Mechanics of Uniform Circular Motion

Uniform Circular Motion (UCM)

Motion in a circular path at a constant speed.

Speed vs. Velocity

Speed is the magnitude of velocity and is constant; velocity changes due to direction changes.

Tangential Velocity

The velocity vector that is tangent to the circular path.

Period (T)

The time taken to complete one full revolution, measured in seconds.

Frequency (f)

The number of revolutions completed per second, measured in Hertz.

Relationship between Period and Frequency

T = 1/f and f = 1/T.

Tangential Speed (v)

Calculated using v = 2πr/T or v = 2πrf.

Centripetal Acceleration (a_c)

Acceleration directed toward the center of a circular path, calculated as a_c = v^2/r.

Direction of Centripetal Acceleration

Always points toward the center of the circular path.

Centripetal Force

The net force required to keep an object moving in a circle, not a physical force.

Newton’s Second Law for Rotation

ΣFradial = mac = mv^2/r - describes forces acting in circular motion.

Static Friction

The force providing centripetal force when a car is turning on a flat road.

Maximum Speed without Slipping

Derived as vmax = √(μs g r) from frictional forces.

Conical Pendulum

A ball on a string moving in a horizontal circle, creating a cone shape.

Critical Velocity (v_critical)

Minimum speed required at the top of a vertical loop to complete the loop.

Normal Force (F_N) at Bottom of Loop

F_N = mg + mv^2/r; it points upward toward the center.

Normal Force (F_N) at Top of Loop

F_N + mg = mv^2/r; it points down toward the center.

Centrifugal Force Myth

The misconception of an outward force when turning; it is actually inertia.

Centripetal Force Source in Satellites

Gravity (F_g) provides the centripetal force for satellites orbiting Earth.

Maximum Static Friction (f_{s,max})

Calculated as f{s,max} = μs F_N; determines the turning capability.

Static Friction for Car Turning

Must act radially inward toward the center of the turn for effective turning.

Effects of Mass on Maximum Speed

Mass cancels out when calculating maximum speed on a turn; heavy and light vehicles slip at the same speed with equal tires.

Mistake: $F_c$ in FBDs

Students often mistakenly draw $F_c$ instead of representing it as the sum of other forces.

Acceleration in Uniform Circular Motion

Acceleration is present even if speed is constant, due to changing direction.

Friction Direction in Turns

Friction acts inward toward the center of the turn, not backward or outward.