Full Review: Uniform Circular Motion — Calculus-Based

Topic Alignment

This bundle follows the approved independent Physics Sensei unit specification. Use it to reinforce key concepts, prepare for coursework, or review before an assessment.

BEST USED

✓ After studying the unit

✓ Before starting homework

✓ Before a quiz or exam

Physics Sensei is an independent educational resource built from the approved Physics Sensei unit specification.

Your Review Plan

Complete these six stages in order. Each stage builds on the previous one and prepares you for the final readiness check.

6 Stages • Approximately 45-60 minutes.

Warm-Up Check

Before you begin, take a moment to see what you already remember. Do not worry about getting everything right. This is only a starting point.

ACTIVITY 1

Recall Activity 1

Use the circular position function and its derivatives.

ACTIVITY 2

Recall Activity 2

Connect angular position, period, and frequency.

ACTIVITY 3

Recall Activity 3

Write Newton’s second law in radial vector form.

Ready to strengthen your understanding?

You've refreshed what you already know. Next, you'll reinforce the essential concepts that will help you solve problems with confidence. Need to see the learning path again?

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Core Concepts

Let's rebuild the key ideas one step at a time. Focus on understanding the relationships before worrying about solving problems.

KEY CONCEPT 1

Circular kinematics: period, frequency, angular speed, and linear speed 1

Represent the motion with r(t)=r cos(ωt)i+r sin(ωt)j. Differentiation gives tangent velocity with constant magnitude v=ωr.

r(t)=r cos(ωt)i+r sin(ωt)j; v(t)=−ωr sin(ωt)i+ωr cos(ωt)j; |v|=ωr.

The dot product r(t)·v(t)=0, showing radial position and tangent velocity are perpendicular.

Linear speed and angular speed are related but not identical. Points at different radii can share the same angular speed while having different linear speeds.

KEY CONCEPT 2

Centripetal acceleration: changing direction at constant speed 2

Differentiating velocity gives a(t)=−ω²r cos(ωt)i−ω²r sin(ωt)j. The minus sign shows that acceleration is opposite the outward radial direction.

a(t)=−ω²r(t); |a|=ω²r=v²/r.

The second derivative of the circular position function is proportional to −r(t).

A statement about how acceleration changes with radius is incomplete unless you also state what is held constant: speed, angular speed, or period.

KEY CONCEPT 3

Radial force modeling: the inward net force 3

Use the vector relation ΣF=−mω²r in the radial direction and combine it with independent vertical or tangential component equations when needed.

ΣF=−mω²r in radial vector form.

For a conical pendulum, vertical balance and radial acceleration are separate component conditions.

“Centripetal” describes the required inward net-force direction. It is not the name of an additional physical interaction.

Ready to apply these ideas?

You've reinforced the essential concepts. Now it's time to put them into practice by working through guided examples and building your problem-solving confidence. Need a quick reminder?

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Guided Practice

Now it's time to apply what you've reviewed.

Work through each activity in order. The examples become gradually more challenging, and each one prepares you for the final readiness check.

PRACTICE 1

Worked Example

Connect period, angular speed, and linear speed; keep units visible.

PRACTICE 2

Guided Problem

Draw the free-body diagram, identify the inward direction, and then apply the radial equation.

PRACTICE 3

Independent Problem

Separate vertical balance from radial acceleration.

Ready to check your understanding?

You've practiced the essential skills with guidance. Now it's time to solve a few short problems on your own and confirm you're ready to move forward. Need a quick reminder?

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Confidence Check

You've rebuilt the key ideas and practiced them with guidance. Now try these short questions on your own to check your understanding before moving on.

QUICK CHECK 1

Direction check

Answer without drawing a separate centripetal-force arrow.

QUICK CHECK 2

Scaling check

State what is held constant before giving the factor.

QUICK CHECK 3

Force-model check

Name the real interaction and the radial equation.

How did it go?

You've checked your understanding. Take one final look at the essential ideas before deciding what to do next. Need a quick reminder?

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Summary

Before moving on, take one final look at the most important ideas from this review.

KEY TAKEAWAY 1

Connect T, f, ω, and v before calculating

One revolution is 2π radians and a distance 2πr. Use f=1/T, ω=2π/T, and v=ωr to translate between cycle, angular, and linear descriptions.

KEY TAKEAWAY 2

Velocity is tangent; acceleration is inward

Constant speed can coexist with nonzero acceleration because velocity direction changes. Use a_c=v²/r=ω²r and always identify the center first.

KEY TAKEAWAY 3

Model the real forces, then impose the radial requirement

Draw the free-body diagram with actual interactions. Their inward components must satisfy ΣFᵣ=mv²/r; never add a duplicate “centripetal force.”

Ready for your next step?

You've reviewed the essential ideas one last time. Now choose the resource that best matches how confident you feel. Need a quick reminder?

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Next Step

Great work!

You've completed this review. Choose the next resource that best matches how confident you feel.

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