SHM Equations and Graphs | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

SHM Equations and Graphs

The displacement, velocity and acceleration equations describe the same object at the same time.

First find the angular frequency. Then decide where the object is when the clock starts.

Choose sine or cosine from the starting point

x=x_0\sin(\omega t)

Symbols

$x$ = displacement at time t (m)
$x_0$ = amplitude (m)
$\omega$ = angular frequency (rad s⁻¹)
$t$ = time (s)

x=x_0\cos(\omega t)

Symbols

$x$ = displacement at time t (m)
$x_0$ = amplitude (m)
$\omega$ = angular frequency (rad s⁻¹)
$t$ = time (s)

Starts at equilibrium

Use the sine form shown above.

It first moves towards positive displacement.

Starts at the positive extreme

Use the cosine form shown above.

It first moves back towards equilibrium.

Common mistake

The angle used by the calculator is in radians. Put the calculator in radian mode before substituting a value of time.

Read displacement, velocity and acceleration together

Swipe left or right to see the whole diagram →

One motion, three graphs. v leads x by a quarter cycle (v is biggest where x crosses zero), and a is the exact mirror of x — that mirror is the a = −ω²x definition drawn out in time.

Positive extreme

Speed is zero.

Acceleration is greatest towards equilibrium; the object turns back.

Equilibrium

Speed is greatest.

Acceleration is zero; the object passes through.

Negative extreme

Speed is zero.

Acceleration is greatest towards equilibrium; the object turns back.

Velocity is one quarter of a cycle ahead of displacement. Acceleration is exactly half a cycle from displacement.
On a graph, velocity is the gradient of displacement, and acceleration is the gradient of velocity.

SHM Equations and Graphs

The displacement, velocity and acceleration equations describe the same object at the same time.

First find the angular frequency. Then decide where the object is when the clock starts.

Choose sine or cosine from the starting point

x=x_0\sin(\omega t)

Symbols

$x$ = displacement at time t (m)
$x_0$ = amplitude (m)
$\omega$ = angular frequency (rad s⁻¹)
$t$ = time (s)

x=x_0\cos(\omega t)

Symbols

$x$ = displacement at time t (m)
$x_0$ = amplitude (m)
$\omega$ = angular frequency (rad s⁻¹)
$t$ = time (s)

Starts at equilibrium

Use the sine form shown above.

It first moves towards positive displacement.

Starts at the positive extreme

Use the cosine form shown above.

It first moves back towards equilibrium.

Common mistake

The angle used by the calculator is in radians. Put the calculator in radian mode before substituting a value of time.

Read displacement, velocity and acceleration together

Swipe left or right to see the whole diagram →

One motion, three graphs. v leads x by a quarter cycle (v is biggest where x crosses zero), and a is the exact mirror of x — that mirror is the a = −ω²x definition drawn out in time.

Positive extreme

Speed is zero.

Acceleration is greatest towards equilibrium; the object turns back.

Equilibrium

Speed is greatest.

Acceleration is zero; the object passes through.

Negative extreme

Speed is zero.

Acceleration is greatest towards equilibrium; the object turns back.

Velocity is one quarter of a cycle ahead of displacement. Acceleration is exactly half a cycle from displacement.
On a graph, velocity is the gradient of displacement, and acceleration is the gradient of velocity.