Uniform Circular Motion

Spin a stone on a string, drive a car round a bend or ride a carousel and see why a rotating body keeps accelerating toward the centre even though its speed never changes. Cut the string — and the body flies off in a straight line. Shadows on the axes, graphs and the readout help you check everything against the textbook formulas.

The idea in 30 seconds

Velocity is an arrow

The body goes round the circle at the same speed, but the velocity arrow keeps turning: it always lies along the tangent to the circle.

A turning arrow means acceleration

Acceleration is any change of velocity, in size or in direction. Here the velocity only turns, and the acceleration points to the centre: a = v²/R. Tick “Where a_c comes from” and you will see it in a drawing.

A force pulls toward the centre

For the body to turn, a force toward the centre must act on it: a string, tyre friction, a carousel seat. Remove it and the body goes straight — it does not “fly outward”.

What to try

Press the red button under “Start” — the force toward the centre disappears. Where does the body fly? Then restore the force.
Press “v × 2”. By what factor did the acceleration change? And after “R × 2”?
Move the R slider at the same v: the radius on the screen stays the same but the acceleration and the arrows change. How do they depend on R?
Choose “Car”, set “Ice” and speed the car up: at what speed does it skid?

Graphs: what changes and what does not

The dot on each curve moves together with the body. Coordinates and velocity components oscillate while the magnitude v stays constant. The lower graphs show how a_c depends on v and on R.

What is rotating

R radius 1,5 m

set via v set via T

v speed 4,00 m/s

T period 2,36 s

m mass 0,5 kg

F_max string strength 30 N

What if…

Slow motion / playback rate

Drag left to slow everything down (the physics does not change). Double-click resets to 1×

1×

0,05×1×4×

Show

Vectors v and a_c

Force F_c

Where a_c comes from (Δv)

Shadows on the x and y axes

Angle θ

Trail

Challenge

The drawing is scaled to fit the scene, but the numbers in the panel are exact — they come straight from the formulas.

In short: in uniform circular motion the speed does not change in size but does change in direction, so there is an acceleration a_c = v²/R pointing to the centre. Example: R = 2 m, v = 3 m/s — the period is T = 2πR/v ≈ 4.19 s, the acceleration is a_c = 4.5 m/s², and for m = 1 kg the force is F_c = 4.5 N. Choose “Stone” and “set via v”, enter these numbers and check them on the readout.

KEY FORMULAS

Centripetal acceleration

$$a_ц = \frac{v^2}{R} = \omega^2 R$$

Acceleration of a body in uniform circular motion: always directed toward the centre and perpendicular to the velocity

▶ Detailed Explanation

In circular motion the velocity changes only its direction, but any change of velocity is an acceleration, and it points to the centre of the circle: a_c = v²/R = ω²R. Its magnitude is constant — on the readout a_c does not change with time, only the arrow turns. The smaller the radius and the greater the speed, the larger the acceleration: at v = 3 m/s and R = 2 m it is 4.5 m/s², almost half of g, while at v = 10 m/s and R = 2 m it is already 50 m/s², about 5g.

Period, frequency and angular velocity

$$T = \frac{2\pi R}{v}, \quad f = \frac{1}{T}, \quad \omega = \frac{2\pi}{T} = 2\pi f, \quad v = \omega R$$

All the relations between v, R, T, f and ω in uniform circular motion

▶ Detailed Explanation

The period T is the time of one revolution: the body covers the circumference 2πR at speed v, so T = 2πR/v. The frequency f = 1/T is the number of revolutions per second. The angular velocity ω = 2π/T = 2πf tells through what angle in radians the body turns each second; the linear speed is v = ωR. The “set via v / T” toggle in the simulator recalculates the other quantity with these same formulas. Check: at R = 2 m and v = 3 m/s you get T ≈ 4.19 s, f ≈ 0.24 Hz, ω = 1.5 rad/s.

Centripetal force

$$F_ц = ma_ц = m\frac{v^2}{R} = m\omega^2 R$$

The force, by Newton’s second law, that turns the velocity of a body on a circle

▶ Detailed Explanation

The centripetal force is not a new kind of interaction but the result of ordinary forces: string tension, friction, the normal force or gravity. By Newton’s second law F_c = ma_c = mv²/R: this is the force that deflects the velocity from straight-line motion at every instant. Set the mass m and F_c appears in the “Measurements” panel: at v = 3 m/s, R = 2 m and m = 1 kg it equals 4.5 N. For a satellite in low Earth orbit this role is played by the Earth’s gravity — the satellite keeps “falling” toward the Earth but, moving sideways, misses it every time. Remove the force and the body goes straight.

Coordinate projections in circular motion

$$\begin{aligned} x(t) &= R\cos\omega t \\ y(t) &= R\sin\omega t \end{aligned}$$

Projections of uniform circular motion onto the axes — two sinusoids shifted in phase by π/2

▶ Detailed Explanation

The starting position is chosen on the x axis, so x = R cos ωt and y = R sin ωt — projections of one and the same uniform motion. Both are harmonic oscillations of the same frequency shifted in phase by π/2 (a quarter of a period). The graphs show it: the x(t) and y(t) sinusoids are identical, but the second lags the first by T/4. Switch on “Shadows on the x and y axes”: the blue and green dots oscillate like a mass on a spring. Describing rotation through two projections underlies the whole topic of oscillations and waves.

Angle of rotation and arc length

$$\theta = \omega t, \qquad s = R\theta = vt, \qquad N = \frac{t}{T}$$

How the angle, the distance travelled and the number of revolutions grow in uniform rotation

▶ Detailed Explanation

The angle of rotation θ grows in proportion to time: θ = ωt. It is measured in radians: 1 rad is the angle at which the arc length equals the radius, and a full turn is 2π rad ≈ 6.28 rad (360°). The arc length covered by the body is s = Rθ; since v = ωR this gives s = vt — the same as in uniform straight-line motion. The number of revolutions is N = t/T. Example: at R = 2 m and v = 3 m/s in t = 10 s the body covers s = 30 m, turns through θ = 15 rad and makes N ≈ 2.4 revolutions. In the simulator the yellow sector shows θ, while t, N and s are on the readout and in the “Measurements” panel.

What uniform circular motion is. A body moves along a circle of radius R and the magnitude of its speed v stays constant. Velocity is a vector, and its direction keeps changing: it always lies along the tangent to the circle, that is, perpendicular to the radius. This is how the tip of a second hand, a stone on a string, a car on a roundabout, a carousel seat and a satellite on a circular orbit move.

Why there is an acceleration. Acceleration means any change of velocity, including a change of direction alone. The velocity turns, so there is an acceleration. It is called centripetal: it points to the centre of the circle, perpendicular to the velocity, so it does not change the speed, it only turns the velocity. Its magnitude is a_c = v²/R = ω²R = 4π²R/T². Tick “Where a_c comes from (Δv)” in the simulator — the drawing shows that the difference of two velocities Δv = v₂ − v₁ points to the centre.

The force toward the centre. By Newton’s second law such an acceleration needs a force F_c = m·a_c = mv²/R pointing to the centre. It is not “one more” force but the result of ordinary ones: the tension of a string for a stone, tyre friction for a car, the push of the seat and handrail for a carousel rider, the Earth’s gravity for a satellite. If this force disappears (the string snaps, the road is icy), the body does not “fly outward” — it moves in a straight line along the tangent at the same speed. That is the law of inertia. Press the red button under “Start” and see for yourself.

Period, frequency, angular velocity. The period T is the time of one revolution: T = 2πR/v. The frequency f = 1/T is the number of revolutions per second. The angular velocity ω = 2π/T = 2πf tells through what angle (in radians) the body turns each second; a full turn is 2π rad ≈ 6.28 rad. The angle grows uniformly: θ = ωt, the arc length is s = Rθ = vt, and the linear speed is v = ωR.

Projections and oscillations. The body’s shadows on the axes follow x = R cos ωt and y = R sin ωt. These are two identical sinusoids shifted by a quarter of a period: that is how uniform rotation is linked to harmonic oscillations.

What the acceleration depends on. It grows as the square of the speed (twice as fast — 4 times larger) and falls with the radius (a bend twice as sharp — twice as large). The lower graphs and the “v × 2”, “R × 2” buttons show this by experiment. Mass does not affect the acceleration, but the force F_c depends on it.

How the simulator computes. Uniform rotation has an exact formula θ = ωt, so nothing is approximated: the frame rate and the playback speed do not affect the numbers. The drawing is scaled to fit the scene, while the numbers in the panel are exact. A car skids when a_c > μg, and the string snaps when F_c > F_max; after that the body moves in a straight line at constant speed (braking while sliding is ignored).