A mass m is pushed against the end of a spring, compressing it by an amount d fr

Question

A mass m is pushed against the end of a spring, compressing it by an amount d from its relaxed position. The spring has spring constant k. The mass is released from rest and slides along a flat frictionless surface. It then slides into a frictionless semicircle of radius R (see diagram). (a) Find an expression for the speed of the mass as a function of ?, m, d, k, R, and g. (b) What is the largest angle the mass reaches before it stops? (c) What is the normal force when the mass reaches its largest angle obtained in part (b)?

Explanation / Answer

a). applying energy conservation (suppose m is at a point that makes an angle ? from vertical.) 0.5kd^2 = 0.5mv^2 + mg(R-Rcos ?) (1) v = [{0.5kd^2 - mg(R-Rcos ?)}/(0.5m)]^0.5 b). by putting v = 0 in equation (1) 0.5kd^2 =0 + mg(R-Rcos ?) (0.5kd^2)/(mg) = (R-Rcos ?) cos ? = { R- (0.5kd^2)/(mg)}/R ? = cos-1({ R- (0.5kd^2)/(mg)}/R) c).normal force should be zero at that point because the mass has stopped at that point.

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