A pinball machine has balls of mass 134 g. The springs have a spring constant of 93.4N/m and a relaxed length of 10 cm. To what length must the spring be compressed if it is to give a ball a speed of 2.0 m/s?
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To solve this problem, you can use the principles of mechanical energy conservation and Hooke's Law.
Here's how you can find the compressed length of the spring:
1. Determine the initial and final positions of the ball:
- As the spring is relaxed, the initial position of the ball is at a compressed length of 10 cm.
- The final position of the ball is when the spring is compressed to an unknown length.
2. Calculate the potential energy stored in the compressed spring:
- The potential energy (PE) stored in a spring is given by the equation PE = (1/2) kx^2, where k is the spring constant and x is the displacement from the relaxed length.
- In this case, the displacement is the difference between the initial compressed length (10 cm) and the final compressed length (unknown).
- Therefore, the potential energy is PE = (1/2) k (x^2).
3. Use the principle of mechanical energy conservation:
- At the initial position, the ball has no kinetic energy, so its total mechanical energy is equal to the potential energy of the spring:
ME_initial = PE_initial
- At the final position, the ball has kinetic energy due to its velocity of 2.0 m/s, and the spring has potential energy due to its compression:
ME_final = KE_ball + PE_spring
4. Equate the initial and final mechanical energies and solve for x:
(1/2) k (x^2) = (1/2) m v^2
(1/2) k (x^2) = (1/2) (134 g) (2.0 m/s)^2
k (x^2) = (134 g) (2.0 m/s)^2
x^2 = [(134 g) (2.0 m/s)^2] / k
x = √{[(134 g) (2.0 m/s)^2] / k}
5. Substitute the given values and solve for x:
x = √[(134 g) (2.0 m/s)^2] / (93.4 N/m)
x = √[(0.134 kg) (2.0 m/s)^2] / (93.4 N/m)
x ≈ 0.0344 m (rounded to four decimal places)
Therefore, the spring must be compressed to a length of approximately 0.0344 meters in order to give the ball a speed of 2.0 m/s.