Characteristics of circular motion :

• It is an accelerated motion: As the direction of velocity changes at every instant, it is an accelerated motion.
• It is a periodic motion: During the motion, the particle repeats its path along the same trajectory. Thus, the motion is periodic.

Linear velocity of a particle performing circular motion:

OR

Relation between the linear velocity and the angular velocity of a particle performing circular motion :

Consider a particle performing circular motion in an anticlockwise sense, along a circle of radius r. In a very small time interval δt the particle moves from point A to point B through a distance δs and its angular position changes by δθ.

δθ = \(\frac{arc\, AB}{radius} = \frac{δs}{r} \)

As \(\vec{δt}\) → 0, B will be very dose to A and displacement \(\vec{AB}=\vec{δs}\)   will be a straight Line perpendicular to radius vector \(\vec{OA}=\vec{r}\)By the right hand rule of cross product.

\(\vec{δs}=\vec{δθ}×\vec{r}\)

∴ \(\lim\limits_{δt\to 0} \frac{\vec{δs}}{δt} = \lim\limits_{δt\to 0} \frac{\vec{δθ}}{δt}×\vec{r}\)

∴ \( \frac{\vec{δs}}{δt} = \frac{\vec{δθ}}{δt}×\vec{r}\)  ............ (1)

The linear velocity \(\vec{v}\) of the particle is the time rate of displacement and its angular velocity \(\vec{ω}\)  is the time rate of angular displacement.

\(∴ \vec{v}=  \frac{\vec{δs}}{δt} and \, \vec{ω} = \frac{\vec{δθ}}{δt}\)

Therefore, from Eq. (1),

  \(∴ \vec{v}=\vec{ω}×\vec{r}\)

Since \(∴ \vec{ds}\)  is tangential, the instantaneous linear velocity \(\vec{v}\)  of a particle performing circular motion is along the tangent to the path, in the sense of motion of the particle.

\(\vec{v}, \vec{ω} \,and\, \vec{r} \) are mutually perpendicular, so that in magnitude, v = ω r

Relation between linear velocity and angular velocity of a particle in circular motion:

Linear velocity \(\vec{v}=\vec{ω}×\vec{r}\)   where \(\vec{ω}\) is the angular velocity and \(\vec{r}\)is the radius vector.

At every instant, \(\vec{v}, \vec{ω} \,and\, \vec{r} \) are mutually perpendicular, so that in magnitude, v = ω r

Acceleration of a particle in UCM: (Expression for the acceleration)

A particle in uniform circular motion (UCM) moves in a circle or circular arc at constant linear speed v. The instantaneous linear velocity  of the particle is along the tangent to the path in the sense of motion of the particle. Since \(\vec{v}\) changes in direction, without change in its magnitude, there must be an acceleration that must be (i) perpendicular to; (ii) constant in magnitude (iii) at every instant directed radially inward, i.e., towards the centre of the circular path.

Such a radially inward acceleration is called a centripetal acceleration.

\(\vec{a}=\frac{\vec{dv}}{dt}=\vec{a_r}\)

If \(\vec{ω}\) is the constant angular velocity of the particle and r is the radius of the circle,

\(\vec{a_r}=-ω^2\vec{r}\)

where ω = |\(\vec{ω}\)| and the minus sign shows that the direction of \(\vec{a_r}\)  is at every instant opposite to that of the radius vector \(\vec{r}\) In magnitude,

\(\vec{a_r} = ω^2\vec{r} = \frac{v}{r} = ωv\)

Diagram showing the linear velocity, angular velocity and radial acceleration of a particle performing circular motion with radius r.

Explanation :

If a particle performing circular motion is speeding up or slowing down, its angular speed w to and linear speed v both change with time.

Tangential acceleration when linear speed is (a) increasing (b) decreasing

The linear acceleration that produces a change only in the linear speed must be along \(\vec{v}\) . Hence, it is called the tangential acceleration \(\vec{a_t}\)  . In magnitude, \(\vec{a_t}=\frac{dv}{dt}\)

(ii) If the linear speed v of the particle is increasing, \(\vec{a_t}\) is in the direction of \(\vec{v}\). If v is decreasing \(\vec{a_t}\)  directed opposite to \(\vec{v}\). see Fig.

Differences between a nonuniform circular motion and a uniform circular motion:

Nonuniform circular motion :

(i) The angular and tangential accelerations are non-zero, so that linear and angular speeds both change with time.

\(\vec{α} =\frac{\vec{dω}}{dt}\)  and \(\vec{a_t} =\frac{\vec{dv}}{dt}=\vec{α}×\vec{r}\)

If a particle in nonuniform circular motion is speeding up, \(\vec{α}\)  is in the direction of \(\vec{ω}\)and \(\vec{a_t}\) is in the direction of \(\vec{v}\) if the particle is slowing down, \(\vec{α}\)  is opposite to \(\vec{ω}\)  and \(\vec{a_t}\) is opposite to \(\vec{v}\) 

(ii) The net linear acceleration, being the resultant of the radial and tangential accelerations, is not radial.\(\vec{a}=\vec{a_t}+\vec{a_r} \)

(ii) The magnitudes of the centripetal acceleration and the centripetal force are not constant.

Example : Motion of the tip of a fan blade when the fan is speeding up or slowing down.

Uniform circular motion :

(i) The angular and tangential accelerations are zero, so that linear speed and angular velocity are constant.

(ii) The net linear acceleration is radially inward, i.e., centripetal.

(iii) The magnitudes of the centripetal acceleration and the centripetal force are also constant.

Example : Motion of the tips of the hands of a mechanical clock.

Kinematical equations for circular motion in analogy with linear motion:

For circular motion of a particle with constant angular acceleration a

average angular speed, ω_av = (ω₀ + ω)/2

a = (ω - ω₀)/t and θ - θ₀ = ω_av × t

where ω₀ and w are the initial and final angular speeds, t is the time, may the average angular speed and θ₀ and θ the initial and final angular positions of the particle.

Then, the angular kinematical equations for the circular motion are (in analogy with linear kinematical equations for constant linear acceleration)

ω = ω₀ + αt

q - q₀ = ω₀ t +  αt²

ω² = ω₀² + 2α(θ - θ₀)

Explanation :A uniform circular motion is an accelerated motion, with a radially inward (i.e.centripetal) acceleration \(\frac{-v}{r}\breve{r}\)  or — ω² \(\vec{r}\), where \(\vec{r}\) is the radius vector and \(\breve{r}\) is a unit vector  the direction of \(\vec{r}\)  . Hence, a net real force must act on the particle to produce this acceleration. This force, which at every instant must point radially towards the centre of the circle, is called the centripetal force.

If m is the mass of the particle, the centripetal force is  or \(-m\frac{v}{r}\breve{r}\)  or — mω² \(\vec{r}\)

Examples of centripetal force :

• For an Earth-satellite in a circular orbit, the centripetal force is the gravitational force exerted by the Earth on the satellite.
• In the Bohr atom, the centripetal force on an electron in circular orbit around the nucleus is the attractive Coulomb force of the nucleus.
• When an object tied at the end of a string is revolved in a horizontal circle, the centripetal force is the tension in the string.
• When a car takes a turn in a circular arc on a horizontal road with constant speed, the force of static friction between the car tyres and road surfaces is the centripetal force.

Explanation: A uniform circular motion is an accelerated motion, with a centripetal acceleration of magnitude v²/r or ω²r. A frame of reference attached to the particle also has this acceleration and, therefore, is an accelerated or noninertial reference frame. The changing direction of the linear velocity appears in this reference frame as a tendency to move radially outward. This is explained by assuming a fictitious centrifugal, i.e., radially outward, force acting on the particle. Since the particle is stationary in its reference frame, the magnitude of the centrifugal force is mv²/r  or mω²r, the same as that of the centripetal force on the particle.

Examples of centrifugal force :

• A person in a merry-go-round experiences a radially outward force.
• Passengers of a car taking a turn on a level road experience a force radially away from the centre of the circular road.
• A coin on a rotating turntable flies off for some high enough angular speed of the turntable.
• As the Earth rotates about its axis, the centrifugal force on its particles is directed away from the axis.
• The force increases as one goes from the poles towards the equator. This leads to the bulging of the Earth at the equator.

Difference  between centripetal force and centrifugal force.