Uniform Circular Motion

Uniform circular motion defines the motion of an object traveling at a constant speed around a fixed center point or axis. The object travels around a curved path and maintains a constant radial distance from the center point at any given time. Realistically speaking, a perfect circle does not exist, but it is useful to study the case of a perfect circle in order to understand how an object might move around an ellipse and to approximate the motion of an object that is almost circular in nature. Some examples of this type of motion are the orbit of a planet, a car going around a circular track or a conical pendulum.

Revolution is a type of circular motion where the object moves around a fixed center point called the axis of revolution given that the axis is some distance away from the object, such as how the earth revolves around the sun.

Rotation is another type of circular motion where an object moves around a point called the axis of rotation which passes through the object, such as how the earth rotates around an axis. Rotation can be viewed as the revolution of all of the particles that make up the object .

Radius \((r)\) is the constant distance that the object remains away from the center point as it revolves around it. However, if thought of as a vector with the center point of the circle being the initial point and the point lying on the circumference of the circle being the terminal point, any two radii vectors are not the same. Two vectors \(\vec{OA} \) and \(\vec{OB}\) which are two radii with the same magnitude are in different directions as seen below.

\[\vec{r}=\vec{r}\cos\theta + \vec{r}\sin\theta\]

\[\] Position \((s)\) describes the location of the object on the circle and is given by the radius vector

\[ \color{green}{\vec{s}}= ( \color{blue}{\Delta x}, \color{red}{\Delta y}).\]

Angular Displacement \((\theta)\) is an angle between any two points on the circle relative to the axis of revolution and is given in units of radians. \(\theta\) is given as a function of time t:

\[\displaystyle \theta(t) = \omega t.\]

Angular Velocity \((\omega)\) is the change in angular displacement over the change in time and is in units of \(\frac{rad}{s}:\)

\[\displaystyle \omega = \frac{d\theta}{dt}.\]

Angular Acceleration \((\alpha)\) is the change in angular velocity over the change in time and is in units of \(\frac{rad}{s^{2}}:\)

\[\displaystyle \alpha = \frac{d\omega}{dt}.\]

\[\]

An object that revolves in a clockwise direction can be thought of in a two-dimensional space as revolving to the right. Traditionally clockwise motion is thought to be in the negative direction.

An object that revolves in a counterclockwise direction can be thought of in a two-dimensional space as revolving to the left. Counterclockwise motion is thought to be in the positive direction.

\(\vec{\omega }\) is a vector perpendicular to the plane of orbit and parallel to the axis of revolution.

The angular displacement of the particles of a rotating object is irrespective of the radius. A particle that is \(10\text{ m}\) out from the axis of rotation will have the same angular velocity as an object that is \(100\text{ m}\) from the axis of rotation. This is easily observed by looking out at the stars at night. Constellations are formed of stars that are all different distances from the Earth but yet we see them all move at the same speed.

Displacement \((d)\) is the change in position of the object relative to some starting point and the term distance is reserved for the measure of the path traveled along the perimeter of the circle. A fraction of the perimeter is called an arc, and has a length \(l\) measured in meters \((\text{m}).\) Arc length is given by

\[l= \theta r,\]

where \(r\) is the radius of the curvature.

Displacement is given as a vector

\[ \color{green}{\vec{d}}= ( \color{blue}{\Delta x}, \color{red}{\Delta y}).\]

If the object makes exactly one revolution then the displacement is zero, and the distance traveled is the circumference of the circle

\[C=2\pi r.\]

Period (\(T)\) is the amount of time \(t\) in seconds that it takes for the object to make one revolution. Given that \(2 \pi\) is the angular displacement of one revolution and \(\omega\) is the angular velocity, the formula for \(T\) is as follows:

\[T=\frac{2\pi}{\omega}.\]

Frequency \((f)\) is the number of revolutions per second in units of Hertz \((\text{Hz}).\) It is the inverse of the period:

\[f=\frac{1}{T}.\]

A tangent is a line which touches exactly one point on the circle. The significance of the tangent is that at this point the circle and the line share the same slope, or rate of change. Although we cannot find the slope of a curve, we can find the slope of the tangent, or the instantaneous rate of change.

Instantaneous or tangential velocity \((v)\) is the velocity of the revolving object at a given point along its path of motion. The magnitude of the velocity, or the speed, remains constant, but in order for the object to travel in a circle, the direction of the velocity must change.

The tangential speed of the object is the product of its angular counterpart and the magnitude of the radius:

\[|\vec{v}|=\omega r.\]

The direction of the tangential velocity is orthogonal to the radius vector for that given point.

Vectors can be used to find the velocity but the direction can also be given by the right hand rule:

\[\vec{v}=\vec{\omega} \times \vec{r}.\]

The velocity can also be viewed as the change in position \(s\) over the change in time:

\[\begin{align} s(t)&= r(\cos\theta , \sin\theta )\\ \\ v(t)&= \frac{\delta s}{\delta t} \\ &= r\left(\cos\theta \frac{\delta \theta }{\delta t} , \sin\theta \frac{\delta \theta }{\delta t} \right)\\ &= r\omega (\cos\theta , \sin\theta )\\ &= r\omega\left( \sqrt{\sin^{2}\theta +\cos^{2}\theta }\right)\\ &=r\omega. \end{align}\]

Centripetal acceleration is the change in the tangential velocity and is orthogonal to the tangential velocity and the angular velocity. It can be found using the cross product

\[\vec{\alpha_c} =\vec{v} \times \vec{\omega}.\]

But \(\vec{v}=\vec{\omega} \times \vec{r}\). Therefore,

\[\vec{\alpha_c} =\vec{\omega} \times \vec{r} \times \vec{\omega}.\]

In terms of magnitude this gives

\[\alpha_c =r\omega^{2}.\]

But we also know the following to be true:

\[\begin{align} \vec{v}&=\vec{\omega} \times \vec{r}\\ \vec{v}^{2}&=\vec{\omega}^{2} \times \vec{r}^{2}\\ \vec{\omega}^{2}&= \frac{\vec{v}^{2}}{ \vec{r}^{2}}. \end{align}\]

Therefore,

\[\begin{align} \vec{\alpha_c} &= \vec{r} \frac{\vec{v}^{2}}{ \vec{r}^{2}}\\ &= \frac{\vec{v}^{2}}{ \vec{r}}. \end{align}\]

There is another wiki which goes into further detail on the topic of centripetal acceleration. (Click here to view the wiki on centripetal acceleration.)

Centripetal force is the force which pulls the object towards the center and keeps the object moving in a circle despite the direction of its velocity. Centripetal force is in the direction opposite of the radius. Centripetal force is often caused by other forces such as gravity, tension, or electromagnetism:

\[F_c =m\alpha_c,\]

where \(m\) is the mass of the revolving object. Mass \((m)\) is the quantity which is solely dependent on the inertia of an object, or its ability to resist changes in its state of motion.

Moment of inertia \((I),\) sometimes called the second moment of area or the the angular mass, is proportional to the mass and the square of the radius:

\[I=mr^2.\]

Angular momentum \((L)\) can be thought of as how hard it is to change the state of motion of an object with regards to both its angular mass and velocity:

\[L=I\omega.\]

Kinetic energy is the amount of energy that an object has due to its current state of motion. Kinetic energy is generally given by the following formula:

\[k=\frac{1}{2}m|\vec{v}|^{2}.\]

When considering rotational movement, however, we know the equivalent of the mass to be the moment of inertia, \(I,\) and the velocity to be the angular velocity, \(\omega:\)

\[k=\frac{1}{2}I\omega^{2}.\]