l \cdot V \cdot g = (4 \cdot \rho_b) \cdot V \cdot g = 4W \]
This shows that the buoyant force is four times the weight of the ball.

### Step 5: Apply the condition of constant velocity
Since the ball rises with a constant velocity, the net force acting on it is zero. Therefore, the upward forces (buoyant force and viscous force) must balance the downward forces (weight of the ball):
\[ F_b - W - F_v = 0 \]
This can be rearranged to find the viscous force:
\[ F_v = F_b - W \]

### Step 6: Substitute the values
Substituting the expressions for \( F_b \) and \( W \):
\[ F_v = 4W - W = 3W \]

### Step 7: Determine the factor by which the viscous force is greater than the weight
From the equation \( F_v = 3W \), we can see that the viscous force is three times the weight of the ball:
\[ F_v = 3 \cdot W \]

### Conclusion
Thus, the frictional force of the liquid on the rising ball is greater than the weight of the ball by a factor of **3**.

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