A block is connected between two ideal springs on a horizontal

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

A 0.25 kg0.25\,\text{kg} block is connected between two ideal springs on a horizontal, frictionless track. Spring 1 (with constant k1=60 N m−1k_{1}=60\,\text{N\,m}^{-1}) is attached to a wall on the left; Spring 2 (with constant k2=40 N m−1k_{2}=40\,\text{N\,m}^{-1}) is attached to a wall on the right. The relaxed lengths of both springs correspond to the block being at x=0x=0. A coordinate system is chosen so that positive xx is to the right. (a) Derive an expression for the net restoring force Fnet(x)F_{\text{net}}(x) acting on the block when it is displaced a distance xx to the right of equilibrium but remains attached to both springs. (b) Show that the block undergoes simple harmonic motion and determine the effective spring constant keffk_{\text{eff}}. (c) If the block is released from rest at x=0.050 mx=0.050\,\text{m}, calculate its maximum speed during the ensuing motion. (d) At the instant the block passes through x=−0.030 mx=-0.030\,\text{m} moving leftward, determine the direction of the net force and explain how that direction ensures continual oscillation about equilibrium.

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