Rotational Kinematics

Learning Goals

After this lesson, you should be able to:

• describe angular position and displacement in radians;
• choose and consistently use a clockwise or counterclockwise sign convention;
• calculate average angular velocity and angular acceleration;
• distinguish angular velocity direction from speeding up or slowing down;
• apply constant-angular-acceleration equations;
• connect slopes and signed areas among \(\theta\), \(\omega\), and \(\alpha\) graphs;
• decide when a rigid-system model is necessary; and
• design a video experiment to measure rotational motion.

1. Rigid-System Rotation

A rigid system maintains its shape while rotating about an axis. Different points move in different instantaneous directions, so the entire rotating system cannot always be replaced by one point object. Nevertheless, every point fixed in the same rigid system sweeps through the same angular displacement during the same interval.

The axis must be specified. A door rotating about its hinges and the same door rotating about a line through its center have different motions even when the angle turned is equal.

2. Angular Position and Radians

Angular position \(\theta\) locates a reference line fixed in the rotating system relative to a chosen zero direction. Angular displacement is

Use radians in rotational equations:

Example: A wheel turns \(3.25\) revolutions counterclockwise. If counterclockwise is positive,

4. Choose a Sign Convention

For rotation about one axis, choose one direction as positive. A common convention is counterclockwise positive and clockwise negative, but the reverse convention also works if used consistently.

The sign of angular acceleration alone does not say “speeding up” or “slowing down.” Compare the signs of \(\omega\) and \(\alpha\).

5. Average Angular Velocity

Average angular velocity is the angular displacement per elapsed time:

Example: Angular position changes from \(+1.2\,\mathrm{rad}\) to \(-5.1\,\mathrm{rad}\) in \(2.1\,\mathrm s\):

The negative result indicates net clockwise angular displacement under the selected convention. It does not reveal every direction change that may have occurred within the interval.

6. Average Angular Acceleration

Average angular acceleration measures how angular velocity changes:

If \(\omega\) changes from \(+8.0\) to \(-4.0\,\mathrm{rad/s}\) in \(3.0\,\mathrm s\),

The system first slows its positive rotation, stops momentarily, and then speeds up in the negative direction. A negative final angular velocity cannot be interpreted as merely “slower.”

7. Translational–Rotational Analogy

The analogy transfers mathematical structure, not units. Replace \(x\to\theta\), \(v\to\omega\), and \(a\to\alpha\) while keeping the same sign logic and constant-acceleration assumptions.

8. Constant Angular Acceleration Equations

When \(\alpha\) is constant,

Select an equation containing the known quantities and the desired unknown. These relationships are not valid with a single average \(\alpha\) when angular acceleration varies significantly through the interval.

9. Worked Example: A Fan Speeds Up

Question: A fan begins at \(\omega_i=4.0\,\mathrm{rad/s}\) and has constant angular acceleration \(+3.0\,\mathrm{rad/s^2}\) for \(6.0\,\mathrm s\). Find its final angular velocity and angular displacement.

Answer: \(\omega_f=22\,\mathrm{rad/s}\) and \(\Delta\theta=78\,\mathrm{rad}\), about \(12.4\) revolutions. The average angular velocity check gives \([(4+22)/2](6)=78\,\mathrm{rad}\).

10. Worked Example: Braking to Rest

Question: A wheel rotates counterclockwise at \(24\,\mathrm{rad/s}\). Braking produces constant angular acceleration \(-3.0\,\mathrm{rad/s^2}\). Find the stopping time and angle turned before rest.

Answer: The wheel stops after \(8.0\,\mathrm s\), turning \(96\,\mathrm{rad}\), or about \(15.3\) revolutions. Extending the equation beyond \(8.0\,\mathrm s\) would predict reversal, which may not match a brake that simply holds the wheel at rest.

11. Reading an Angular-Position Graph

• The vertical value is angular position, not distance traveled.
• The secant slope over an interval is average angular velocity.
• A tangent slope represents instantaneous angular velocity.
• A horizontal tangent means momentary rest.
• Increasing positive slope means positive angular acceleration.
• Negative slope with increasing steepness means negative angular velocity and negative angular acceleration.

A graph may cross \(\theta=0\) without the system stopping; crossing the reference angle is not the same as having zero slope.

12. Reading Angular-Velocity and Acceleration Graphs

Signed area under an angular-velocity graph gives angular displacement:

Slope of an \(\omega\)-versus-\(t\) graph is angular acceleration, while signed area under an \(\alpha\)-versus-\(t\) graph is angular-velocity change:

Area below the time axis subtracts. A negative \(\omega\) value describes clockwise motion under the usual convention; a negative slope describes negative angular acceleration.

13. Worked Graph Example

Angular velocity rises linearly from \(0\) to \(12\,\mathrm{rad/s}\) during \(4.0\,\mathrm s\), remains at \(12\,\mathrm{rad/s}\) for \(3.0\,\mathrm s\), then decreases linearly to zero during \(2.0\,\mathrm s\). The total angular displacement is the sum of two triangular areas and one rectangle:

The angular acceleration is \(+3.0\,\mathrm{rad/s^2}\), then zero, then \(-6.0\,\mathrm{rad/s^2}\). The wheel never reverses because \(\omega\) never becomes negative.

14. Investigation: Video Analysis of a Turntable

Use a rotating platform or wheel, a visible radial marker, a phone camera, and a timing reference.

1. Mark the axis and a reference line, then state the positive rotation direction.
2. Record the system starting, rotating, and slowing while keeping the camera aligned with the axis.
3. Extract angular position at equal time intervals; unwrap angles beyond one revolution rather than resetting every \(2\pi\).
4. Plot \(\theta(t)\), calculate interval slopes for \(\omega(t)\), and calculate velocity slopes for \(\alpha(t)\).
5. Test whether \(\omega(t)\) is linear during any interval and use a best-fit slope to estimate constant \(\alpha\).
6. Compare graph area with measured \(\Delta\theta\).
7. Discuss frame rate, marker width, parallax, axis wobble, and angle-unwrapping uncertainty.

15. Common Traps and Key Takeaways

• Use radians in rotational kinematic equations.
• State the axis and positive direction before assigning signs.
• Angular displacement is final minus initial position, not automatically total angle traveled.
• Negative angular velocity indicates direction, not negative speed.
• Compare \(\omega\) and \(\alpha\) signs to decide whether angular speed increases.
• Use constant-\(\alpha\) equations only when justified.
• Slope and signed area have different graph meanings.
• Do not reset angular position every revolution when total displacement is required.