4.2 Uniform Circular Motion (Centripetal/Centrifugal)

Uniform Circular Motion (Centripetal/Centrifugal)

Not all Module 2 motion is in a straight line. Propellers, rotors, wheels, gyro rotors, and aircraft in a level turn follow circular (or approximately circular) paths. Uniform circular motion (UCM) means motion at constant speed along a circle of fixed radius. Speed is constant, but velocity is not, because direction changes every instant. Changing velocity means acceleration — and that acceleration points toward the centre of the circle.

Velocity on a Circle

At any instant the velocity vector is tangential to the path (perpendicular to the radius). After a short time the body has moved around the arc and the tangent has a new direction. The magnitude of velocity (speed) can stay the same while the direction rotates; that continuous change of direction is the essence of centripetal acceleration.

Arc length for one full revolution is the circumference 2πr. If the period (time for one revolution) is T, then:

v = 2πr / T

If frequency f is revolutions per second, v = 2πrf.

Angular Speed, Period, and Frequency

Angular speed (angular velocity magnitude) ω measures how fast the angle at the centre sweeps, in radians per second (rad/s):

ω = θ / t  (for constant ω)

One full revolution is 2π radians, so:

ω = 2π / T = 2πf

Period T is the time for one complete revolution (seconds). Frequency f = 1/T is revolutions per second, unit hertz (Hz). Engine and propeller speeds are often quoted in revolutions per minute (rpm); convert carefully:

f (Hz) = rpm / 60  and  ω = 2π × (rpm/60)

Linear (tangential) speed and angular speed are linked by:

v = ω r

Points farther from the axis move faster for the same ω. That is why propeller tip speed is higher than speed at a mid-blade station, and why tip Mach number is a design constraint.

Worked example — propeller tip speed. A propeller of diameter 2.0 m (radius r = 1.0 m) turns at 2400 rpm.

f = 2400/60 = 40 Hz  ω = 2π × 40 ≈ 251 rad/s  v = ωr ≈ 251 × 1.0 ≈ 251 m/s

(About 0.74 times the sea-level speed of sound ≈ 340 m/s — a typical order of magnitude for high-speed props; real limits depend on design and altitude.)

Centripetal Acceleration

The acceleration that keeps velocity changing direction toward the centre is the centripetal acceleration:

a_c = v² / r = ω² r

Direction: always toward the centre of the circle (“centre-seeking”). Magnitude grows with the square of speed and shrinks as radius increases for fixed v. For fixed ω, larger r means larger centripetal acceleration (and higher tip speed).

Worked example. A mass on a 0.50 m string moves in a horizontal circle at 4.0 m/s.

a_c = v²/r = 16 / 0.50 = 32 m/s² toward the centre.

Centripetal Force

Newton’s second law requires a net force toward the centre to produce a_c:

F_c = m a_c = m v² / r = m ω² r

Centripetal force is not a new kind of force in nature — it is the name for the net force role that must point inward. The physical agent might be:

• Tension in a string or cable
• Friction toward the centre (car on a flat curve)
• Horizontal component of lift (aircraft in a banked turn)
• Normal / structural reaction (bearing loads on a spinning shaft, rim of a rotating wheel)

If that inward force is removed, the body continues in a straight line tangent to the path at the release point (Newton’s first law) — it does not “fly outward along a radius.”

Worked example — string tension. A 0.20 kg mass whirled on a light string of length 0.80 m at 3.0 m/s in a horizontal circle (idealised, neglect gravity’s effect on tension for a purely horizontal case):

F_c = mv²/r = 0.20 × 9.0 / 0.80 = 2.25 N (string tension provides this).

Centrifugal Force — Reaction and Apparent Force

Centrifugal means “centre-fleeing.” In the inertial (non-rotating) description used for most Module 2 free-body diagrams, you draw the real inward centripetal force on the moving body; you do not add an extra outward “centrifugal force” on that same free body. By Newton’s third law, the body pulls outward on the string, seat, or bearing that supplies the inward force — that reaction on the constraint is sometimes loosely called centrifugal. In a rotating frame (sitting in a turning cabin), occupants feel an apparent outward force; that is a frame effect, not an additional fundamental interaction.

Exam discipline: centripetal force/acceleration toward the centre; centrifugal as reaction or apparent outward effect — not a second simultaneous real force on the same free-body diagram in inertial analysis.

Aircraft Turning Flight

In a coordinated level turn, the aircraft’s path is (approximately) a horizontal circle. Lift L is tilted inward by bank angle φ. The vertical component balances weight: L cos φ = mg. The horizontal component supplies centripetal force:

L sin φ = m v² / r  ⇒  r = v² / (g tan φ) (from combining with L cos φ = mg)

Faster true airspeed or shallower bank ⇒ larger turn radius. Load factor n = L/W = 1/cos φ increases with bank. Module 2 expects the centripetal idea and the role of the horizontal lift component more than full performance formulae, but the structure is the same: something real pulls (or pushes) toward the centre of the path.

Rotating Machinery and Tip Speeds

Propeller blades, helicopter rotors, fans, and turbines experience large mv²/r loads at the root and along the blade. Mass near the tip contributes heavily because v is largest there. Designers limit rpm and diameter so tip speed and centrifugal stress stay within material and compressibility limits. Imbalance (mass not symmetric about the axis) produces a rotating radial force felt as vibration — linking circular motion to the resonance topics of the next section.

Formula Sheet

Uniform circular motion is constant speed, never constant velocity. Keep that distinction sharp, supply centripetal force with a real agent, and convert rpm carefully — those habits prevent most circular-motion errors on Module 2.