H2 Physics 9 - Oscillations: Simple Harmonic Motion of a Mass-Spring System

Step 1 - Set up the problem

A 0.5 kg mass is on a horizontal spring with k = 20 N/m.

Step 1 - Set up the problem

It is pulled 0.1 m from equilibrium and released from rest.

Step 1 - Set up the problem

Find omega, the period, maximum speed, and maximum acceleration. Assume no friction.

Step 2 - Find the angular frequency and period

For SHM on a spring: omega = square root of k over m.

Step 2 - Find the angular frequency and period

omega = square root of (20/0.5) = square root of 40.

Step 2 - Find the angular frequency and period

omega = 6.32 rad/s.

Step 2 - Find the angular frequency and period

Period T = 2 pi / 6.32 = 0.99 s, approximately 1 s.

Step 3 - Find the maximum speed

Maximum speed occurs at the equilibrium position.

Step 3 - Find the maximum speed

v_max = omega times A = 6.32 times 0.1.

Step 3 - Find the maximum speed

v_max = 0.632 m/s.

Step 4 - Find the maximum acceleration

Maximum acceleration occurs at the extreme positions.

Step 4 - Find the maximum acceleration

a_max = omega squared times A = 40 times 0.1.

Step 4 - Find the maximum acceleration

a_max = 4.0 m/s squared.

Step 5 - Write displacement as a function of time

Released from amplitude with zero velocity - use cosine.

Step 5 - Write displacement as a function of time

x(t) = 0.1 cos(6.32t), with x in metres and t in seconds.

Step 5 - Write displacement as a function of time

Summary: omega = 6.32 rad/s, T = 1.0 s, v_max = 0.63 m/s, a_max = 4.0 m/s squared.