More practice questions

4. Q4. The same force F stretches two springs, one stiff (k₁) and one soft (k₂ < k₁). Which stores more energy?

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   The stretch is F/k, so the energy is $\tfrac{1}{2}k(F/k)^{2}$, that is F²/(2k). For the same force, the softer spring (smaller k) stretches more and stores more energy. (For the same stretch, it is the other way round: the stiffer spring stores more, $\tfrac{1}{2}kx^{2}$.)

5. Q5. A 1 kg block is dropped from 0.5 m above the top of a vertical spring of spring constant 1000 N m^⁻¹. Find the greatest compression of the spring. Take g = 9.8 m s⁻².

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   At the greatest compression x the block is momentarily at rest. It has fallen 0.5 + x, so the gravitational potential energy lost equals the spring energy gained:

   $$\begin{aligned} &1 \times 9.8 \times (0.5 + x) \\ &= \tfrac{1}{2} \times 1000 \times x^{2} \end{aligned}$$ $$500x^{2} - 9.8x - 4.9 = 0$$ $$\begin{aligned} x &= \frac{9.8 + \sqrt{9.8^{2} + 4 \times 500 \times 4.9}}{1000} \\ &= \frac{9.8 + 99.48}{1000} \approx 0.109\ \text{m} \end{aligned}$$

   about 10.9 cm.

6. Q6. A spring stores 2 J of energy when stretched by 4 cm. How much energy does it store when stretched by 6 cm, and what is its spring constant?

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   k = 2V/x² = 4/0.0016, that is 2500 N m^⁻¹. At 6 cm the energy is $\tfrac{1}{2}(2500)(0.06)^{2}$⁠, that is 4.5 J (it goes as x²: (6/4)² × 2 = 4.5).