All surfaces are frictionless except for the patch of length d = 1.0 m, which ha

Question

All surfaces are frictionless except for the patch of length d = 1.0 m, which has a sprinkling of dust giving it a coefficient of friction mu_k = 0.001. m is not attached to the spring - only pressed against it. The mass m (500 g) is pressed against spring 1 (k_1 = 100 N/m), compressing it 10 cm. Released from rest, m slides across to the other side of the platform (wall-to-wall distance 10.0 m) and bounces off of spring 2 (k_2 = 200N/m). (a) What is the maximum compression of spring 2 on the first bounce off the right hand side? (b) What is the maximum compression of spring 1 on the first bounce off the left hand side? (c) How many times does the mass bounce off of spring 1 before coming to rest?

Explanation / Answer

initial energy = 0.5 k1(0.1)2 = 0.5 J

energy used against friction in one time crossing d , W = umgd = 0.001 x 0.5 x 9.8 x 1 = 0.0049 J

so (a) 0.5 k2y2 = 0.5 - 0.0049 = 0.4951

=> y = 0.07036 m = 7.036 cm

(b) 0.5 k1y'2 = 0.5 - (2 x0.0049) = 0.4902

=> y' = 0.099015 m = 9.9015 cm

(c) between any bounce of spring 1 energy lost = 0.0098 J

so total bounces from spring 1 = 0.5/0.0098 = 51.02,,,, so 51 bounces

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