Uniform circular motion
A particle follows a circle of fixed radius. In uniform circular motion, speed and angular velocity remain constant while the direction of velocity changes continuously. The period T is the time for one revolution, and frequency f is the number of revolutions per second.
One revolution equals 2π rad or 360°. Revolutions per minute convert as 60 rpm = 2π rad/s, a frequency of 1 Hz and a period of 1 s. At rest, frequency is zero and the period is undefined.
Constant angular acceleration
When angular acceleration α is constant, the initial angular velocity ω0 and elapsed time determine:
Counterclockwise is positive, and the initial angle is measured from the +x axis. Speed v=r∣ω∣ is always nonnegative. The unit circle relates the angle to the particle’s position.
When α=0, the time per revolution changes, so there is no constant global period. Nor does 2π/∣ω(t)∣ give the exact time of the next revolution.
Centripetal acceleration and force
Changing the direction of velocity requires acceleration toward the center. Its magnitude depends on speed and radius; Newton’s second law gives the required radial force.
“Centripetal” describes the inward component of the net force. It can result from tension, gravity, friction or other interactions. It must not be added as an extra force to the interactions that produce it.
At fixed speed, doubling the radius halves ac. At fixed angular velocity, doubling it doubles ac, because speed also doubles.
Tangential force and net force
Tangential acceleration changes speed. Its component aθ is positive counterclockwise, even when the particle moves clockwise. Radial and tangential components are perpendicular:
The net force is the vector sum of both components. The diagram places the force arrows tip to tail to illustrate this sum. When the tangential arrow is labeled “enlarged,” its length has been increased to show its direction; the results give the magnitudes.
The inclined plane provides a concrete force system. In projectile motion, gravitational acceleration keeps its direction; in circular motion, the radial direction rotates with the particle.
Braking, stopping and reversing
Negative angular acceleration does not always slow the particle. When α and ω have the same sign, speed increases; opposite signs reduce it. For constant nonzero angular acceleration, the stopping time is:
At that instant, speed and centripetal force are zero, but tangential force can remain. If acceleration continues, the particle reverses direction. Permanent rest requires angular acceleration to become zero.
Displacement, distance and energy
Angular displacement Δθ keeps its sign; net revolutions equal Δθ/(2π). Distance adds every part of the path, including travel after a reversal, so it can be positive even when angular displacement is zero.
Centripetal force is perpendicular to velocity and does no work. Tangential force changes kinetic energy; its work is negative during braking.
Worked example: braking and reversal
A 3 kg particle moves at 2 rad/s around a circle of radius 2 m. It starts at θ0=0 and maintains α=−1 rad/s2 for 4 s.
1. Stopping time. t∗=−2/(−1)=2 s. At that instant, Δθ=2 rad and speed is zero. Tangential force remains Fθ=3⋅2⋅(−1)=−6 N.
2. State at 4 s.
The particle returns to its initial position with the direction of rotation reversed. It travels 4 m before stopping and 4 m afterward: the total distance is 8 m, although net revolutions are zero.
3. Energy and work. Initially and finally, K=21⋅3⋅42=24 J. Net work over 4 s is zero: work up to the stop is −24 J and work on the return is +24 J.
Scope of the relationships
- Particle. Mass, radius and center are constant, and motion is observed from an inertial frame. The energy of a whole disk requires its moment of inertia.
- Required forces. String tension or track normal force is determined by including the other forces and contact conditions. Gravity also matters in a vertical circle.
- Classical mechanics. These equations describe speeds small compared with the speed of light. Displayed digits do not replace the precision of measurements.
When solving Fc=mv2/r, speed is nonnegative and radius and mass must be positive. If force and speed are both zero, any positive radius or mass satisfies the equation. If exactly one is zero, no finite positive radius or positive mass satisfies it.
The projection of uniform circular motion onto an axis is simple harmonic motion. A circular Kepler orbit adds a gravitational condition: v2=μ/r, where μ is the central body’s gravitational parameter.