Unit 2 · Lesson 5 of 11

An oscillation's slope has its own phase.

A vibrating mass stops momentarily at a turning point, yet moves fastest near the center. Follow displacement and velocity together.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its signed velocity.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase time? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

PositionVelocity (numeric overlay) · dashed

Time1 sAngular frequency1 rad/s

signed velocity

v(1)=2(1)\cos(1)=1.0806\,\mathrm{m/s}

At displacement extrema velocity is zero. Passing the middle can have the largest speed.

Assumptions, units, and limits

Sinusoidal displacement with amplitude 2 m; radians and time in seconds.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

The sine derivative is cosine in radians. The inner angle's time rate contributes an angular-frequency factor. Differentiating again gives acceleration directed toward the equilibrium displacement.

s(t)=2\sin(\omega t),\quad v(t)=2\omega\cos(\omega t)

Keep the reasoning visible

s''=-\omega^2s

At displacement extrema velocity is zero. Passing the middle can have the largest speed.

What makes this model useful?

Sinusoidal displacement with amplitude 2 m; radians and time in seconds.

Zero velocity at a displacement extremum does not mean zero acceleration. Units and phase matter.

A worked example

Calculate the starting scenario

Use time = 1 s; angular frequency = 1 rad/s.
$v(1)=2(1)\cos(1)=1.0806\,\mathrm{m/s}$
At displacement extrema velocity is zero. Passing the middle can have the largest speed. The checked quantity is signed velocity.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the signed velocity using time 1 s, angular frequency 1 rad/s. State the meaning of its sign and units.

Checked quantity: signed velocity · m/s. Follow the definition for finite-cutoff and numerical-model questions.

Hint

The sine derivative is cosine in radians. The inner angle's time rate contributes an angular-frequency factor. Differentiating again gives acceleration directed toward the equilibrium displacement.

Explore the visual solution

PositionVelocity (numeric overlay) · dashed

Time1 sAngular frequency1 rad/s

signed velocity

v(1)=2(1)\cos(1)=1.0806\,\mathrm{m/s}

At displacement extrema velocity is zero. Passing the middle can have the largest speed.

Assumptions, units, and limits

Sinusoidal displacement with amplitude 2 m; radians and time in seconds.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

The sine derivative is cosine in radians. The inner angle's time rate contributes an angular-frequency factor. Differentiating again gives acceleration directed toward the equilibrium displacement.

s(t)=2\sin(\omega t),\quad v(t)=2\omega\cos(\omega t)

Calculate and explain

Apply the idea to a new scenario

At $t=2$ s and angular frequency 0.5 rad/s, find velocity of the amplitude-2 m oscillator.

Checked quantity: signed velocity · m/s. Follow the definition for finite-cutoff and numerical-model questions.

Hint

The sine derivative is cosine in radians. The inner angle's time rate contributes an angular-frequency factor. Differentiating again gives acceleration directed toward the equilibrium displacement.

Explore the visual solution

PositionVelocity (numeric overlay) · dashed

Time2 sAngular frequency0.5 rad/s

signed velocity

v(2)=2(0.5)\cos(1)=0.540302\,\mathrm{m/s}

At displacement extrema velocity is zero. Passing the middle can have the largest speed.

Assumptions, units, and limits

Sinusoidal displacement with amplitude 2 m; radians and time in seconds.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

The sine derivative is cosine in radians. The inner angle's time rate contributes an angular-frequency factor. Differentiating again gives acceleration directed toward the equilibrium displacement.

s(t)=2\sin(\omega t),\quad v(t)=2\omega\cos(\omega t)

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

PositionVelocity (numeric overlay) · dashed

Time1 sAngular frequency1 rad/s

signed velocity

v(1)=2(1)\cos(1)=1.0806\,\mathrm{m/s}

At displacement extrema velocity is zero. Passing the middle can have the largest speed.

Assumptions, units, and limits

Sinusoidal displacement with amplitude 2 m; radians and time in seconds.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. The sine derivative is cosine in radians. The inner angle's time rate contributes an angular-frequency factor. Differentiating again gives acceleration directed toward the equilibrium displacement.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.