Motion · Topic 4 of 17

Rotation & Uniform Circular Motion

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Physics · rotation

Circling at constant speed is still accelerating, because direction keeps changing. Both a_c = v²/r and the net force point to the center.

01 · The key ideaCircle at constant speed = accelerating

02 · Accelerationa_c = v²/r, toward the center

03 · ForceF_c = mv²/r, pointing inward

Constant speed in a circle is still acceleration (direction keeps changing) must know

Centripetal force comes from a real force (tension, friction, gravity)

Centripetal acceleration a_c = v² ÷ r, toward the center must know

Centripetal force F_c = mv² ÷ r, the net inward force must know

1 revolution = 360° = 2π rad

1 rad ≈ 57.3°; π rad = 180°

Angular velocity ω = angle turned ÷ time (rad/s)

Linear speed v = rω — the rim moves faster than near the axle

Angular acceleration α = Δω ÷ time (rad/s²)

Torque τ = r × F (force at right angles), in N·m

Arc length s = rθ (θ in radians)

More torque: bigger force, farther from pivot, at 90°

Faster or tighter turn → more centripetal acceleration (2× speed = 4×)

1.5 kg, 4 m/s, r = 0.8 m: a_c = 16 ÷ 0.8 = 20 m/s², F_c = 30 N

30 N at 0.25 m, at right angles: τ = 0.25 × 30 = 7.5 N·m

Commonly confused

Inward force lost → straight-line path along the tangent, not outward

Show 5 more facts

Net torque = 0 → no change in rotation (lighter person sits farther out)

a_c = rω² (same as v²/r)

General torque τ = rF sin θ

Centrifuge: big centripetal acceleration separates by density

Counterclockwise torque +, clockwise − (common convention)

Pearl

Centripetal means center-seeking: in circular motion the acceleration and the net force both point inward.

Practice this topic, then test the whole section.

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