Uniform Circular Motion

Uniform Circular Motion

• Object travels in a circle at constant speed.
• Example: Points on a spinning propeller, hands of a watch.
• Points on rotating objects are accelerating, even if the rotation rate is constant.

Centripetal Acceleration

• In 2D/3D kinematics, constant speed can still mean acceleration on a curved path.
• Particle moving counterclockwise on a circular path of radius $r$.
• Velocity vector is tangent to the path.
• Change in velocity is proportional to change in radius: $\frac{\Delta v}{v} = \frac{\Delta r}{r}$.
• Centripetal acceleration: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{v^2}{R}$.

Period of Circular Motion

• Period (T): Time for one complete revolution.
• $T = \frac{2\pi R}{v}$

Angular Speed

• Rate of change of angle with respect to time: $\Omega = \frac{2\pi}{T}$.
• $v = \Omega R$
• Centripetal acceleration in terms of angular speed: $A_C = \frac{v^2}{R} = \Omega^2 R$

Sample Problem: Earth's Orbit

• Earth's revolution: 365 days ($3.15 \times 10^7$ seconds).
• Distance from Earth to Sun: $1.496 \times 10^{11}$ meters.
• Centripetal acceleration: $A_C = \frac{v^2}{R} = 0.00595 \frac{m}{s^2}$
• Angular speed: $\Omega = \frac{2\pi}{T} = 1.995 \times 10^{-7} \frac{rad}{s}$

Tangential and Radial Acceleration

• Non-uniform circular motion involves changing speed.
• Tangential acceleration ($A<em>T$): Acceleration tangent to the circle, due to changing speed. $A</em>T = \frac{dV}{dt}$
• Centripetal acceleration ($A_R$) is always directed inward.
• Total acceleration: $A = A<em>T + A</em>R$.
• Magnitude of total acceleration: $A = \sqrt{A<em>T^2 + A</em>R^2}$

Sample Problem: Car on a Rise

• Constant tangential acceleration: $0.3 \frac{m}{s^2}$.
• Radius of curvature: 500 meters.
• Velocity at top: $6 \frac{m}{s}$.
• Centripetal acceleration: $A_R = \frac{v^2}{R} = 0.072 \frac{m}{s^2}$.
• Total acceleration: $A = \sqrt{A<em>R^2 + A</em>T^2} = 0.3085 \frac{m}{s^2}$