A spring which has a spring constant of k, needs 14 J of energy to be compressed [math: x] m. How much energy is needed for the same spring to be compressed by 5[math: x] m?*Hint: work out the energies at each velocity separately.

Question

BlackTom AI · Accepted Answer

Elastic potential energy for a spring is U = (1/2) k x^2, so energy depends on the square of the compression.

1. Relevant knowledge points and concepts
- Hooke’s law energy formula for a spring: U = (1/2) k x^2.
- If displacement is multiplied by a factor, energy scales with the square of that factor.

2. Step-by-step reasoning
- Given: U1 = 14 J when displacement = x.
- Use U = (1/2) k x^2 to relate k and x, then compute U for 5x.

3. Intermediate calculations
- From U1 = 14 J: (1/2) k x^2 = 14
  => k x^2 = 28.         (multiply both sides by 2)
- For displacement 5x: U2 = (1/2) k (5x)^2
  = (1/2) k (25 x^2)
  = 25 * (1/2) k x^2.
- Substitute known (1/2) k x^2 = 14:
  U2 = 25 * 14 = 350 J.

4. Why the correct answer is correct
- Energy depends on x^2, so increasing the compression by factor 5 increases energy by factor 5^2 = 25. Therefore 14 J × 25 = 350 J.

5. Why the other options are incorrect
- 175 J would be 14 × 12.5, implying a non-square scaling factor (incorrect).
- 70 J equals 14 × 5, as if energy scaled linearly with displacement (wrong; it scales with the square).
- 35 J equals 14 × 2.5, again not matching the quadratic scaling.
All three ignore that U ∝ x^2, so they are incorrect.

6. Conclusion
- Using U = (1/2) k x^2 and the 5× displacement gives U = 25 × 14 = 350 J. This matches the provided correct answer: 350 J (option c).

A spring which has a spring constant of k, needs 14 J of energy to be compressed [math: x] m. How much energy is needed for the same spring to be compressed by 5[math: x] m?*Hint: work out the energies at each velocity separately.Single choice

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