27.3 Content
27.3.1 Uniform Circular Motion
In uniform circular motion, an object moves in a circle at constant speed. Although the speed is constant, the velocity is changing because its direction changes continuously.
Any change in velocity requires acceleration. In circular motion, this acceleration points toward the center and is called centripetal acceleration.
27.3.2 Interactive: Circular Motion Visualisation
27.3.3 Centripetal Acceleration
The centripetal acceleration always points toward the center of the circle:
\[a_c = \frac{v^2}{r}\]
where: - \(a_c\) = centripetal acceleration (m/s²) - \(v\) = speed (m/s) - \(r\) = radius of circle (m)
Using \(v = 2\pi r/T\), centripetal acceleration can also be written as: \[a_c = \frac{4\pi^2 r}{T^2}\]
27.3.4 Speed and Period
For an object completing one revolution:
\[v = \frac{2\pi r}{T}\]
where \(T\) is the period (time for one complete revolution).
Frequency (\(f\)) is the number of revolutions per second: \[f = \frac{1}{T}\]
27.3.5 Centripetal Force
By Newton’s second law, centripetal acceleration requires a centripetal force:
\[F_c = ma_c = \frac{mv^2}{r}\]
Centripetal force is NOT a new type of force. It is the name for the net inward force that causes circular motion. It could be: - Tension (mass on a string) - Friction (car on a bend) - Gravity (satellites) - Normal force (banked tracks)
27.3.6 Interactive: Forces in Circular Motion
27.3.7 Angular Velocity
Angular velocity (\(\omega\)) measures how fast an angle changes:
\[\omega = \frac{\Delta\theta}{t}\]
Relationship to linear speed: \[v = \omega r\]
Angular velocity is measured in radians per second (rad/s). To convert from degrees: multiply by \(\pi/180\).
27.3.8 Applications: Cars on Bends
Flat bend: Friction provides the centripetal force.
\[\mu_s mg \geq \frac{mv^2}{r}\]
Maximum safe speed: \(v_{max} = \sqrt{\mu_s gr}\)
Banked bend: Normal force component provides centripetal force.
For ideal banking (no friction needed): \[\tan(\theta) = \frac{v^2}{gr}\]
27.3.9 Interactive: Car on Banked Track
27.3.10 Energy and Work in Circular Motion
In uniform circular motion: - Centripetal force is perpendicular to velocity - No work is done by the centripetal force - Kinetic energy remains constant
\[W = F \cdot d \cdot \cos(90°) = 0\]
27.3.11 Torque
Torque measures the effectiveness of a force in causing rotation:
\[\tau = r_\perp F = rF\sin(\theta)\]
where: - \(\tau\) = torque (N·m) - \(r\) = distance from pivot to force - \(F\) = applied force - \(\theta\) = angle between \(r\) and \(F\)
Torque is maximum when force is perpendicular to the lever arm (\(\theta = 90°\)).