### Correct Option: C. It increases by a factor of four ### Detailed Explanation: In a mass-spring system undergoing simple harmonic motion (SHM), the total mechanical energy \( E \) of the system is given by the formula: \[ E = \frac{1}{2} k A^2 \] where: - \( E \) is the total mechanical energy, - \( k \) is the spring constant, - \( A \) is the amplitude of the motion. #### Step-by-Step Analysis: 1. **Understanding the Formula**: The total mechanical energy in a mass-spring system is directly proportional to the square of the amplitude \( A \). This means that if you change the amplitude, the energy will change according to the square of that change. 2. **Doubling the Amplitude**: If the amplitude is doubled, we can express the new amplitude as: \[ A' = 2A \] 3. **Calculating New Energy**: We can now substitute this new amplitude into the energy formula: \[ E' = \frac{1}{2} k (A')^2 = \frac{1}{2} k (2A)^2 \] Simplifying this gives: \[ E' = \frac{1}{2} k (4A^2) = 4 \left(\frac{1}{2} k A^2\right) = 4E \] This shows that the new total mechanical energy \( E' \) is four times the original energy \( E \). 4. **Conclusion**: Therefore, when the amplitude is doubled, the total mechanical energy of the system increases by a factor of four. ### Why Other Options Are Incorrect: - **Option A: It remains the same**: This is incorrect because the energy depends on the square of the amplitude. Doubling the amplitude changes the energy significantly, not leaving it unchanged. - **Option B: It doubles**: This option is also incorrect. While it might seem intuitive that doubling something would double the energy, the relationship is quadratic (squared), not linear. Thus, the energy increases by a factor of four, not two. - **Option D: It decreases by half**: This option is incorrect as well. Increasing the amplitude cannot lead to a decrease in energy. In fact, increasing the amplitude increases the potential energy stored in the spring, thus increasing the total mechanical energy. ### Common Pitfalls: - **Misunderstanding the Relationship**: Students often confuse linear and quadratic relationships. Remember that energy in SHM is proportional to the square of the amplitude. - **Forgetting the Formula**: Always recall the formula for total mechanical energy in a mass-spring system. It’s crucial for solving problems related to energy changes. ### Revision Summary: - The total mechanical energy \( E \) in a mass-spring system is given by \( E = \frac{1}{2} k A^2 \). - Doubling the amplitude \( A \) results in the new energy being \( E' = 4E \). - The energy increases by a factor of four when the amplitude is doubled. - Remember the quadratic relationship between amplitude and energy in SHM to avoid common mistakes.