A massless spring of constant k = 63.4 N/m is fixed on the left side of a level

Question

A massless spring of constant k = 63.4 N/m is fixed on the left side of a level track. A block of mass m = 0.50 kg is pressed against the spring and compresses it a distance of d, as in the figure shown below. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius R = 1.5 m.The entire track and the loop-the-loop are frictionless, except for the section of track between points A and B. Given that the coefficient of kinetic friction between the block and the track along AB is k = 0.32,and that the length of AB is 2.5 m, determine the minimum compression d of the spring that enables the block to just make it through the loop-the-loop at point C. Hint: The force exerted by the track on the block will be zero if the block barely makes it through the-loop-the-loop.

m

Explanation / Answer

Use Conservation of Energy

(1/2)kd2 - ukmgL = mg(2R)

=> (1/2)63.4d2 - 0.32(0.5 x 9.8)(2.5) = (0.5 x 9.8)(2 x 1.5)

=> (1/2)63.4d2 = 0.32(0.5 x 9.8)(2.5) + (0.5 x 9.8)(2 x 1.5)

=> d = 0.7664 m = 76.64 cm.

this is the compression needed in the spring.

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