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Physics Unit 5: Circular motion, gravity and rotation

Objects moving in circles need a center-seeking force. Rotation has its own versions of force (torque), mass (rotational inertia) and momentum.

Covers: Centripetal acceleration and force, universal gravitation, torque, rotational inertia, angular momentum

Study guide

An object moving in a circle at constant speed is still accelerating, because its direction keeps changing. This centripetal acceleration points toward the center of the circle, with magnitude a = v²/r. It requires a centripetal force of F = mv²/r, provided by an actual force such as tension in a string, friction for a car on a curve, the normal force on a roller coaster, or gravity for orbits. Centripetal force isn't a new type of force; it's the net inward force. If the force disappears, the object moves off in a straight line tangent to the circle.

Newton's law of universal gravitation, F = Gm₁m₂/r², explains orbits. For a satellite in a circular orbit, gravity provides the centripetal force, so the orbital speed depends on the planet's mass and the orbit's radius, not on the satellite's mass. Gravitational field strength at a planet's surface is g = GM/r², and objects in orbit feel weightless because they're in continuous free fall.

Rotational motion has its own versions of linear quantities. Torque, τ = rF sin θ, is the rotational effect of a force: it's largest when the force is applied far from the pivot and perpendicular to the lever arm, which is why long wrenches help. An object is in rotational equilibrium when torques balance.

Rotational inertia (moment of inertia, I) is resistance to changes in rotation and depends on how mass is distributed: mass farther from the axis means more inertia. Newton's second law for rotation is τ_net = Iα.

Angular momentum, L = Iω, is conserved when no external torque acts. That's why a skater spins faster when pulling in their arms: I decreases, so ω increases.

At the top of a vertical loop, gravity and the normal force both point down, toward the center: N + mg = mv²/r. The block stays on the track only if N ≥ 0, so its speed at the top must be at least √(gr). Combine this with energy conservation to find the release height needed. A rotating object has kinetic energy K = ½Iω². The work done by a torque (force × string length unwound) equals the change in rotational kinetic energy, so for the same work, a smaller rotational inertia gives a larger ω. A rolling object has both translational and rotational kinetic energy.

Angular position θ (in radians), angular velocity ω and angular acceleration α work like x, v and a. For constant α the same kinematic equations apply, such as ω = ω₀ + αt. A point at radius r moves with linear speed v = rω and tangential acceleration a = rα. For an extended object, gravity acts at the center of mass, so a uniform beam's weight acts at its midpoint. A hinge or pivot can push in any direction, so include a pivot force. Taking torques about the pivot removes that unknown force, because its lever arm is zero.

Key ideas

• Centripetal acceleration a = v²/r points toward the center.
• Centripetal force isn't a new force; gravity, tension or friction provides it.
• Gravity weakens with the square of distance.
• Torque τ = rF sin θ; a longer lever arm gives more torque.
• With no outside torque, angular momentum is conserved (spinning skaters speed up when they pull in).

Formulas and rules

Common mistakes

• Drawing a centrifugal force outward on a free-body diagram.
• Forgetting gravity depends on 1/r², not 1/r.

Words to know

Centripetal acceleration

v²/r, toward the center.

Torque

τ = rF sin θ.

Inverse-square law

Force ∝ 1/r²: double the distance, 1/4 the force.

Conservation of angular momentum

Iω stays constant; less rotational inertia means faster spin.

Centripetal force

The net inward force keeping an object in circular motion.

Uniform circular motion

Constant speed in a circle; velocity direction changes.

Centripetal force formula

F = mv²/r

Orbital speed

Set by gravity providing the centripetal force; independent of the satellite's mass.

Gravitational field strength

g = GM/r²

Apparent weightlessness

Being in continuous free fall, as in orbit.

Rotational inertia (I)

Resistance to changes in rotation; larger when mass is farther from the axis.

Newton's second law for rotation

τ_net = Iα

Angular momentum

L = Iω

Rotational equilibrium

Net torque is zero.

Sample practice questions

17 more questions like these, from level 1 to 5, are in the practice set.