A2 Physics, Circular Motion and Oscillation, 1-2

  • 1 Circular Motion

Circular Motion

Motion in a Circular Path

Angular measurements are normally made using the SI unit of radians, where a radian is defined as the angle subtended at the centre of a circle by an arc that is equal in length to the radius. This gives the equation

\begin{equation} \theta=\frac{s}{r},\nonumber \\ \end{equation}

where \(s\) is the arc length. \(\theta\) is equal to the angular displacement. Also, \(T=\frac{1}{f}\) and vice versa.

The angular velocity, \(\omega\), is the angular displacement per unit time, in rad s-1. This is given by

\begin{equation} \omega=\frac{2\pi}{T}=2\pi f.\nonumber \\ \end{equation}

Linear/orbital speed, \(v\), is the distance moved per unit time, and is in a direction tangential to the circular path.

\begin{equation} v=\omega r\nonumber \\ \end{equation}

Centripetal Acceleration and Force

Due to Newton’s law, there must be a constant force for an object to be travelling in a circular path. Thus, the acceleration of an object moving at a constant speed in a circular path is the centripetal acceleration.

The direction of the acceleration is dependent on the vector representing the velocity change. This is always towards the centre of the circle, and this is the centripetal force. This is given by:

\begin{gather*} a=\frac{v^{2}}{r}=\omega^{2}r \\ F=\frac{mv^{2}}{r}=m\omega^{2}r. \\ \end{gather*}

The centripetal force is the sum of forces acting in the direction of the centre of the circle.

Motion of a Conical Pendulum

Centripetal force is given by the horizontal component of tension in the string. By equation the weight of the mass, and both components of the tension, the following can be attained:

\begin{equation} T\text{sin}\theta=\frac{mv^{2}}{r}{}\\ \nonumber \\ \end{equation} \begin{equation} T\text{cos}\theta=mg{}\\ \nonumber \\ \end{equation} \begin{equation} \frac{\text{(1)}}{\text{(2)}}=\text{tan}\theta=\frac{v^{2}}{rg}\nonumber \\ \end{equation}

Banked Tracks

A banked track makes curving easier and allows for a greater speed. The centripetal force will be due to the horizontal components of the normal reaction and lateral friction.

The principle of banked tracks can also be applied to a banked aircraft. Assuming no air resistance, the centripetal force is given by the horizontal component of the lift force. If this is denoted as \(L\), then:

\begin{equation} L\text{sin}\theta=\frac{mv^{2}}{r}{}\\ \nonumber \\ \end{equation} \begin{equation} L\text{cos}\theta=mg{}\\ \nonumber \\ \end{equation} \begin{equation} \frac{\text{(1)}}{\text{(2)}}=\text{tan}\theta=\frac{v^{2}}{rg}\nonumber \\ \end{equation}

Motion in a Vertical Circle

For an object travelling in a vertical circle, the weight must be considered. If it is a bucket on a string, then the centripetal force will be greatest at the top of the circle, as the force will be due to tension and weight. At the bottom, it will be lowest, as it will be the tension minus the weight. For a whirling bucket, the water will fall out if the tension becomes zero.

For a car or similar, then the circle could be considered to be a bumb, bridge, or similar. In this case, the centripetal force is due to the weight, minus the contact force. If the contact force becomes zero, the car will leave the road - this would be due to an increase in velocity.


Simple Harmonic Motion

Oscillations are a type of periodic motion, repeating regularly in a given time interval. Mechanical oscillation requires a resultant force in the direction of the equilibrium position.
In simple harmonic motion (SHM), the restoring force is directly proportional to the displacement, and in the opposite direction - towards the equilibrium position.
When \(x=\) displacement, and \(A=\) amplitude,

\begin{gather*} x=A\cos(\omega t) \\ \omega=\frac{2\pi}{T}=2\pi f \\ \end{gather*}
SHM graphs