Understanding Circular Motion

Expert reviewed • 22 November 2024 • 9 minute read

What is Circular Motion?

Circular motion is a cornerstone of physics, describing the movement of objects along a circular path. This concept is crucial in understanding various phenomena, from planetary orbits to the operation of simple machines.

What is Uniform Circular Motion?

Uniform circular motion occurs when an object moves in a circular path at a constant speed. Despite the constant speed, the object is continuously accelerating due to its constantly changing direction. This unique type of motion is characterised by several important quantities:

Centripetal Acceleration
Centripetal Force
Period and Frequency
Angular Speed

Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, always directed towards the centre of the circle. It’s given by the equation:

a_c = \frac{v^2}{r}

Where:

$a_c$ is the centripetal acceleration (m/s²)
$v$ is the linear speed of the object (m/s)
$r$ is the radius of the circular path (m)

Centripetal Force

To maintain circular motion, a force must be applied towards the centre of the circle. This force, known as centripetal force, is responsible for the centripetal acceleration. Using Newton’s Second Law $(F=ma)$ , we can derive the equation for centripetal force:

F_c = m a_c = \frac{mv^2}{r}

Where:

$F_c$ is the centripetal force (N)
$m$ is the mass of the object (kg)
$v$ is the linear speed of the object (m/s)
$r$ is the radius of the circular path (m)

It’s important to note that centripetal force is not a new type of force, but rather the net force causing circular motion. This force can be provided by various means, such as tension in a string, friction between tires and a road, or gravitational attraction.

Practice Question 1

A car with a mass of 1500 kg is traveling at 20 m/s on a circular track with a radius of 100 m. Calculate the centripetal force required to keep the car on its circular path.

Using the equation $F_c = \frac{mv^2}{r}$ , we can substitute the given values:

F_c = \frac{1500 \cdot (20 )^2}{100 } \\= 6000 \text{ N}

Therefore, a centripetal force of 6000 N is required to keep the car on its circular path.

What is Period and Frequency?

The period $(T)$ of circular motion is the time taken for one complete revolution. It’s related to the linear speed and radius of the circular path:

T = \frac{2\pi r}{v}

Frequency $(f)$ is the number of revolutions per unit time, typically measured in Hertz $(Hz)$ . It’s the reciprocal of the period:

f = \frac{1}{T} = \frac{v}{2\pi r}

Practice Question 2

A satellite orbits Earth in a circular path with a radius of 42,164 km. If its linear speed is 3.07 km/s, calculate its period and frequency.

First, let’s calculate the period using: $T = \frac{2\pi r}{v}$

T = \frac{2\pi \cdot 42164 }{3.07 } \\= 86,294.53 \\\approx23.97 \:\:\text{hours}

Now, we can calculate the frequency:

f = \frac{1}{T} \\= \frac{1}{ 86294.53} \\\approx 1.159 \times 10^{-5} \text{ Hz}

This satellite completes one orbit every 24 hours, which is known as a geosynchronous orbit.

Angular Speed

Angular speed $(\omega)$ is the rate of change of angular position. It’s measured in radians per second (rad/s) and is related to linear speed and radius:

\omega = \frac{v}{r} = \frac{2\pi}{T}

We can also express linear speed in terms of angular speed:

v = ωr

Practice Question 3

A Ferris wheel with a radius of 10 m makes one complete rotation every 30 seconds. Calculate its angular speed and the linear speed of a passenger at the edge of the wheel.

First, let’s calculate the angular speed:

\omega = \frac{2\pi}{T}\\ = \frac{2\pi}{30 } \\\approx 0.209 \text{ rad/s}

Now, we can calculate the linear speed of a passenger:

v = ωr \\= 0.209  \times 10 m \\= 2.09 m/s

What are the Factors Affecting Centripetal Force?

From the equation $F_c = \frac{mv^2}{r}$ , we can deduce that:

Centripetal force is directly proportional to the mass of the object.
Centripetal force is directly proportional to the square of the velocity.
Centripetal force is inversely proportional to the radius of the circular path.

Understanding these relationships is crucial for analysing circular motion in various scenarios.

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Return to Module 5: Advanced Mechanics