Question

A particle executed simple harmonic motion with an amplitude of 1.67 cm. At what positive displacement from the midpoint of its motion does its speed equal half of its maximum speed. Answer in units of cm.

Answer 1

Solution:

Simple harmonic motion has constant mechanical energy

1/2mv² + 1/2kx² = C

When the spring is at amplitude x = A, velocity is zero
When the spring is at zero deflection velocity is at maximum
therefore
C = 1/2kA² = 1/2m(vmax)²

Because the velocity is squared, when velocity is half of its maximum, kinetic energy is one quarter of the total energy therefore spring potential is three quarters of the total energy.

0.75C = 1/2kx²
0.75 1/2 kA² = 1/2kx²
0.75A² = x²

x = A√0.75
x = 1.67(√(0.75)
x = 1.45 cm .

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