Interactive classroom lab · SHM

See Simple Harmonic Motion

One motion, two views: a particle oscillates periodically between A and B, while moving paper records displacement against time as a sinusoidal trace.

Amplitude A3.0 m

Period T4.0 s

Initial phase φ0°

Paper speed1.0×

1A particle oscillates between A–O–B

Displacement x is measured from O. The green arrow shows velocity; the red acceleration arrow always points towards O.

Displacement+3.00 mTo the right of O

Velocity0.00 m/sInstantaneously at rest

Acceleration−7.40 m/s²Leftwards, towards O

2Moving paper records the oscillation

The pen moves vertically in simple harmonic motion while the paper moves uniformly to the right. Vertical position represents displacement x; horizontal position represents time t.

Amplitude AThe maximum displacement from the mean position.

Period TThe time taken to complete one full oscillation.

Paper speedFaster paper increases the distance between adjacent peaks, but the period is unchanged.

3The defining relation

a = −ω²xAcceleration is proportional to displacement and opposite in direction

xx(t) = A cos(ωt + φ)

vv(t) = −Aω sin(ωt + φ)

aa(t) = −Aω² cos(ωt + φ)

Always: acceleration has the opposite sign to displacement because a = −ω²x. At O, both x and a are zero.

Circular-projection view

θ = 0°Imagine a point moving uniformly around a circle. Its projection onto the horizontal diameter undergoes simple harmonic motion.

Display settings

Show velocity vector v Show acceleration vector a Show moving grid

What should you notice?

The acceleration vector always points towards O. At A and B, speed is zero and |a| is greatest; at O, acceleration is zero and speed is greatest.

The same rule holds throughout every cycle