What is Circular Motion? Formula & Examples, useful important fact

What is Circular Motion?

Circular motion means moving around in a circle. When something goes round and round on the same path, we call it circular motion.

Think about the hands of a clock, a fan spinning on the ceiling, or the wheel of a bicycle. They all move in a circle. That’s circular motion!

Now, circular motion can happen in two ways:

1. Uniform Circular Motion – This means the object is moving at the same speed all the time. For example, if a toy train moves around a circular track without speeding up or slowing down, it is in uniform circular motion.
2. Non-Uniform Circular Motion – This happens when the speed keeps changing. Maybe the object goes faster or slower at different points while moving in the circle.

When something moves in a circle, its direction keeps changing all the time, even if the speed stays the same. That’s different from straight-line motion, where the direction doesn’t change unless something makes it.

Some simple examples of circular motion:

• A satellite going around the Earth
• A ceiling fan spinning
• A bicycle wheel turning
• The blades of a windmill
• The gears inside machines

In circular motion, even a small object follows a round path, like when you tie a stone to a string and swing it in a circle.

So, circular motion is just when something keeps moving around a fixed point in a circle. It may go fast or slow, but it always turns round and round.

1. Motion on a Circle

When something moves along a circle, it stays the same distance from the center of that circle all the time. This distance is called the radius (we call it R). Even though the object moves in two directions — left-right and up-down — we often treat it as moving along just one curved path.

2. Measuring the Position

There are two easy ways to say where the object is on the circle:

• One way is to say how far it has moved along the edge of the circle. We call that s.
• The other way is to talk about the angle it has made as it moves — we call that theta (θ).

Here’s something cool:
If the object moves a little along the edge of the circle, the angle also changes. And there’s a very simple connection between them:

θ = s / R

This means if we know how far it has moved on the circle and the radius of the circle, we can find the angle.

3. Going Around the Circle

If the object goes all the way around the circle, it travels a total distance of:

s = 2πR

And the angle it covers is:

θ = 2π radians (which is the same as one full circle or 360 degrees).

4. Where Is It Exactly?

We can also find the exact spot of the object using two things:

• How far it is to the right or left (x),
• How far it is up or down (y).

If we know the angle θ, we can write:

x = R × cos(θ)
y = R × sin(θ)

That’s how we can always know where the object is on the circle.

5. Speed and Velocity

Now let’s talk about how fast the object is moving.

We use something called velocity to show both how fast and in what direction the object is moving.

In circular motion, the object’s velocity is always pointing along the edge of the circle, like a line that just touches the circle — this is called a tangent. And the amazing thing is:

• The object’s velocity is always perpendicular (at a right angle) to the position vector (the line from the center to the object).

If the angle is changing with time, the velocity becomes:

vx = –R × sin(θ) × (dθ/dt)
vy = R × cos(θ) × (dθ/dt)

And the speed (just how fast it is going) is:

v = R × (dθ/dt)

We often call this angular speed:

ω (omega) = dθ/dt

So:

v = R × ω

6. What If the Object Speeds Up or Slows Down?

If the object speeds up or slows down as it goes around the circle, we say it has acceleration.

There are two types of acceleration in circular motion:

a. Tangential Acceleration (along the path):

This tells us how the speed is changing — faster or slower. It’s like the object pushing forward or being pulled back.

We write it as:

aₜ = R × α
Where α (alpha) is how fast the angle is changing — called angular acceleration.

b. Centripetal Acceleration (towards the center):

This is always there, even if the object is moving at a steady speed. It keeps pulling the object towards the center so it doesn’t fly off the circle.

We write it as:

aₙ = v² / R = R × ω²

It always points inward, straight toward the center of the circle.

7. Final Summary – All Key Formulas in One Place

In Simple Words:

• Circular motion means moving around a circle.
• We can measure the position using distance or angle.
• The object always wants to go straight, but a force (like the string in our toy example) pulls it toward the center to keep it on the path.
• There is always a pull toward the center called centripetal acceleration.
• If the object changes speed, there’s also tangential acceleration along the circle’s edge.

Radial Vector

Imagine a point that is moving around in space. Let’s say it starts from the centre—this centre point is called the origin.

Now, draw a straight line from the origin to the current position of the moving point. This straight line is called a radial vector (also known as a radius vector).

It simply shows where the point is and how far it is from the centre. As the point moves, this vector also changes, because the direction and distance from the origin are changing.

So in easy words:
A radial vector tells us the direction and distance of a moving point from the starting point (the origin).

Uniform Circular Motion

When something moves in a circle and keeps the same speed the whole time, we call it uniform circular motion.

When an object moves in a circle at a constant speed, its motion is called uniform circular motion

It moves around a fixed point, which we call the center. Even though the speed doesn’t change, the direction keeps changing all the time. And because the direction changes, the object is said to be accelerating, even though it’s not going faster or slower.

Imagine a small ball tied to a string. If you swing it around in a circle with your hand, the ball moves in a round path. Your hand is the centre, and the ball moves around it at the same speed. That’s a perfect example of uniform circular motion.

Now, let’s talk about what’s really happening:

• The ball’s position at any moment can be shown by a straight line from the centre to the ball. This line is called the radius vector.
• The length of this line stays the same. It’s just the direction that keeps changing as the ball moves around the circle.
• At every point on the circle, the ball has a velocity. This means the direction in which it is moving right at that moment.
• The velocity is always a straight line tangent to the circle – that means it touches the circle at just one point and goes off straight from there.
• The speed of the ball stays the same, but the direction of the velocity changes at every moment. That’s why we say the motion has acceleration, even though the speed is constant.

In simple words:
Even if the speed doesn’t change, the object in circular motion is still changing how it moves. It keeps turning — and that turning needs a special kind of push (called centripetal force), which always pulls the object toward the center.

Angular Displacement

When something moves along the edge of a circle, the angle it turns from its starting point is called its angular displacement.

Imagine a boy walking around a big round garden. He starts from one point and begins to walk along the edge of the circle. As he moves, he slowly turns from his starting direction. The amount he turns is what we call angular displacement.

Let’s say he starts at point P₀ and walks in the anticlockwise direction. Now, draw a straight line from the center of the circle to where he started — that’s the radius vector. As he walks, this radius vector turns along with him. The angle it makes with its original position is called angular displacement.

So, angular displacement simply tells us how much the direction has changed as the person or object moves around the circle. It doesn’t tell how far the person has walked — just how much the direction has turned.

For example:

• If the person is at position θ₁ at time t₁, and then moves to position θ₂ at time t₂, the angular displacement is the difference between θ₂ and θ₁.

Imagine a small particle moving along a round path, like the edge of a circle. It starts moving at time 11 and keeps going until time 12. This time gap is just 1 unit.

In this short time, the particle moves a little distance along the edge of the circle. Let’s call this small distance As. Since the particle is moving along the circle, this distance As is called the linear displacement. It tells us how far the particle has moved along the path.

At the same time, the particle also turns or rotates around the center of the circle. Let’s say it starts at angle θ₁ and ends at angle θ₂. The difference between these two angles is the amount it has turned. This is called the angular displacement and we write it like this:

Δθ = θ₂ – θ₁

Now, there is a beautiful connection between the linear displacement and the angular displacement. If the particle moves a distance As on the circle, and the radius of the circle is r, then:

Angular displacement (Δθ) = As / r

So,

Angle = Arc / Radius

This shows the relation between the angle turned and the distance moved along the circle.

The unit we use for angular displacement is called the radian.

One radian is very simple to understand. If the arc (curved part of the circle) is as long as the radius, then the angle made at the center is 1 radian.

What is Angular Velocity?

In circular motion, the rate of change of angular displacement of a particle with time is called the ‘angular velocity’

Imagine a particle moving around the edge of a circle, like a toy car going around a round track. As it moves, it changes its position on the circle. This change in position, measured in angles, is called angular displacement.

Now, if we look at how fast this angular displacement is changing with time, we get something called angular velocity.

In Simple Words:

Angular velocity means how quickly something is turning or rotating around a point.

We use the Greek letter ω (omega) to show angular velocity.

How to Understand It:

Suppose the angle the object moves through is a small amount, and we know how much time it took. Then:

Average Angular Velocity = Change in Angle ÷ Change in Time

Or in short:

ω = Δθ / Δt

Now, if the time is very, very small—almost zero—we get the instantaneous angular velocity, which tells us how fast the object is rotating at a single moment.

We write it like this:

ω = dθ / dt

Unit of Angular Velocity:

We measure angular velocity in radians per second.

That means:

• Radian is the unit for measuring angles.
• Second is the unit for time.

So, angular velocity = radians / second

Dimensional Formula:

The dimensional formula of angular velocity is [T⁻¹], which tells us it depends only on time.

In Short:

• Angular velocity shows how fast something rotates.
• It is written as ω.
• Formula: ω = dθ / dt
• Unit: radian/second
• Dimensional formula: [T⁻¹]
  

Angular velocity

Angular velocity is a special kind of speed. But instead of telling us how fast something is moving in a straight line, it tells us how fast something is spinning or going around in a circle.

It is not just about speed — it also has direction, which makes it a vector.

Now, how do we know the direction of angular velocity?

There is a simple trick called the right-hand thumb rule. Imagine you are holding a paper flat in front of you.

• If something is moving in a circle anticlockwise on that paper, point the fingers of your right hand in the direction it is moving.
• Now, stick out your thumb like you’re giving a thumbs up. Your thumb will point upward, out of the paper. That is the direction of the angular velocity.
• If the object goes clockwise, your thumb will point downward, into the paper.

So, the direction is always perpendicular (at a right angle) to the circular path.

One Complete Round

When something makes one full circle, we say it has moved through an angle of 2π radians (which is the same as 360 degrees).

Let’s say the time taken to make this full circle is T seconds. We call this time the period.

Now, to find out how fast it is spinning on average, we use this formula:

Average Angular Velocity = 2π / T

That means: if something takes less time, it has more angular velocity — it spins faster!

What If It Spins Again and Again?

If the object keeps going round and round, the number of rounds it makes in one second is called its frequency. We write this as n.

And there’s another simple formula:

Angular Velocity = 2π × n

This shows that if an object makes more rounds every second, its angular velocity becomes greater.

In Simple Words:

• Angular velocity tells us how fast and in which direction something spins.
• Its direction is given by your right-hand thumb.
• If it spins once in T seconds, then angular velocity is 2π divided by T.
• If it spins n times in one second, then angular velocity is 2π multiplied by n.
  
  

Linear Velocity

In rectilinear motion, the rate of change of linear displacement of a particle with time is called the ‘linear velocity’ of that particle.

When something moves in a straight line, it changes its position over time. How fast it changes its position is called its linear velocity.

Imagine you have a little toy car moving straight on the floor. If the car moves a little bit in a very short time, the distance it covers divided by that time is its velocity.

For example, if the car moves 2 meters in 1 second, its velocity is 2 meters per second.

If we call the small change in position “Δs” and the small time it takes “Δt,” then the average linear velocity (v) is:

v = Δs ÷ Δt

When we look at a very, very tiny amount of time—so small it’s almost zero—the average speed of something moving during that time becomes the same as the speed at that exact moment. This speed at a single instant is called the instantaneous linear velocity.

In simple words:
Instantaneous linear velocity means how fast something is moving right now.

Instantaneous linear velocity,
v = lim (Δs / Δt) = ds / dt

Here,

• s is the distance traveled,
• t is the time taken,
• ds/dt means how fast the distance changes with time at that instant.

The unit we use to measure linear velocity is meters per second (m/s).

This means the velocity is the small change in distance divided by the small change in time.

The unit we use for linear velocity is meters per second (m/s).

Also, linear velocity is a vector. That means it has both a size (how fast) and a direction (which way).

Relation Between Angular Velocity and Linear Velocity

Imagine a child sitting on a merry-go-round. When it spins, the child moves in a circle.

Now think about this:

• The merry-go-round spins at a certain speed. This spinning speed is called angular velocity.
• The child sitting on it moves along the edge of the circle. The speed at which the child moves along the circle is called linear velocity.

How Are They Related?

Let’s make it simple:

• If you are sitting near the center, you don’t move much, even if the merry-go-round is spinning fast.
• But if you are sitting at the edge, you move a lot more for the same spinning speed.

So, the farther you are from the center, the faster you move.

This means:

Linear velocity (v) = Angular velocity (ω) × Radius (r)

Or simply:

v = ω × r

What Does This Mean?

• ω (omega) is the angular velocity — how fast something spins.
• r is the distance from the center (radius).
• v is the linear velocity — how fast the object moves in a straight path along the circle.

So, if the spinning speed is the same for everyone on the merry-go-round, the one sitting farther out moves faster in a straight path.

Time Period and Speed

Now, when the particle (or the child on the merry-go-round) makes one full round, the time it takes is called the time period.

To find the time period, we use:

Time Period (T) = Total distance of one round / Speed

The distance of one full round of a circle is:

2πr

So,

T = 2πr / v

If we use v = ω × r, we can also write:

T = 2π / ω

When a hard object (called a rigid body) spins around a fixed line (called the axis), something interesting happens.

Every tiny part of the object starts moving in a circle. These circles are not separate, they are like layers – all spinning together. No matter where a particle (small part) is in the object, it rotates through the same angle in the same time. That means their angular velocity (how fast they turn) is equal.

But here’s something important: not all parts are at the same distance from the axis. Some are close to the center, and some are far away. Because of this, their linear velocity (how fast they move along their circular path) is different.

• The farther a particle is from the center,
  the faster it moves in a straight line (along the circle).
  So, more distance = more speed in the circle.

This gives us a very important relation:
v = r × ω

Where:

• v is the linear velocity (how fast it moves on the circle),
• r is the distance from the center (radius),
• ω is the angular velocity (how fast it turns).

All three – v, r, and ω – are vector quantities. This means they have both size and direction.

Now imagine a round paper lying flat on a table. That’s your spinning plane.

• The radial vector (r) points from the center of the paper out to the particle.
• The angular velocity vector (ω) goes straight upward, coming out of the paper, if the object spins anticlockwise.
• The linear velocity (v) goes along the edge of the circle, pointing in the direction the particle is moving.

These directions are all related through the right-hand rule. It’s like magic, but it’s just how rotation works.

Examples of Circular Motion (Centripetal Force)

Circular motion means something is moving in a round path. But to keep moving in a circle, a special force is needed. This force is called centripetal force. It always pulls the object towards the center of the circle. Without this force, the object would not go in a circle — it would move straight.

Let’s look at some easy-to-understand examples:

1. A Car Turning on the Road

Imagine a car is moving on a straight road. Now, it comes to a curve or turn. To make that turn, the car needs a force that pulls it towards the center of the curve. This force comes from friction between the car’s tires and the road.

• If the road is dry and the tyres are good, the car turns safely.
• But if the tyres are old or the road is wet or slippery, the friction is less.
• If there’s not enough friction, the car can’t turn properly and may slide straight ahead instead of going around the curve.

So, the centripetal force here is the friction between the tyres and the road.

2. A Ball Tied to a String and Spun in a Circle

Now, take a light ball and tie it to one end of a string. Hold the other end and spin the ball around in a circle in front of you.

• What keeps the ball moving in a circle? It is the pull of the string — that’s the centripetal force.
• If you suddenly let go of the string, the ball will not go in a circle anymore. Instead, it will fly off in a straight line.
• This happens because the pulling force is gone. The ball wants to keep moving — and it goes in the direction it was heading at that moment (this is called a tangent to the circle).

A similar thing happens with mud on a fast-moving bicycle. Mud sticks to the tyre but gets thrown off in a straight direction because it is not getting any inward force. That’s why mudguards are used — to stop the mud from flying onto the rider.

Circular Motion in Nature

Look around you. Many things in nature move in a round or circular way. One of the best examples is our Earth. The Earth moves around the Sun in a big circle. This movement is called circular motion. But to keep moving in a circle, the Earth needs a special kind of force. This force is always pulling the Earth towards the Sun. We call this a centripetal force. The Sun pulls the Earth using its gravity, and this pull gives the Earth the force it needs to stay in its path.

In the same way, the Moon goes around the Earth in a nearly circular path. The Earth pulls the Moon with its gravity. This pulling force gives the Moon the power to keep going around the Earth. So, both the Earth and the Moon move in circles because of gravity.

Circular Motion in an Atom

Now let’s go even smaller – into the world of atoms. An atom is very tiny, but even here, we can see circular motion. Inside an atom, there is a center part called the nucleus. Around this nucleus, tiny particles called electrons move in small circles.

To keep moving in a circle, each electron also needs a force. The nucleus has a positive charge, and the electrons have a negative charge. Because opposite charges attract each other, the nucleus pulls the electrons towards it. This pulling force is called an electrostatic force, and it acts just like the centripetal force that keeps the electron moving in a circle.

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