Rotational Kinematics: Study Guide (Physics 1 5.1)

This guide is for the algebra-based Physics 1 course. Unit 5 builds rotational versions of the ideas you used in Units 1 and 2. If you are comfortable with displacement, velocity and acceleration along a line, you already know most of the mathematics here. No calculus is needed.

When a point is not enough: rigid systems

Until now we have used the object model: a car or a block is treated as a single point. That works when every part of the thing moves the same way.

A spinning wheel breaks this model. While the wheel turns, a point at the top moves one way and a point at the bottom moves the opposite way. The centre of the wheel may not move at all. No single point describes the motion.

So we use a new model, the rigid system:

• it keeps its shape (the distances between its parts do not change);
• different points on it move in different directions as it rotates;
• it rotates about an axis, a line that the whole body turns around.

A rigid system cannot be modelled as an object while you care about its rotation.

The object model still has a place. If the rotation does not affect the question, you can describe the motion of the system’s centre of mass and treat it as a point. A cricket ball spins as it flies, but if you only want its range you can model it as one object at its centre of mass. The spin matters only if the question is about the spin itself (or about effects that come from it, such as the curving path of a football kicked with spin).

Angular position and angular displacement

To describe rotation, choose:

1. An axis of rotation, for example the axle of a wheel.
2. A reference line, from the axis out to a starting direction, where θ = 0.
3. A positive direction: clockwise or counterclockwise. Either is fine, but write it down, for example “counterclockwise positive”.

The angular position θ of a point on the body is the angle between the reference line and the line from the axis to that point. The angular displacement is the change in angular position:

Δθ = θ − θ₀

Δθ has a sign. With counterclockwise positive, a wheel that turns a quarter-turn clockwise has Δθ = −π/2 rad.

Measure angles in radians

The radian is the natural unit for rotation. One full revolution is 2π rad.

Revolutions	Degrees	Radians
1 rev	360°	2π rad ≈ 6.28 rad
½ rev	180°	π rad
¼ rev	90°	π/2 rad ≈ 1.57 rad
1/(2π) rev	about 57.3°	1 rad

Rotation rates are often quoted in revolutions per minute (rpm). To convert to rad/s, multiply by 2π (rad per rev) and divide by 60 (s per min). For example, 45 rpm = 45 × 2π ÷ 60 ≈ 4.71 rad/s.

Angular velocity and angular acceleration

These are defined exactly like their linear partners.

Average angular velocity: ω_avg = Δθ / Δt (unit rad/s)

Average angular acceleration: α_avg = Δω / Δt = (ω − ω₀) / Δt (unit rad/s²)

ω (omega) and α (alpha) carry the same sign convention as θ. With counterclockwise positive, a wheel turning clockwise has negative ω.

Every point on a rigid body has the same θ change, the same ω and the same α at a given instant. A point near the axle and a point on the rim both turn through one revolution when the wheel does. (How fast each point moves along its path does depend on its distance from the axis; that is Topic 5.2.)

Speeding up or slowing down?

The sign rule from Topic 1.2 carries straight over:

ω	α	What happens to the rate of spin
+	+	speeding up (turning counterclockwise, faster)
−	−	speeding up (turning clockwise, faster)
+	−	slowing down
−	+	slowing down
any	0	constant angular velocity

(The table assumes counterclockwise is positive.)

A negative α does not by itself mean the rotation is slowing.

Instantaneous values

An instantaneous angular velocity is the average angular velocity over a very short interval. Suppose a turntable starts from rest and its angle is θ = (0.75 rad/s²)t², so θ = 3.0 rad at t = 2.0 s. Average angular velocities starting at t = 2.0 s:

Interval	Δθ (rad)	Δt (s)	ω_avg (rad/s)
2.0 s → 3.0 s	3.75	1.0	3.75
2.0 s → 2.5 s	1.6875	0.5	3.375
2.0 s → 2.1 s	0.3075	0.1	3.075
2.0 s → 2.01 s	0.030075	0.01	3.0075

The values settle on 3.0 rad/s, the instantaneous angular velocity at 2.0 s. On an angle–time graph it is the slope of the tangent line at that instant.

The analogy with linear motion

Rotation about one fixed axis follows the same mathematics as motion along one line. Swap the symbols and every Unit 1 tool still works.

Linear (one dimension)	Rotational (one axis)
position x (m)	angular position θ (rad)
displacement Δx	angular displacement Δθ
velocity v_x (m/s)	angular velocity ω (rad/s)
acceleration a_x (m/s²)	angular acceleration α (rad/s²)
v_x = v_x0 + a_x t	ω = ω₀ + αt
x = x₀ + v_x0 t + ½a_x t²	θ = θ₀ + ω₀t + ½αt²
v_x² = v_x0² + 2a_x(x − x₀)	ω² = ω₀² + 2α(θ − θ₀)

The three equations on the right are valid only when α is constant. As in Unit 1, choose the equation that leaves out the quantity you neither know nor need. For constant α you can also use Δθ = ½(ω₀ + ω)t, the area under a straight-line ω–t graph.

Functional dependence: predicting changes

Exam questions often ask how one quantity changes when another does. For a rotation that slows to rest with constant α, the third equation gives 0 = ω₀² + 2αΔθ, so the size of the stopping angle is

|Δθ| = ω₀² / (2|α|)

• Double the starting angular velocity (same α): the stopping angle becomes 2² = 4 times larger.
• Double the size of α (same ω₀): the stopping angle halves.

The stopping time is t = ω₀ / |α|, so doubling ω₀ only doubles the time. Write the symbolic expression first, then read off the factor.

Graphs of rotation

The graph rules of Topic 1.3 transfer directly:

• Slope of an angle–time graph = angular velocity.
• Slope of an angular velocity–time graph = angular acceleration.
• Area under an ω–t graph = angular displacement (area below the axis is negative).
• Area under an α–t graph = change in angular velocity.

Reading Figure 2: from 10 s to 13 s, ω is positive and α is negative, so the drum is slowing down. The total angle turned is 114 rad ÷ 2π ≈ 18.1 rev.

Worked example 1: a potter’s wheel speeding up

Question. Take counterclockwise as positive (seen from above). A potter’s wheel starts from rest and speeds up steadily, reaching 90 rpm counterclockwise after 6.0 s. Find (a) its final angular velocity in rad/s, (b) its angular acceleration and (c) how many revolutions it makes in the 6.0 s.

1. Convert: ω = 90 rev/min × 2π rad/rev ÷ 60 s/min = 3π rad/s ≈ +9.42 rad/s.
2. Angular acceleration: α = (ω − ω₀) / t = (9.42 − 0) ÷ 6.0 = π/2 ≈ +1.57 rad/s².
3. Angular displacement (constant α): Δθ = ½(ω₀ + ω)t = ½ × (0 + 9.42) × 6.0 ≈ 28.3 rad.
4. Revolutions: 28.3 rad ÷ 2π rad/rev = 4.5 rev.

Check. Use a different equation: ω² = 2αΔθ gives 2 × 1.57 × 28.3 ≈ 88.8, and 9.42² ≈ 88.8. A second check: the average rate is 45 rpm, and 45 rev/min for 0.10 min is 4.5 rev.

Worked example 2: a grinding wheel slowing down clockwise

Question. Take counterclockwise as positive. A bench grinding wheel spins clockwise at 42 rad/s. It is switched off and slows at a constant rate, turning through 63 rad before it stops. Find (a) its angular acceleration, (b) the time it takes to stop and (c) the number of revolutions.

1. Signed values: ω₀ = −42 rad/s (clockwise), ω = 0, Δθ = −63 rad (the wheel still turns clockwise while it slows).
2. Time is not given or needed for (a), so use ω² = ω₀² + 2αΔθ: 0 = (−42)² + 2α(−63), so α = 1764 ÷ 126 = +14 rad/s².
3. Time: t = (ω − ω₀) / α = (0 − (−42)) ÷ 14 = 3.0 s.
4. Revolutions: 63 ÷ 2π ≈ 10 rev.

Interpretation. α is positive while the wheel slows down, because ω is negative. Opposite signs mean slowing down. “Positive α” here means “acting counterclockwise”, against the spin.

Check. Δθ = ½(ω₀ + ω)t = ½ × (−42 + 0) × 3.0 = −63 rad. ✓

Investigating rotation in the lab

A typical question asks how you would test whether a wheel’s angular acceleration is constant. One sound plan:

1. Fix a small marker to the rim of the wheel and film it from the front, with a timer or known frame rate in view.
2. Choose counterclockwise as positive and a reference line (for example, horizontal to the right).
3. From the video, record the marker’s angular position every fixed time interval, counting whole turns so θ keeps increasing (do not reset at 2π).
4. For each interval find ω_avg = Δθ / Δt and plot it against the midpoint time of the interval.
5. A straight line means constant α; its slope is α and its intercept is ω₀.

Repeat the run and average, and use short intervals so the averages are close to instantaneous values. A photogate counting spokes as they pass is another good way to measure ω.

Common misconceptions

• “Points on the rim spin faster, so they have a bigger ω.” Every point on a rigid body has the same ω. Rim points have greater linear speed (Topic 5.2).
• “Negative α means slowing down.” Compare signs (Worked example 2).
• Mixing angle units. Quoting rpm in an equation with rad/s², or degrees with rad/s, gives nonsense. Convert to radians first.
• Resetting the angle after each turn. Over many turns Δθ keeps growing: 3 rev is 6π rad, not 0.
• Using the constant-α equations when α changes. Split the motion into stages, as in Figure 2.
• “A spinning object can always be treated as a point.” Only when its rotation does not matter to the question.
• Doubling ω₀ doubles the stopping angle. It quadruples it, because |Δθ| = ω₀² / (2|α|).

Where this leads

Topic 5.2, Connecting Linear and Rotational Motion, links θ, ω and α to the distance, speed and tangential acceleration of each point on the body. Later topics ask what causes angular acceleration: torque (Topic 5.3) and rotational inertia (Topic 5.4). Try the practice questions now, then use the revision notes and the checklist. You can also return to the course roadmap.