Circular Motion

Learning Goals

After this lesson, you should be able to:

• identify tangential velocity and inward centripetal acceleration at any point;
• calculate centripetal acceleration using speed, radius, or period;
• write Newton's second law in radial and tangential directions;
• identify the real force or force components providing radial acceleration;
• analyze cars on flat and frictionless banked curves;
• compare forces at the top and bottom of a vertical circle;
• derive the minimum speed needed at the top of a vertical loop; and
• interpret force, speed, radius, and mass relationships experimentally.

1. Constant Speed Does Not Mean Zero Acceleration

Uniform circular motion has constant speed but continuously changing velocity direction. Its centripetal, or radial, acceleration points toward the instantaneous center of curvature:

The velocity is tangent to the path, so \(\vec v\perp\vec a_c\) at every instant during uniform circular motion.

2. Period and Angular-Speed Forms

One revolution covers circumference \(2\pi r\). If the period is \(T\),

With angular speed \(\omega=v/r=2\pi/T\),

Use consistent units: seconds for period, meters for radius, and radians per second for angular speed.

4. “Centripetal Force” Is a Role, Not a New Interaction

Newton's second law in the inward radial direction is

The left side contains real forces or their radial components. Depending on the situation, tension, gravity, friction, normal force, or several forces together may provide the required inward net force. Do not add a separate \(F_c\) arrow after drawing those interactions.

5. Radial and Tangential Components

For nonuniform circular motion, separate acceleration into perpendicular components:

The radial component changes velocity direction. The tangential component changes speed. Their magnitudes combine as \(|\vec a|=\sqrt{a_r^2+a_t^2}\).

6. Worked Example: Centripetal Acceleration

Question: A sensor travels at \(8.0\,\mathrm{m/s}\) around a circular path of radius \(5.0\,\mathrm m\). Find its centripetal acceleration.

Answer: Its acceleration magnitude is \(13\,\mathrm{m\,s^{-2}}\), directed inward. The object's velocity remains tangent to the circle.

7. Horizontal Circle with a String

A \(0.50\,\mathrm{kg}\) puck moves in a horizontal circle of radius \(1.2\,\mathrm m\) at \(3.0\,\mathrm{m/s}\). If tension is the only horizontal radial force,

The string pulls inward. It does not push the puck forward along the tangent during uniform circular motion.

8. Car on a Flat Curve

For a car turning on level ground without slipping, static friction may supply the horizontal radial force. At the limiting speed,

In this simple model mass cancels. Greater available friction or a larger curve radius permits a greater speed without sliding.

9. Frictionless Banked Curve

On a banked road of angle \(\theta\), the horizontal component of the normal force can supply the radial acceleration while the vertical component balances weight:

Dividing the equations gives the design relation

This speed needs no friction in the ideal model; at other speeds, friction may be required and its direction must be determined from the sliding tendency.

10. Vertical Circle: Top and Bottom

Choose inward as positive separately at each location. For a mass on a string:

Gravity points inward at the top but outward relative to the center at the bottom. A single memorized sign pattern cannot replace a free-body diagram.

11. Minimum Speed at the Top of a Loop

At the minimum speed that maintains a taut string or just maintains contact with an inside track, tension or normal force at the top becomes zero. Gravity alone supplies the inward force:

A negative calculated tension or normal force is unphysical; it signals that the assumed circular contact has already been lost.

12. Proportional Reasoning

From \(a_c=v^2/r\) and \(F_{r,\mathrm{net}}=mv^2/r\):

• doubling speed multiplies \(a_c\) and required radial force by 4;
• doubling radius at fixed speed halves both;
• doubling mass changes required radial force by 2 but does not change \(a_c\);
• halving period at fixed radius multiplies \(a_c\) by 4.

State which quantities are held constant before making a comparison.

13. What Happens When the Constraint Breaks?

If a string breaks or a track ends, the constraint force vanishes. The object retains its instantaneous velocity, so its initial path is tangent to the circle. Other remaining forces, such as gravity, then curve the subsequent trajectory.

An apparent outward tendency in a rotating frame is not an extra real force on an inertial-frame free-body diagram.

14. Investigation: Test \(F_r\propto v^2/r\)

Use a securely mounted rotating platform or a low-mass stopper attached through a guide to a force sensor.

1. Measure rotating mass and radius from the rotation axis to the mass's center.
2. Use video timing for several revolutions to find period and speed.
3. Record the inward force after the motion becomes approximately steady.
4. At fixed \(m\) and \(r\), plot force against \(v^2\).
5. At fixed \(m\) and similar speed, compare force with \(1/r\).
6. Use the best-fit slope and uncertainty to test \(F_r=mv^2/r\).

Keep the rotation plane clear, secure all connectors, use eye protection, and never exceed the apparatus rating.

15. Common Traps and Key Takeaways

• Constant speed can still mean acceleration because velocity direction changes.
• Velocity is tangent and centripetal acceleration is inward.
• Do not draw centripetal force as an extra interaction; identify the real radial forces.
• Use \(v^2\), so speed changes have a squared effect.
• At each point in a vertical circle, redefine inward before assigning signs.
• At the minimum top speed, the contact or tension force is zero, not gravity.
• When the inward constraint disappears, motion begins along the tangent, not radially outward.