Circular motion is a motion in which an object moves around a fixed point in a circular motion. It can be both uniform and non-uniform. If there is no tangential component of acceleration, then it is a uniform circular motion, and if the tangential component of acceleration is present, then it is a non-uniform circular motion.
⇒There are lots of examples around us in our daily lives where bodies perform circular motions every day. From the hands of the clock to a car turning on a banked road. All these are examples of circular motion. This motion can be classified into two categories - uniform circular motion and non-uniform circular motion.
⇒If the tangential component of acceleration is present, the motion becomes non-uniform circular motion. In non-uniform circular motion, the net acceleration is the vector sum of radial and tangential accelerations. Radial acceleration points towards the center, while tangential acceleration affects the particle's speed along the path.
⇒When observing a particle in uniform circular motion from an inertial frame of reference, we can apply Newton’s second law of motion. Since the particle experiences acceleration, the net force on it must be non-zero.
⇒In uniform circular motion, the particle moves with constant speed, but it continuously changes direction, implying an acceleration towards the center of the circle, which is given by a=v2/r where 'v' is the particle's speed and 'r' is the radius of the circular path.
⁛ Now, considering the mass of the particle to be mmm, according to Newton’s second law, the net force acting on the particle is:
F=ma
F=m(v2/r)
⇒This force is directed inward, towards the center of the circular path. This inward force is termed centripetal force, a specific force that acts to keep the object in a stable, circular motion.
⇒The nature of centripetal force can vary depending on the situation: it could arise due to tension in a string (in the case of a pendulum), gravitational pull (as in planetary orbits), or friction (such as a car turning around a bend). Despite its different sources, centripetal force always acts inwards, maintaining the object's motion along the circular trajectory.
What is Circular Motion?
An object moving in a circular trajectory around a point is said to be performing a circular motion. For example, a car turning on the road is performing a circular motion around some centre. The circular motion of continued is periodic in nature. Periodic motions are motions that repeat themselves after a certain period of time. Based on the velocity and the acceleration, this motion can be classified into two categories -
- Uniform Circular Motion: When a body moves at constant speed in a circular motion along the circumference of the circular trajectory, it is called Uniform Circular Motion.
- Non-Uniform circular motion: When a body moves in a circular motion along the circumference of a circular trajectory in such a way that its speed keeps changing. It is called a non-uniform circular motion.
Dynamics of Circular Motion
In this motion, the body is moving at a constant speed. Let's say the radius of the circular trajectory on which the body is moving is “r”, and the speed of the body is v m/s. The figure shows the body going from point A to point B in time “t”. The length of the arc from point A to point B is denoted by “s”. In this, the angle covered by the object is given by,
θ = s / r
The image given below tells us about the Dynamics of Circular Motion,
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The angular velocity of the body is defined as the rate of change of angle. It's similar to velocity in the case of straight-line motion. It is denoted by the Greek symbol ω.
ω = dθ / dt
Using the relation given above for the angle covered.
ω = d / dt (s/r)
ω = ds / dt (1/r)
where,
s is the length of the arc (distance covered by the body)
v = ds / dt
where,
v is the speed of the body
Substituting the value in the equation,
ω = ds / dt (1/r)
ω = v (1/r)
v = rω
(or)
ω = v /r
The image given below shows the formula for angular velocity.
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Uniform Circular Motion
Bodies have a tendency to move in a straight line. For the bodies making a circular motion at a constant speed, there must be some force that keeps them on a circular path. Such a force is called the centripetal force. The reaction of this force is called centrifugal force. This means that both these forces are equal and opposite in direction.
Centrifugal force is given by,
F = mv2 / r
We known that, ω = v / r
Substituting this relation into the equation,
F = mv2 / r
F = m(rω)2 / r
F = mrω2
Circular turnings on Roads
Vehicles turning on the road travel along a circular arc the forces acting on the vehicle are given below,
- Weight (Mg) of the car
- Normal reaction force (N)
- Friction
In a horizontal road, the weight of the car and normal reaction force cancel each other. The only force acting in the radial direction is friction force which provides the radial acceleration here, the force of friction acts as the centripetal force.
F = m(v2/r)
As, there is a limit to the maximum value of friction, for a safe turn,
m(v2/r) ≤ μsN
where,
µs is the coefficient of static friction
Right-Hand Rule
While holding the axis of rotation and gripping the fingers in the direction of motion of the object. If the thumb is arranged perpendicular to the curvature of the fingers then it represents the angular displacement vector. The velocity in this motion is called angular velocity.
⇒Angular velocity is the rate of change of angular displacement. It is denoted by ω. It is a vector quantity.
ω = v/r
⇒In the Right-Hand rule, the thumb represents the direction of angular velocity.
⇒If a body performs anticlockwise rotation, then according to the right-hand rule, the direction of 'ω' is along the axis of a rotation and directed upwards, while for clockwise rotation, 'ω' is directed downwards.
Solved Examples - Dynamics Of Circular Motion
Example 1: Find the angular velocity of a body that is moving at a speed of 10 m/s in a circle of radius 2 m.
Solution:
Formula for angular velocity is given by,
ω = v/r
Given:
v = 10m/s and r = 2 m.
ω = v/r
ω = 10/2
ω = 5 rad/sec
Example 2: Find the force acting on a particle of mass 5 kg moving in a circle of radius 4 m at an angular speed of 20 rad/s.
Solution:
Formula for centripetal force is given by,
F = mrω2
Given:
ω = 20 rad/sec, r = 2 m and m = 2Kg
F = mrω2
F = 2×2×(20)2
F = 1600 N
Example 3: An insect moves in a circle of 4 m radius and completes 20 revolutions per second. Find the angular velocity, linear velocity, and acceleration.
Solution:
The body moves at 20 revolutions per second.
ω = Angle Cover per Second
= 20(2π)
= 40πAngular velocity is 40π rad/s
Linear velocity is given by,
v = rω
Given:
r = 4 m
v = (4)(40π)
v= 160π m/s
Acceleration will be given by,
a = v2 / r
a = (160π)2 / 4
a = 64000 π2 m/s2
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