Edexcel A Level (IAL) Physics-5.22 Displacement–Time Graph for an Oscillator- Study Notes- New Syllabus

Displacement–Time Graphs for an Object Oscillating in Simple Harmonic Motion

A displacement–time graph shows how the displacement of an oscillating object varies with time. For simple harmonic motion (SHM), this graph has a sinusoidal shape.

 Shape of the Displacement–Time Graph

• The graph is a smooth sine or cosine wave.
• Displacement oscillates between \( +A \) and \( -A \).
• The motion is periodic and repeats after each time period \( T \).

The displacement in SHM can be written as:

\( x = A \sin \omega t \)

Key Features on the Graph

• Amplitude \( A \): maximum displacement from equilibrium.
• Period \( T \): time taken for one complete oscillation.
• Equilibrium position: where \( x = 0 \).

Gradient of the Displacement–Time Graph

The gradient of a displacement–time graph gives the velocity of the object:

\( v = \dfrac{dx}{dt} \)

Therefore:

• Steep gradient → large velocity.
• Zero gradient → zero velocity.
• Positive gradient → velocity in the positive direction.
• Negative gradient → velocity in the negative direction.

Velocity at Key Points

• At maximum displacement (\( x = \pm A \)): gradient = 0 → velocity = 0.
• At equilibrium (\( x = 0 \)): gradient is maximum → velocity is maximum.

This matches the SHM velocity equation:

\( v = -A\omega \cos \omega t \)

Interpreting Direction of Motion

• If the curve is rising, the object is moving in the positive direction.
• If the curve is falling, the object is moving in the negative direction.
• At turning points, the object changes direction.

 Common Exam Interpretation Points

• Velocity is NOT proportional to displacement.
• Velocity depends on the gradient, not the height of the graph.
• Maximum speed occurs at zero displacement.
• Zero speed occurs at maximum displacement.