Physics

Circular Motion Kinematics

Study of circular motion. Angular quantities (position, velocity, and acceleration), relationship with linear quantities, Uniform Circular Motion (UCM) and Uniformly Accelerated Circular Motion (UACM).

The motion of an object along a circular path is defined by angular quantities:

  • Angular Position θ\theta: It is the angle traversed measured from the positive X-axis in radians (rad). The positive direction is counterclockwise. It corresponds to position in linear motion.
  • Angular Displacement Δθ\Delta\theta: is the difference between two angular positions Δθ=θf−θi\Delta\theta = \theta_f - \theta_i (rad). It corresponds to displacement in linear motion.
  • Angular Velocity ω\omega: measures the angular displacement per unit of time (rad/s). It corresponds to velocity in linear motion.
  • Angular Acceleration α\alpha: represents the change in angular velocity with respect to time (rad/s²). It corresponds to acceleration in linear motion.
Angular quantities in circular motion
Revolutions per minute (rpm): unit of angular velocity. Conversion factor between rpm and rad/s
1 rpm→2π rad60 s1\text{ rpm} \rightarrow \frac{2\pi\text{ rad}}{60\text{ s}}
Revolutions per second (rps):
1 rps→2π rad1 s1\text{ rps} \rightarrow \frac{2\pi\text{ rad}}{1\text{ s}}
  • When a body undergoes circular motion, it traverses an arc of length s=2πRs=2\pi R when rotating through an angle of θ=2π (rad)\theta=2\pi \text{ (rad)}, therefore the arc length ss along the path is related to the rotated angle by:

    s=θ⋅Rs = \theta \cdot R
  • Likewise, the relationship between linear (tangential) velocity and angular velocity, as well as linear (tangential) acceleration and angular acceleration;

    v=ω⋅Rv = \omega \cdot R

    at=α⋅Ra_t = \alpha \cdot R
Relationship between linear and circular quantities
  • Centripetal or normal acceleration is the acceleration component that keeps the object describing the curve, directed towards the center of the curve.
an=v2R=ω2Ra_n = \frac{v^2}{R} = \omega^2 R
The equations of circular motion are analogous to linear equations, making the following substitutions:
s→θv→ωat→αs \rightarrow \theta \qquad v \rightarrow \omega \qquad a_t \rightarrow \alpha
In Uniform Circular Motion, the object moves with constant angular velocity:
ω=cte\omega = cte

The equation of motion is:
θ=θ0+ω0⋅t\theta = \theta_0 + \omega_0 \cdot t

Linear velocity is also constant. Tangential acceleration will be 0, and centripetal acceleration will be constant:
v=cteat=0an=v2R=ctev = cte \qquad a_t = 0 \qquad a_n = \frac{v^2}{R} = cte
Sign convention
Period: The time taken to complete one full revolution, in seconds (s):
T=2πωT = \frac{2\pi}{\omega}
Frequency: The number of revolutions per unit of time. It is the reciprocal of the period. Measured in hertz (Hz) which is equivalent to (rps)
f=1Torω=2πff = \frac{1}{T} \quad \text{or} \quad \omega = 2\pi f
An object moves with a constant velocity of 5 m/s along a circular path of radius 10 m. Calculate:
a) angular velocity in rad/s and rpm,
b) the period and frequency of the motion,
c) the number of revolutions and the distance covered in 10 min.
a)
ω=vR=510=0.5 rad/s\htmlClass{text-primary}{\omega = \frac{v}{R} = \frac{5}{10} = 0.5\text{ rad/s}}
ω=0.5rads⋅1 rev2π rad⋅60 s1 min=4.77 rpm\htmlClass{text-primary}{\omega = 0.5\frac{\text{rad}}{\text{s}} \cdot \frac{1\text{ rev}}{2\pi\text{ rad}} \cdot \frac{60\text{ s}}{1\text{ min}} = 4.77\text{ rpm}}
b)
T=2πω=12.56 s\htmlClass{text-primary}{T = \frac{2\pi}{\omega} = 12.56\text{ s}}
f=1T=0.079 Hz\htmlClass{text-primary}{f = \frac{1}{T} = 0.079\text{ Hz}}
c) Applying the UCM equation of motion with t=600s:
θ=θ0+ω0⋅t  ⟹  θ=0.5⋅600=300 rad\htmlClass{text-primary}{\theta = \theta_0 + \omega_0 \cdot t \implies \theta = 0.5 \cdot 600 = 300\text{ rad}}
The number of revolutions will be:
N=θ2π=3002π=47.74 rev\htmlClass{text-primary}{N = \frac{\theta}{2\pi} = \frac{300}{2\pi} = 47.74\text{ rev}}
The distance, or arc length covered:
s=θ⋅R=300⋅10=3000 m\htmlClass{text-primary}{s = \theta \cdot R = 300 \cdot 10 = 3000\text{ m}}
In Uniformly Accelerated Circular Motion, the object moves with a constant angular acceleration: α=cte\alpha = cte
The equations of motion are:
{θ=θ0+ω0⋅t+12α⋅t2ω=ω0+α⋅t\begin{cases} \theta = \theta_0 + \omega_0 \cdot t + \frac{1}{2}\alpha \cdot t^2 \\ \omega = \omega_0 + \alpha \cdot t \end{cases}
Sign convention
Tangential acceleration will also be constant, and centripetal or normal acceleration will be variable.
at=α⋅R=ctea_t = \alpha \cdot R = cte
an=v2Ra_n = \frac{v^2}{R}
Combining both equations of motion yields Torricelli's Equation for rotation:
ω2=ω02+2α(θ−θ0)\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)