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 regarding the total force acting on a mass in a damped oscillator, we need to analyze the forces involved. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Forces Acting on the Mass
In a damped oscillator, there are two primary forces acting on the mass:
1. **Spring Force (Restoring Force)**: This force is proportional to the displacement from the equilibrium position and acts in the opposite direction of the displacement. It can be expressed as:
\[ F_{\text{spring}} = -kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

2. **Damping Force**: This force opposes the motion of the mass and is proportional to the velocity of the mass. It can be expressed as:
\[ F_{\text{damping}} = -b v \]
where \( b \) is the damping coefficient and \( v \) is the velocity of the mass.

### Step 2: Write the Total Force Equation
The total force \( F_{\text{total}} \) acting on the mass is the sum of the spring force and the damping force. Therefore, we can write:
\[ F_{\text{total}} = F_{\text{spring}} + F_{\text{damping}} \]
Substituting the expressions for the forces, we get:
\[ F_{\text{total}} = -kx - bv \]

### Step 3: Final Expression for Total Force
Thus, the total force acting on the mass at any time \( t \) can be expressed as:
\[ F_{\text{total}} = -kx - b \frac{dx}{dt} \]
where \( \frac{dx}{dt} \) represents the velocity \( v \).

### Conclusion
The total force acting on the mass in a damped oscillator is given by:
\[ F_{\text{total}} = -kx - bv \]