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Circular motion is one of the most elegant topics in A-Level Physics. A body moving in a circle is a perfect example of accelerated motion in which the speed is constant but the velocity is changing all the time. Before we can talk about the forces involved, we need a new language — the language of angular quantities: angle measured in radians, angular displacement, angular velocity, frequency and period.
This lesson is the foundation of OCR A-Level Physics A Module 5.2 (Circular motion). Everything else in the topic — centripetal force, banked tracks, vertical loops — depends on being able to move fluently between the linear world (metres and metres per second) and the angular world (radians and radians per second).
You have been using degrees since primary school. Why, at A-Level, do we suddenly switch to radians?
The honest answer is: calculus. Differentiating the trigonometric functions gives the neatest possible formulae only when the angle is measured in radians. Engineers and physicists need those neat formulae, so they use radians by default and leave degrees to navigators and carpenters.
The practical answer is: arc length. With radians, the relationship between the arc length s, the radius r and the angle θ is simply
s = rθ
No awkward factor of π/180. The radian was invented specifically so this equation would have no fudge factor in it.
Exam Tip: Whenever an OCR Physics question gives you an angle without explicitly saying "degrees", assume it is in radians. Make sure your calculator is in radian mode for any circular motion or SHM question.
Imagine a circle of radius r. Mark two radii so that the arc between them has length s. The angle θ between the radii, measured in radians, is defined as
θ = s / r
In words: one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Because both s and r are lengths, the radian is dimensionless — it is really just a ratio.
graph LR
C((Centre)) --- A[Point A]
C --- B[Point B]
A -. arc s .- B
If the arc length equals the radius (s = r), the angle is exactly 1 rad. If the arc length is the full circumference (s = 2πr), the angle is 2π rad, which corresponds to 360°.
| Degrees | Radians |
|---|---|
| 360° | 2π |
| 180° | π |
| 90° | π/2 |
| 60° | π/3 |
| 45° | π/4 |
| 30° | π/6 |
| 1° | π/180 ≈ 0.01745 rad |
| 1 rad | 180/π ≈ 57.30° |
Conversion rules:
- Degrees → radians: multiply by π/180
- Radians → degrees: multiply by 180/π
Angular displacement θ is the angle, measured in radians, through which a rotating body has turned from some reference direction. It is the angular equivalent of linear displacement s.
Just as linear displacement is a vector, angular displacement is also a vector — by convention, its direction lies along the axis of rotation and is given by the right-hand rule. For OCR A-Level purposes you only need to treat it as a signed scalar: positive for anticlockwise, negative for clockwise (or vice versa, by convention).
A car tyre of radius 0.33 m rolls through 12 complete revolutions without slipping. Calculate the distance travelled.
One revolution = 2π rad, so 12 revolutions = 24π rad.
s = rθ = 0.33 × 24π = 24.88 m
Notice how painless the calculation is in radians. In degrees you would have needed 12 × 360 = 4320°, then converted to radians anyway.
Just as linear velocity is the rate of change of linear displacement, angular velocity ω (the Greek letter omega) is the rate of change of angular displacement:
ω = θ / t (for uniform circular motion) ω = dθ/dt (instantaneous angular velocity)
The SI unit is radians per second (rad s⁻¹). Numerically ω is a scalar for our purposes, but strictly speaking it is also a vector along the axis of rotation.
A body that travels once around a circle (θ = 2π) in a time T — the period — has angular velocity
ω = 2π / T
Rearranging,
T = 2π / ω
The period tells you how many seconds each revolution takes.
Frequency f is the number of complete revolutions per second, measured in hertz (Hz). Since one revolution takes T seconds, there are 1/T revolutions per second, so
f = 1 / T ω = 2πf
You will use ω = 2πf constantly, both in circular motion and later in simple harmonic motion.
If a body moves in a circle of radius r with angular velocity ω, in time t it sweeps through an angle ωt and covers an arc length
s = rθ = r(ωt)
Its linear (tangential) speed is therefore
v = s / t = rω
This is arguably the single most important equation in circular motion. It is the bridge between the angular world and the linear world.
| Angular quantity | Linear equivalent | Relationship |
|---|---|---|
| θ (rad) | s (m) | s = rθ |
| ω (rad s⁻¹) | v (m s⁻¹) | v = rω |
| α (rad s⁻²) | a_tangential (m s⁻²) | a_t = rα |
A vinyl record spins at 33⅓ rpm (revolutions per minute). A speck of dust sits 0.12 m from the centre.
(a) Angular velocity: 33⅓ rpm = 33.33 / 60 = 0.556 rev s⁻¹
ω = 2πf = 2π × 0.556 = 3.49 rad s⁻¹
(b) Period:
T = 2π / ω = 2π / 3.49 = 1.80 s
(which checks out: 60 s / 33.33 rev ≈ 1.80 s per revolution).
(c) Linear speed of the dust:
v = rω = 0.12 × 3.49 = 0.419 m s⁻¹
Estimate the linear speed of a point on the Earth's equator. Take the Earth's radius as 6.37 × 10⁶ m and its period as 24 hours.
T = 24 × 3600 = 86 400 s
ω = 2π / T = 2π / 86 400 = 7.27 × 10⁻⁵ rad s⁻¹
v = rω = 6.37 × 10⁶ × 7.27 × 10⁻⁵ = 463 m s⁻¹
So you and your chair are moving eastward at roughly 463 m s⁻¹ (about Mach 1.4) without feeling a thing, because the ground beneath you is moving at exactly the same speed.
Exam Tip: OCR frequently asks candidates to compute the linear speed of a satellite, spinning wheel or ride at a funfair using
v = rω. Make sure your angular velocity is in rad s⁻¹, not revolutions per minute.
A body moving at constant speed in a circle has a constant |v| but a continuously changing direction of v. Because velocity is a vector, any change in direction is a change in velocity, and any change in velocity is — by Newton's first law — an acceleration.
The velocity vector is always tangential to the circle. At the top of the circle it points one way; a quarter-turn later it points perpendicular to the first. The speed is unchanged, but the velocity is not.
graph TD
subgraph Circle
P1[Position 1<br/>v tangential, east]
P2[Position 2<br/>v tangential, north]
P3[Position 3<br/>v tangential, west]
P4[Position 4<br/>v tangential, south]
end
This observation is the gateway to the next lesson, where we will quantify this "changing velocity" as the centripetal acceleration and use Newton's second law to deduce the force required to keep the object in its circular path.
ω = 1500 × 2π / 60 = 157 rad s⁻¹, not 1500.s = rθ with θ in degrees. The equation is only valid in radians.T = 1/f, not T = f.| Quantity | Symbol | SI Unit | Formula |
|---|---|---|---|
| Angular displacement | θ | rad | s/r |
| Period | T | s | 1/f |
| Frequency | f | Hz | 1/T |
| Angular velocity | ω | rad s⁻¹ | θ/t = 2π/T = 2πf |
| Linear (tangential) speed | v | m s⁻¹ | rω |
Master these five relationships and you will never be stuck on a circular-motion calculation. The next lesson builds directly on them to derive centripetal acceleration — the acceleration required to keep an object moving in a circle.