Kinematics of Uniform Circular Motion (A-Level Physics 9702)
Hello future physicist! Welcome to one of the most exciting topics in A-Level Physics: Circular Motion. Up until now, you have mostly studied motion in a straight line (linear motion). But the real world is full of curves! Think about satellites orbiting Earth, cars turning corners, or a spinning Ferris wheel. Understanding how objects move in circles is crucial, and that's exactly what this chapter is about. Don't worry if it seems tricky; we will break it down using everyday examples!
The key difference here is that even if an object moves at a constant speed in a circle, its *velocity* is constantly changing, meaning it is always *accelerating* and experiencing a *resultant force*.
12.1 Describing Circular Motion: Kinematics
What is Uniform Circular Motion (UCM)?
Uniform Circular Motion (UCM) is the motion of an object moving in a circle or circular arc at a constant speed.
- Speed (Magnitude): Constant (e.g., always 10 m/s).
- Velocity (Vector): Changing constantly because its direction is always changing.
The Radian: A New Way to Measure Angles
When dealing with rotational motion, degrees are often impractical. We use a more fundamental unit called the radian.
Definition of the Radian:
One radian (rad) is the angle subtended at the center of a circle when the arc length \(s\) is equal to the radius \(r\) of the circle.
The relationship between arc length, radius, and angular displacement \(\theta\) is:
$$ s = r\theta $$
Where \(\theta\) must be measured in radians.
Converting between Radians and Degrees:
- A full circle is $360^{\circ}$.
- A full circle has an arc length equal to the circumference, $s = 2\pi r$.
- Since $s = r\theta$, we have $2\pi r = r\theta$.
- Therefore, a full circle angle $\theta$ is $2\pi$ radians.
$$ 360^{\circ} = 2\pi \text{ radians} $$
$$ 180^{\circ} = \pi \text{ radians} $$
Quick Tip for Radians:
If you ever need to convert an angle in degrees to radians, multiply by \(\pi/180\).
Angular Displacement and Angular Speed (\(\omega\))
When an object moves around a circle, we describe its position using angular displacement, \(\theta\), the angle swept out by the radius.
The rate at which this angle is swept out is called the Angular Speed (\(\omega\)).
Definition of Angular Speed:
Angular speed is the angular displacement (\(\theta\)) per unit time (\(t\)).
$$ \omega = \frac{\theta}{t} $$
Unit: radians per second (rad s\(^{-1}\)).
Connecting Angular Speed to Period and Frequency
The Period (T) is the time taken for one complete revolution. In one period, the object sweeps through an angle of $2\pi$ radians.
Using the definition \(\omega = \theta/t\):
$$ \omega = \frac{2\pi}{T} $$
The Frequency (f) is the number of revolutions per unit time, where $f = 1/T$. We can also write the angular speed as:
$$ \omega = 2\pi f $$
Relating Linear Speed (\(v\)) and Angular Speed (\(\omega\))
For an object in UCM, its linear speed ($v$) is constant and is directed tangent to the circle.
We know that speed is distance divided by time. In one revolution, the distance travelled is the circumference $s = 2\pi r$, and the time taken is the period $T$.
$$ v = \frac{\text{Distance}}{T} = \frac{2\pi r}{T} $$
Since we found that $\omega = 2\pi/T$, we can substitute $\omega$ into the speed equation:
$$ v = r\omega $$
This is a critical equation! It tells us that for a fixed angular speed, points further away from the center (larger \(r\)) must travel faster linearly. Think about a Merry-Go-Round: the people on the edge travel a much greater distance per spin than those near the center, even though they complete the revolution in the same time.
Quick Review 12.1 Key Takeaway
In UCM, we use radians. Angular speed (\(\omega\)) tells us how fast the object rotates, and the linear speed (\(v\)) is proportional to the radius ($v = r\omega$).
12.2 Centripetal Acceleration and Force
The Paradox of Constant Speed but Changing Velocity
Recall Newton's First Law: an object maintains constant velocity unless acted upon by a resultant force. If an object is moving in a circular path, even though its speed might be constant, its direction is always changing.
Changing direction means changing velocity.
Changing velocity means there is an acceleration.
This acceleration requires a resultant force.
Centripetal Acceleration (\(a\))
The acceleration responsible for changing the direction of the velocity vector (pulling the object inwards) is called Centripetal Acceleration.
Key Understanding (Syllabus 12.2.1 & 12.2.2):
- The acceleration is always directed towards the center of the circle. (Centripetal literally means "center-seeking").
- This acceleration vector is always perpendicular to the instantaneous velocity vector (which is tangential).
- A force of constant magnitude that is always perpendicular to the direction of motion causes this centripetal acceleration, resulting in circular motion with constant angular speed.
Formulas for Centripetal Acceleration
Centripetal acceleration (\(a\)) can be expressed in terms of linear speed (\(v\)) or angular speed (\(\omega\)):
$$ a = \frac{v^2}{r} $$
or, using the substitution $v = r\omega$:
$$ a = r\omega^2 $$
Why is the acceleration inwards?
Imagine a car turning a corner. If the car continued in its original straight line, it would move tangentially. To pull the car onto the curve, the tires must push inwards. This inward push results in the necessary inward acceleration.
Centripetal Force (\(F\))
According to Newton's Second Law ($F = ma$), since there is an acceleration, there must be a resultant force causing it. This resultant force is the Centripetal Force (\(F_c\)).
Key Fact: Centripetal force is NOT a new type of force!
The centripetal force is simply the resultant force acting on the object, provided by an existing force (like tension, friction, or gravity).
Direction: The centripetal force is always directed towards the center of the circular path, parallel to the centripetal acceleration.
Formulas for Centripetal Force
We use $F = ma$, substituting the acceleration formulas derived above ($a = v^2/r$ and $a = r\omega^2$):
$$ F = m \frac{v^2}{r} $$
or
$$ F = m r\omega^2 $$
Important Note: In solving problems, always identify which physical force (or combination of forces) is providing the required centripetal force.
Example scenarios:
- Ball tied to a string moving horizontally: Centripetal force is provided by the tension in the string.
- Car rounding a flat corner: Centripetal force is provided by friction between the tires and the road.
- Satellite orbiting Earth: Centripetal force is provided by gravitational attraction.
Common Mistake: The Centrifugal Force Myth
You may hear people talk about "centrifugal force" (center-fleeing force). In your syllabus (and in Newtonian physics applied to an inertial frame), this term is often misleading or incorrect when discussing the forces acting ON the object moving in a circle.
The force causing the circular motion is the centripetal force (inwards). The "centrifugal effect" is simply the tendency of the object to continue in a straight line (Newton’s First Law) when the centripetal force is insufficient or removed. When you feel pushed outwards on a roundabout, you are feeling the reaction force to the wall pushing you inwards, or your inertia trying to keep you moving straight.
Did you know?
The largest rotating object we regularly interact with is the Earth! Although we don't usually feel it, everything on the equator is moving at about 460 m/s relative to the center, requiring a slight centripetal force provided by the effective difference between gravity and the normal force.
Step-by-Step Problem Solving Strategy
- Identify UCM: Does the object move in a circle at a constant speed?
- Identify Given Variables: $r, v, \omega, T, m$ or $f$. Use $v = r\omega$ or $\omega = 2\pi/T$ to find missing kinematic values.
- Determine the Required Force: The resultant force must be the centripetal force ($F_c$). Use $F_c = mv^2/r$ or $F_c = mr\omega^2$.
- Identify the Provider: Determine which actual physical force(s) (tension, friction, gravity) act towards the center of the circle to provide this $F_c$.
- Apply Newton's Second Law: Set the sum of the inward forces equal to the calculated centripetal force.
Quick Review 12.2 Key Takeaway
UCM involves centripetal acceleration ($a = v^2/r$) directed inwards. This acceleration is caused by the centripetal force ($F = mv^2/r$), which is the resultant force acting towards the center, provided by tension, gravity, or friction.