Theory — Angular Motion
Angular Displacement
When an object moves around a circle, its position is described by an angle θ measured from a starting line, in radians. The change in this angle is the angular displacement. One full revolution is 2π radians, so the angular displacement can be many radians if the object goes around several times.
θ = angular displacement (rad)
ω = angular velocity (rad/s)
t = time (s)
Angular Velocity and Angular Acceleration
The angular velocity ω is how fast the angle changes (radians per second). The angular acceleration α is how fast the angular velocity changes. If the object goes around at a steady rate — a constant angular velocity — then ω does not change, so the angular acceleration is zero.
If ω is constant, Δω = 0
Linking to Linear Quantities
A point on the rim at radius r has a tangential speed v = ωr directed along the circle. Even when the angular velocity is constant, the direction of this velocity keeps changing, so there is a centripetal acceleration ac = ω²r pointing toward the centre. This centripetal acceleration is a linear acceleration that changes the direction of motion; it is not the same as angular acceleration, which would change the rate of spin.
a_c = ω² · r (centripetal acceleration, m/s²)
More time
At constant ω, the angular displacement grows in direct proportion to time (θ = ωt).
Constant spin
If ω does not change, the angular acceleration α is zero — no matter how fast the spin.
Still accelerating
The bug has centripetal acceleration ω²r toward the centre even when α = 0.
| Quantity | Symbol | Equation / meaning |
|---|---|---|
| Angular displacement | θ | θ = ωt at constant ω (rad) |
| Angular velocity | ω | rate of change of angle (rad/s) |
| Angular acceleration | α | α = Δω/Δt; zero if ω constant |
| Tangential speed | v | v = ωr (m/s) |
| Centripetal acceleration | a_c | a_c = ω²r (m/s²) |
| Revolutions | N | N = θ/(2π) |
Instructions — Running the Virtual Experiment
The Rotation tab spins the ladybug at a chosen angular velocity and reads the angle, time, and related quantities; the Graphs tab plots the angle and angular velocity against time. Record every reading in your lab notebook. Angles are shown in radians.
Simulation — Ladybug on a Rotating Track
Controls
Scenarios (ω, run time)
Run a scenario, then watch the graphs
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission.
Angular Displacement and Angular Acceleration
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To investigate the rotational motion of a ladybug on a circular track of diameter 3 m, to measure the angular displacement produced by a constant angular velocity over a set time, and to compare the measured angular displacement with the theoretical value θ = ωt. The experiment also examines why the angular acceleration is zero when the angular velocity is constant.
Theory
For motion at constant angular velocity, the angular displacement increases in direct proportion to time, θ = ωt, where θ is in radians. The angular acceleration is the rate of change of angular velocity, α = Δω/Δt; if the angular velocity is constant then α = 0. A point at radius r has tangential speed v = ωr and centripetal acceleration a_c = ω²r directed toward the centre.
v = ω·r · a_c = ω²·r (r = 1.5 m)
Calculations — Sample: ω = 5 rad/s, t = 12 s
Angular displacement: θ = ωt = 5 × 12 = 60 rad
Revolutions: N = 60 / (2π) = 9.55 rev
Angular acceleration: ω is constant, so α = 0 rad/s²
Tangential speed: v = ωr = 5 × 1.5 = 7.5 m/s; Centripetal accel: a_c = ω²r = 25 × 1.5 = 37.5 m/s²
Results Table (diameter 3 m, r = 1.5 m)
| Scenario | ω (rad/s) | t (s) | θ = ωt (rad) | Revolutions | α (rad/s²) |
|---|---|---|---|---|---|
| 1 | 5 | 12 | 60 | 9.55 | 0 |
| 2 | 6 | 22 | 132 | 21.01 | 0 |
| 3 | 7 | 6 | 42 | 6.68 | 0 |
The angular displacement read from the simulation matched the theoretical θ = ωt for each scenario.
Discussion
The angular displacement measured from the simulation agreed with the theoretical values of 60, 132, and 42 rad, confirming that θ = ωt for motion at constant angular velocity. The θ-versus-time graph was a straight line whose slope equalled the set angular velocity, and the ω-versus-time graph was a flat horizontal line, showing that the angular velocity did not change. Because the angular velocity was constant, the angular acceleration was zero in every scenario (α = Δω/Δt = 0).
It is worth noting that a zero angular acceleration does not mean the ladybug had no acceleration at all: moving in a circle, it always had a centripetal acceleration a_c = ω²r directed toward the centre (37.5, 54, and 73.5 m/s² for the three scenarios). This is a linear acceleration that continually changes the direction of the velocity, distinct from angular acceleration, which would change the rate of spin. The simulation does not display angular acceleration directly, so only the angular displacement was compared between experiment and theory.
Conclusion
The experiment confirmed that angular displacement is proportional to time at constant angular velocity (θ = ωt), giving 60, 132, and 42 rad for the three scenarios, and that the angular acceleration is zero when the angular velocity is constant. The constant-slope θ-vs-time graph and flat ω-vs-time graph supported these conclusions.
Practice Questions
Show all work and include units in your answers. Use a radius of 1.5 m unless told otherwise.