Question

A mass \( m \), suspended vertically by a massless ideal spring with spring constant \( k \), is at rest. The mass is displaced upward by a height \( h \). When released, the kinetic energy of the mass will be proportional to (Neglecting air resistance)

• A. only \( h \)
• B. only \( h^2 \)
• C. \( m \)
• D. a linear combination of terms involving \( h \) and \( h^2 \)
• E. \( k \)

Hint:

In vertical spring systems, always consider both spring and gravitational potential energies.

Answer 1

The Correct Option is D

Concept:
Total energy change involves both spring potential energy and gravitational potential energy. \[ U = \frac{1}{2}kx^2 + mgh \]

Step 1: Initial condition
Mass is displaced upward by \( h \), so: - Spring gets compressed/extended change - Gravitational potential energy increases

Step 2: Energy released
When released, both: \[ \frac{1}{2}k h^2 \quad \text{and} \quad mgh \] contribute to kinetic energy.

Step 3: Expression of kinetic energy
\[ K \propto \frac{1}{2}kh^2 + mgh \]

Step 4: Conclusion
Kinetic energy depends on both \( h \) and \( h^2 \) \[ \boxed{\text{Linear combination of } h \text{ and } h^2} \]