A mass m is attached to a spring that is held stretched a distance x by a force

Question

A mass m is attached to a spring that is held stretched a distance x by a force F (Figure 2), and then released. The spring compresses, pulling the mass. Assuming there is no friction, determine the speed of the mass m when the spring returns.
Figure 2
i) To its normal length (x = 0) (3.5 marks)
ii) To half of its original extension (x / 2) (2.5 marks)
iii) Suppose that there is friction in Figure 2, and the mass on the end of the stretched spring, after being released, comes to rest just when it reaches the spring

Explanation / Answer

m for the mass, x for the distance xmax and F for the force Fmax spring constant k = Fmax / xmax max stored energy in spring = 0.5 * k * xmax ^2 = max KE when the spring returns to its normal length (x = 0) = 0.5 * m * v^2 m * v^2 = k * xmax ^2 v^2 = [Fmax / xmax] * [ xmax ^2 / m] v = square root [Fmax * xmax / m ] answer (m/s) when the spring returns to half its original extension (x = xmax/2) total energy = 0.5 * k * (xmax/2)^2 + 0.5 * m * v^2 = max stored energy in spring = 0.5 * k * xmax ^2 k * (xmax/2)^2 + m * v^2 = k * xmax ^2 m * v^2 = k * xmax ^2 - k (xmax/2)^2 m v^2 = k * xmax ^2 [ 1 - 0.25] v^2 = k * xmax ^2 *0.75 / m v^2 = Fmax / xmax * xmax ^2 *0.75 / m v = square root [ Fmax * xmax *0.75 / m ] answer

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