The total force acting on the mass at any time t, for damped oscillator is given as (where symbols have their usual meanings)

Options

A. `vecF = -k vecx`
B. `vec F = -k vec x - b vec v`
C. `vec F = -k vec x - b vec v + 2 vec x`
D. `vec F = -b vec v - bk^2 vec x`

Answer

To solve the problem of finding the total force acting on a mass in a damped oscillator, we can break down the steps as follows:

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a mass \( m \) connected to a spring with spring constant \( k \).
- The mass oscillates around an equilibrium position \( O \).

**Hint**: Identify the components of the system: mass, spring, and damping.

2. **Restoring Force**:
- When the mass is displaced by a distance \( x \) from the equilibrium position, the restoring force due to the spring is given by Hooke's Law:
\[ F_s = -kx \]
- Here, \( F_s \) is the restoring force, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.

**Hint**: Remember that the restoring force acts in the opposite direction to the displacement.

3. **Damping Force**:
- The damping force \( F_D \) is proportional to the velocity \( v \) of the mass and acts in the opposite direction. It can be expressed as:
\[ F_D = -Bv \]
- Here, \( B \) is the damping coefficient and \( v \) is the velocity of the mass.

**Hint**: Damping force always opposes the motion of the object.

4. **Total Force**:
- The total force \( F \) acting on the mass at any time \( t \) is the sum of the restoring force and the damping force:
\[ F = F_D + F_s \]
- Substituting the expressions for \( F_D \) and \( F_s \):
\[ F = -Bv - kx \]

**Hint**: Combine the forces algebraically, keeping track of their signs.

5. **Final Expression**:
- Therefore, the total force acting on the mass at any time \( t \) is:
\[ F = -Bv - kx \]

**Hint**: Ensure you understand that this equation represents the net force acting on the mass in a damped oscillatory motion.

### Summary:
The total force acting on the mass in a damped oscillator is given by:
\[ F = -Bv - kx \]
where \( B \) is the damping coefficient, \( v \) is the velocity of the mass, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position.