A 5kg mass bounces in simple harmonic motion at the end of a spring. At which point the acceleration of the mass is the greatest?

Options

A. When the spring is fully compressed and when the spring is fully extended
B. When the spring is at its rest length
C. When the spring is halfway between its rest length and is fully extended or compressed length
D. The acceleration is constant.

Answer

To determine at which point the acceleration of a 5 kg mass bouncing in simple harmonic motion at the end of a spring is greatest, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, the restoring force acting on the mass is provided by the spring. This force is proportional to the displacement from the equilibrium position (rest length of the spring).

2. **Spring Force**:
- The force exerted by a spring is given by Hooke's Law:
\[ F = -kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

3. **Acceleration in SHM**:
- The acceleration \( a \) of the mass can be expressed as:
\[ a = \frac{F}{m} = \frac{-kx}{m} \]
Here, \( m \) is the mass of the object (5 kg in this case). The acceleration is directly proportional to the displacement \( x \).

4. **Maximum Displacement**:
- The acceleration will be greatest when the displacement \( x \) is at its maximum value. This occurs when the spring is either fully compressed or fully extended.

5. **Conclusion**:
- Therefore, the points at which the mass experiences the greatest acceleration are when the spring is either fully compressed or fully extended. At these points, the displacement \( x \) is maximum, leading to maximum force and hence maximum acceleration.

### Final Answer:
The acceleration of the mass is greatest when the spring is either fully compressed or fully extended.