Table of Contents
- Introduction to Rotatory Motion
- Rotatory Motion and Circular Motion
- What is Radius Vector?
- Angular Displacement
- Angular Velocity
- Angular Acceleration
- Relation Between Linear Velocity and Angular Velocity
- Comparison of Equations of Circular Motion with Equations of Translatory Motion
- Uniform Circular Motion
- Non-Uniform Circular Motion
- Centripetal Force
- Centripetal Acceleration
- Centrifugal Force
- Dynamics of Non-Uniform Circular Motion
-
Applications of Centripetal Force
- Why is Centripetal Force Necessary When a Cyclist Takes a Turn on the Road?
- Why is Centripetal Force Necessary During the Motion of a Vehicle at a Turn?
- Why is Centripetal Force Necessary When a Vehicle Moves on a Smooth Banked Road?
- Why is Centripetal Force Necessary When a Vehicle Moves on a Rough Banked Road?
- Centripetal Force and Death Well
- Centripetal Force and Conical Pendulum
- Centripetal Force When a Particle is in a Rotating Ball
- Centripetal Force When a Particle Revolves in a Cone
- Centripetal Force and Tension in a Ring
Introduction to Rotatory Motion
In translatory or linear motion, different particles of a rigid body have the same linear displacement and linear velocity. In rotational motion, different particles of a rigid body move in circles, and the centres of all these circles lie on a line called the axis of rotation. The angular displacement and angular velocity are the same for all the particles of a rigid body.
When a fan is switched on, its blades perform rotatory motion. When the Earth revolves around the Sun, it performs orbital or circular motion. When we travel in a car, the wheels of the car perform rolling motion, which is a combination of rotatory motion and translatory motion.
Force is the cause of translatory motion. Torque is the cause of rotatory motion, and in rotational motion it plays the same role as force plays in translatory motion. Thus, torque is regarded as the rotational analogue of force.
Rotatory Motion and Circular Motion
What is Rotatory Motion?
Rotatory motion is the motion of a rigid body in which the body rotates about a fixed axis, and every particle of the body moves along a circular path whose centre lies on the axis of rotation.
Figure: A rigid body rotating about a fixed axis.
In rotatory motion, the particles of a rigid body move in circular paths around the axis of rotation. The centres of these circular paths lie on the axis. Particles located at different distances from the axis travel along circles of different radii.
Although the linear speeds of different particles may be different, all particles of a rigid body rotating about a fixed axis have the same angular displacement and angular velocity at a given instant. A particle farther from the axis travels a greater distance during the same angular rotation and therefore has a greater linear speed.
Examples of Rotatory Motion
1. Rotation of a Ceiling Fan
When a ceiling fan is switched on, its blades rotate about the fixed axis passing through the centre of the fan. Every point on a blade moves along a circular path around this axis. Points farther from the axis travel through larger circular paths.
2. Rotation of a Spinning Top
A spinning top rotates about an axis passing through its body. Points of the top away from the axis describe circular paths as the top spins. The top therefore provides a familiar example of rotational motion about an axis.
3. Rotation of a Bicycle Wheel
When a bicycle wheel rotates about its axle, the wheel undergoes rotational motion about its central axis. Points on the rim move along circular paths centred on the axle. When the bicycle is moving forward without slipping, the wheel simultaneously participates in translational motion, producing rolling motion.
What is Circular Motion?
Circular motion is the motion of a particle or object along a circular path about a fixed point or fixed axis, such that its distance from the centre of the circular path remains constant.
Figure: A particle P moving along a circular path of radius r about the centre O.
In circular motion, the particle continuously changes its direction of motion while moving along the circular path. Even if the magnitude of its speed remains constant, its velocity changes because velocity is a vector quantity and its direction changes continuously.
The line joining the centre of the circular path to the particle is called the radius. During circular motion, the magnitude of this radius remains constant. The velocity of the particle at any instant is directed along the tangent to the circular path, while the required inward acceleration is directed towards the centre.
Circular motion may be uniform or non-uniform. In uniform circular motion, the particle moves with constant speed, whereas in non-uniform circular motion its speed changes with time.
Examples of Circular Motion
1. A Stone Whirled in a Circle
When a stone tied to a string is whirled in a horizontal circle, the stone moves continuously along a circular path around the hand or fixed point about which it is being rotated. Its distance from the centre remains constant if the string remains taut.
2. A Vehicle Moving Around a Circular Track
A vehicle moving around a circular track follows a curved path whose centre remains fixed. As the vehicle travels around the track, its direction of motion continuously changes. If the vehicle moves with constant speed, its motion is an example of uniform circular motion.
3. Motion of the Tip of the Second Hand of a Clock
The tip of the second hand of a clock moves continuously along a circular path about the centre of the clock. Its distance from the centre remains constant, while its direction of motion changes continuously. Since the second hand completes one revolution in a fixed time interval, its motion is an example of uniform circular motion.
Differences Between Rotatory Motion and Circular Motion
Rotatory motion and circular motion may look similar, but they are different types of motion. The following points help us understand the difference clearly.
| Rotatory Motion | Circular Motion |
|---|---|
|
1. Distance from the axis: Each particle of the rotating body remains at the same distance from the axis of rotation. |
1. Distance from the centre: The object remains at the same distance from the centre of its circular path. |
|
2. What happens? The whole body rotates about an axis. |
2. What happens? An object or particle moves along a circular path. |
|
3. Axis: The body rotates about an axis of rotation. |
3. Centre: The object moves around a fixed centre. |
|
4. Number of paths: Different particles of the body move in circles of different radii. |
4. Number of paths: The object moves along one circular path. |
|
5. Angular displacement: All particles of a rigid body have the same angular displacement. |
5. Angular displacement: The particle changes its angular position as it moves around the centre. |
|
6. Angular velocity: All particles of a rigid body have the same angular velocity. |
6. Angular velocity: The particle has angular velocity while moving around the centre. |
|
7. Size of the body: The rotating body has a definite size and contains many particles. |
7. Size of the object: The moving object can be treated as a particle for simple calculations. |
|
8. Example: A ceiling fan rotating about its central axis shows rotatory motion. |
8. Example: A stone tied to a string and whirled in a circle shows circular motion. |
|
9. Direction of motion: Different particles of the body move in different circular paths around the same axis. |
9. Direction of motion: The object continuously changes its direction while moving around the circular path. |
|
10. Common example: The rotating blades of a fan are an example of rotatory motion. |
10. Common example: A car moving around a circular track is an example of circular motion. |
Easy way to remember:
Rotatory motion means a body rotates about an axis.
Circular motion means an object moves around a centre.
What is Radius Vector?
The radius vector is a directed line drawn from the centre of the circular path to the position of the moving object.
It tells us both the distance of the object from the centre and the direction of the object from the centre. It is represented by the symbol r.
In the diagram, C is the centre of the circular path and P is the position of the moving object. The line CP represents the radius vector.
The radius vector always points from the centre towards the object. Therefore, it has both magnitude and direction, so it is a vector quantity.
Important Points
- The radius vector is represented by r.
- It is directed from the centre to the object.
- Its magnitude gives the distance of the object from the centre.
- It has both magnitude and direction.
- Therefore, the radius vector is a vector quantity.
- In uniform circular motion, the magnitude of the radius vector remains constant, but its direction continuously changes.
In simple words: The radius vector is the line joining the centre of the circular path to the moving object, with an arrow pointing towards the object.
Angular Displacement
When an object moves along a circular path, the line joining the centre of the circle to the object is called the radius vector. The angle through which the radius vector turns from its initial position to its final position is called angular displacement.
In the diagram, OP₀ is the initial radius vector and OP is the final radius vector. The angle between them is the angular displacement, represented by θ.
Formula for Angular Displacement
If a particle moves through an arc of length s on a circle of radius r, its angular displacement in radians is given by:
θ = s / r
where θ is the angular displacement, s is the length of the arc travelled, and r is the radius of the circular path.
SI Unit
The SI unit of angular displacement is the radian (rad).
Angular displacement has no dimensions because the ratio of arc length to radius is a ratio of two lengths.
Angular Displacement in One Complete Revolution
When a particle completes one complete revolution, it turns through an angle of 360°, which is equal to 2π radians.
1 revolution = 360° = 2π rad
Therefore, if a particle completes N complete revolutions, its angular displacement is:
θ = 2πN
Angular Displacement of a Rotating Rigid Body
When a rigid body rotates about a fixed axis, all the particles of the body turn through the same angle in the same time. Therefore, all the particles have the same angular displacement.
However, particles at different distances from the axis travel through different lengths of arc. A particle farther from the axis travels through a larger arc than a particle closer to the axis.
In the diagram, points A and B rotate through the same angular displacement θ. But point B is farther from the axis, so it travels through a larger distance than point A.
Direction of Angular Displacement
Angular displacement has a direction along the axis of rotation. Its direction can be determined using the right-hand rule.
Curl the fingers of your right hand in the direction of rotation. Your thumb points in the direction of the angular displacement vector.
Important Points
- Angular displacement is the angle through which the radius vector turns.
- It is represented by the symbol θ.
- Its SI unit is the radian (rad).
- Its dimensional formula is [M⁰L⁰T⁰].
- One complete revolution is equal to 2π radians.
- For N complete revolutions, angular displacement is θ = 2πN.
- For a rigid body rotating about a fixed axis, all particles have the same angular displacement.
- Angular displacement is different from linear displacement. Linear displacement depends on the distance of the particle from the axis.
In simple words: Angular displacement tells us how much an object has turned from its initial position to its final position.
Angular Velocity
The rate of change of angular displacement of a particle is called angular velocity.
In simple words, angular velocity tells us how fast a particle is rotating about a fixed point or axis.
In the diagram, the particle moves from its initial position P₀ to its final position P. During this motion, the radius vector turns through an angular displacement θ.
The angular velocity tells us how quickly this angular displacement takes place.
Mathematical Expression
Angular velocity is mathematically expressed as the change in angular displacement per unit time:
ω = Δθ / Δt
where:
- ω = angular velocity
- Δθ = change in angular displacement
- Δt = time interval
For the angular velocity at a particular instant, the time interval is made very small. Therefore:
ω = dθ / dt
The expression ω = dθ/dt represents the instantaneous angular velocity.
SI Unit
The SI unit of angular velocity is radian per second.
SI unit = rad s−1
Dimensions
Angular displacement is dimensionless because it is the ratio of two lengths. Therefore, angular velocity has the dimensions of inverse time.
[ω] = [T−1]
Therefore, the dimensional formula of angular velocity is:
[M0L0T−1]
Angular Velocity of a Rotating Rigid Body
When a rigid body rotates about a fixed axis, all its particles turn through the same angular displacement in the same time.
Therefore, all the particles of a rigid body rotating about a fixed axis have the same angular velocity.
However, particles at different distances from the axis may have different linear velocities.
Important Points
- Angular velocity tells us how fast a particle is rotating.
- It is represented by the symbol ω.
- Angular velocity is the rate of change of angular displacement.
- Its SI unit is rad s−1.
- Its dimensional formula is [M0L0T−1].
- For a rigid body rotating about a fixed axis, all particles have the same angular velocity.
- Particles at different distances from the axis can have different linear velocities.
In simple words: Angular velocity tells us how fast an object turns or rotates.
Average Angular Velocity
Average angular velocity is the total angular displacement divided by the total time taken.
If a particle changes its angular position from θ₁ to θ₂ in a time interval from t₁ to t₂, its angular displacement is θ₂ − θ₁.
Mathematical Expression
ωavg = Δθ / Δt
ωavg = (θ₂ − θ₁) / (t₂ − t₁)
where:
- ωavg = average angular velocity
- Δθ = angular displacement
- Δt = time interval
SI Unit
The SI unit of average angular velocity is radian per second.
rad s−1
Dimensional Formula
Since angular displacement is dimensionless, the dimensional formula of average angular velocity is:
[M0L0T−1]
Nature of Physical Quantity
Average angular velocity is a vector quantity because it has both magnitude and direction.
In simple words: Average angular velocity tells us the average angular displacement made per unit time during a given time interval.
Instantaneous Angular Velocity
The angular velocity of the particle at a particular instant of time is called instantaneous angular velocity.
ωinst = limΔt→0 (Δθ / Δt)
ωinst = dθ / dt
The SI unit of angular velocity is radian per second. Its dimensional formula is [T−1].
Angular velocity is a pseudo vector. Its direction is given by the right hand screw rule. The direction of the angular velocity will be along the axis of rotation.
Important Notes
i) If the particle moves with a constant speed in a circular path, its instantaneous and average angular velocities are equal.
ii) When a body performs N rotations in a time interval t, then its average angular velocity is:
ω = 2πN / t
Angular velocity is also known as angular frequency.
iii) If a particle makes n rotations per second (rps), its angular velocity is given by:
ω = 2πn
where:
n = N / t
iv) If T is the time period of revolution of a particle, its angular velocity is:
ω = 2π / T
where n = 1 / T
v) Angular velocity of a particle depends on about which point it is determined or about which axis of rotation it is determined.
Application 4.1
As shown in the figure, the angular velocity of the particle about O is
ωO = α / t
and the angular velocity of the particle P about A is
ωA = β / t
ωO ≠ ωA
If a particle is moving on a circle from P to P′ in time t as shown in the figure, the angular velocity with respect to O will be:
ωO = β / t
While with respect to A it will be:
ωA = α / t
But from the geometry of the figure:
β = 2α
Therefore, ωO = 2ωA
Relative Angular Velocity
vi) If two particles are moving on the same circle or different coplanar concentric circles in the same direction with different uniform angular speeds ωA and ωB respectively, the angular velocity of B relative to A for an observer at the centre will be:
ωrel = ωB − ωA
So the time taken by one to complete one revolution around O with respect to the other (i.e., time in which B completes one more or less revolution around O than A) is:
T = 2π / ωrel
T = 2π / (ωB − ωA)
T = T1T2 / (T1 − T2)
In the above case, if the two particles are revolving in opposite directions in a circular path, then:
ωrel = ωB + ωA
T = 2π / ωrel
T = 2π / (ωB + ωA)
T = T1T2 / (T1 + T2)
vii) If two particles are moving on two different concentric circles with different velocities then angular velocity of B relative to A as observed by A will depend on their positions and velocities. When A and B are closest to each other and moving in the same direction:
vrel = |vB − vA| = vB − vA
rrel = |rB − rA| = rB − rA
ωrel = vrel / rrel
ωrel = (vB − vA) / (rB − rA)
viii) When a body is in pure rotation about an axis, all the particles of the body will have the same angular velocity irrespective of their distance from axis of rotation.
Angular Velocity of Hands of a Clock
1. Angular velocity of seconds hand: Seconds hand completes one revolution in 60 s.
ω = 2π / 60
ω = π / 30 rad s−1
2. Angular velocity of minutes hand: Minutes hand completes one revolution in one hour (3600 s).
ω = 2π / 3600
ω = π / 1800 rad s−1
3. Angular velocity of hours hand: Hours hand completes one revolution in 12 hours.
ω = 2π / (12 × 3600)
ω = π / 21,600 rad s−1
4. Spin angular velocity of Earth about its own axis:
ω = 2π / (24 × 3600)
ω = π / 43,200 rad s−1
Angular Acceleration
The rate of change of angular velocity of a particle is called angular acceleration.
In simple words, angular acceleration tells us how quickly the angular velocity of a particle changes.
Mathematical Expression
Angular acceleration is represented by the symbol α.
α = dω / dt
where α is angular acceleration, ω is angular velocity, and t is time.
Average Angular Acceleration
Average angular acceleration is the change in angular velocity divided by the time taken.
Mathematical Expression
αavg = Δω / Δt
αavg = (ω2 − ω1) / (t2 − t1)
where:
- αavg = average angular acceleration
- Δω = change in angular velocity
- Δt = time interval
SI Unit
The SI unit of angular acceleration is radian per second squared.
rad s−2
Dimensional Formula
Since angular velocity has the dimension of inverse time, the dimensional formula of angular acceleration is:
[M0L0T−2]
Nature of Physical Quantity
Angular acceleration is a vector quantity. Its direction is along the axis of rotation and is determined by the right-hand rule.
In simple words: Angular acceleration tells us how quickly the angular velocity changes.
Instantaneous Angular Acceleration
The angular acceleration of a particle at a particular instant of time is called instantaneous angular acceleration.
Mathematical Expression
αinst = limΔt→0 (Δω / Δt)
αinst = dω / dt
where α is angular acceleration and ω is angular velocity.
SI Unit
The SI unit of instantaneous angular acceleration is radian per second squared.
rad s−2
Dimensional Formula
The dimensional formula of instantaneous angular acceleration is:
[M0L0T−2]
Nature of Physical Quantity
Angular acceleration is a pseudo vector quantity. Its direction is along the axis of rotation and is determined by the right-hand screw rule.
Note
1. When the angular velocity increases, the direction of angular acceleration is in the same direction as the angular velocity. When the angular velocity decreases, the direction of angular acceleration is in the opposite direction to the angular velocity.
2. If the particle is moving with constant angular velocity, its angular acceleration is zero.
Relation Between Linear Velocity and Angular Velocity
When a particle moves along a circular path, it has both linear velocity and angular velocity. These two quantities are related to each other by the radius of the circular path.
If a particle is at a distance r from the centre of the circular path and has angular velocity ω, its linear velocity v is given by:
v = rω
In the diagram, the particle moves from its initial position P₀ to the nearby position P. The radius vector turns through a small angular displacement dθ.
The small distance travelled by the particle along the circular path is represented by ds. The linear velocity v acts along the tangent to the circular path at the position of the particle.
Derivation of the Relation
Consider a particle moving through a small angular displacement dθ along a circular path of radius r.
The small arc length ds is related to the angular displacement by:
ds = r dθ
Dividing both sides by the small time interval dt:
ds / dt = r dθ / dt
We know that the linear velocity is:
v = ds / dt
We also know that the angular velocity is:
ω = dθ / dt
Therefore:
v = rω
v = rω
Thus, the linear velocity of a particle moving in a circular path is equal to the product of its distance from the centre and its angular velocity.
Meaning of the Relation
The relation v = rω shows that the linear velocity of a particle depends on its distance from the axis of rotation.
For the same angular velocity, a particle farther from the axis travels a greater distance in the same time. Therefore, it has a greater linear velocity.
v ∝ r when ω = constant
Units and Dimensions
The SI unit of linear velocity is m s−1, while the SI unit of angular velocity is rad s−1.
The dimensional formula of linear velocity is:
[M0L1T−1]
In simple words: The linear velocity of a particle in circular motion is equal to its radius multiplied by its angular velocity.
Comparison of Equations of Circular Motion with Equations of Translatory Motion
The equations of circular motion have a similar mathematical form to the equations of translatory motion. The corresponding equations are compared below.
| Equations of Circular Motion | Equations of Translatory Motion |
|---|---|
|
First equation ω = ω₀ + αt |
First equation v = u + at |
|
Second equation θ = ω₀t + ½αt² |
Second equation s = ut + ½at² |
|
Third equation ω² = ω₀² + 2αθ |
Third equation v² = u² + 2as |
|
Average angular velocity ωavg = (ω₀ + ω) / 2 |
Average linear velocity vavg = (u + v) / 2 |
|
Displacement using average velocity θ = [(ω₀ + ω) / 2]t |
Displacement using average velocity s = [(u + v) / 2]t |
|
Angular displacement in the nth interval θn = ω₁ + ½α(2n − 1) |
Linear displacement in the nth interval Sn = u + ½a(2n − 1) |
Corresponding Quantities
| Circular Motion | Translatory Motion |
|---|---|
| Angular displacement (θ) | Linear displacement (s) |
| Angular velocity (ω) | Linear velocity (v) |
| Angular acceleration (α) | Linear acceleration (a) |
In simple words: Angular quantities in circular motion play a role similar to linear quantities in translatory motion.
Uniform Circular Motion
When a particle moves in a circular path such that its radius vector makes equal angular displacements in equal intervals of time then the particle is said to be in uniform circular motion.
(or)
When a particle moves in a circular path with constant speed then it is said be in uniform circular motion.
Examples:
1) When the moon is revolving round the earth in a circular path, its motion is uniform circular motion.
2) When we sit on a merry-go-round moving with constant angular velocity we perform uniform circular motion.
3) A satellite revolving round the earth in a circular orbit performs uniform circular motion.
Note: In uniform circular motion magnitude of linear velocity (speed), kinetic energy, angular velocity and angular momentum with respect to centre of circle remains constant, but velocity changes due to change in direction.
Note: In uniform circular motion all real vectors like velocity, acceleration, force momentum are constant in magnitude but continuously change in their direction.
Note: In uniform circular motion the angular velocity is constant and angular acceleration of the particle is zero.
Note: A particle moving with constant speed in a circle is not in equilibrium. (As net force is there which is acting as centripetal force)
In uniform circular motion θ = ωt.
Non-Uniform Circular Motion
1. When a particle moves in a circular path such that its radius vector makes unequal angular displacements in equal intervals of time then the particle is said to be in non-uniform circular motion.
OR
If the speed of a particle moving in a circle changes with respect to time, then its motion is said to be non-uniform circular motion.
We know
v⃗ = ω⃗ × r⃗
The linear acceleration is
a⃗ = dv⃗/dt = d/dt (ω⃗ × r⃗)
= dω⃗/dt × r⃗ + ω⃗ × dr⃗/dt
a⃗ = α⃗ × r⃗ + ω⃗ × v⃗
Here α⃗ × r⃗ is the tangential component of acceleration (a⃗t) and ω⃗ × v⃗ is the radial component (a⃗c).
Hence the resultant acceleration in non-uniform circular motion is
a⃗ = a⃗c + a⃗t (1)
In non-uniform circular motion, the velocity of a particle changes both in magnitude and in its direction. So the acceleration of particle has both the radial and tangential components.
The radial component of acceleration is due to change in the direction of velocity of particle.
The radial acceleration (or) centripetal acceleration
ac = v2/r = ω2r
In vector form
a⃗c = ω⃗ × v⃗
The tangential component of acceleration is due to the change in the magnitude of velocity of the particle, tangential acceleration
at = dv/dt = d/dt(rω) = r dω/dt = rα
In vector form
a⃗t = α⃗ × r⃗
The magnitude of net acceleration of a particle in non-uniform circular motion is given by
a = √(ac2 + at2) = √[(v2/r)2 + (rα)2]
The direction of the net acceleration, with radial component is given by
tan φ = at/ac
Note: In uniform circular motion, speed (v) of the particle is constant i.e.,
dv/dt = 0
Thus,
at = 0
and
a = ar = v2/r
2. In accelerated circular motion
dv/dt = positive, i.e., tangential acceleration of particle is parallel to velocity v⃗.
3. In decelerated circular motion
dv/dt = negative, i.e., tangential acceleration is antiparallel to velocity v⃗.
4. In non-uniform circular motion, there are linear acceleration and angular acceleration.
a) Linear acceleration: It has two components. One is radial towards the centre of circle and the other is tangential to the circular path.
b) Angular acceleration (α): It is along the axis of rotation.
Centripetal Force
Centripetal force is the net or unbalanced force required by a body to perform circular motion, and it is always directed towards the centre of the circular path.
OR
Centripetal force is the net or unbalanced force directed towards the centre of the circle at any point in order to deviate the body from its tangential path into a circular path.
Why is Centripetal Force Required?
A body moving in a circular path continuously changes the direction of its velocity. Since acceleration is produced whenever the velocity of a body changes, the body must have an acceleration directed towards the centre of the circle. This acceleration is called centripetal acceleration.
According to Newton’s second law of motion, a force is required to produce this acceleration. The force responsible for producing the centripetal acceleration is called centripetal force.
The centripetal force continuously changes the direction of the velocity of the body and keeps the body moving along a circular path.
Direction of Centripetal Force
The direction of the centripetal force is along the radius towards the centre of the circular path. Therefore, at every point of the circular path, the centripetal force is directed radially inward.
The centripetal force is perpendicular to the instantaneous linear velocity of the body because the velocity is always tangential to the circular path.
Thus, the direction of centripetal force and the direction of instantaneous velocity are mutually perpendicular.
Source of Centripetal Force
Centripetal force is not a separate or new type of force. It is the name given to the net inward force that provides the required centripetal acceleration for circular motion.
Depending on the situation, the required centripetal force may be provided by different physical forces such as gravitational force, tension, friction, electrostatic force, or the lift produced by an aircraft’s wings.
Expression for Centripetal Force
According to Newton’s second law of motion,
\[ F_c=ma_c \]
For circular motion, the centripetal acceleration is
\[ a_c=\frac{v^2}{r} \]
Therefore,
\[ F_c=m\frac{v^2}{r} \]
\[ \boxed{F_c=\frac{mv^2}{r}} \]
where \(m\) is the mass of the body, \(v\) is its linear speed and \(r\) is the radius of the circular path.
Centripetal Force in Terms of Angular Velocity
We know that the relation between linear velocity and angular velocity is
\[ v=r\omega \]
Substituting \(v=r\omega\) in the expression for centripetal force,
\[ F_c=\frac{m(r\omega)^2}{r} \]
\[ \boxed{F_c=mr\omega^2} \]
Centripetal Force in Terms of Linear and Angular Velocity
Since
\[ v=r\omega \]
we can write
\[ F_c=\frac{mv^2}{r} \]
\[ \boxed{F_c=mv\omega} \]
Thus, the important expressions for centripetal force are
\[ \boxed{F_c=\frac{mv^2}{r}=mr\omega^2=mv\omega} \]
Centripetal Force in Terms of Kinetic Energy
The kinetic energy of a particle of mass \(m\), moving with speed \(v\), is
\[ K=\frac{1}{2}mv^2 \]
Therefore,
\[ mv^2=2K \]
Since
\[ F_c=\frac{mv^2}{r} \]
\[ \boxed{F_c=\frac{2K}{r}} \]
Centripetal Force and Centripetal Acceleration
Centripetal force produces centripetal acceleration. The centripetal acceleration is always directed towards the centre of the circular path.
\[ \boxed{a_c=\frac{v^2}{r}=r\omega^2} \]
Therefore,
\[ \boxed{F_c=ma_c} \]
Both centripetal force and centripetal acceleration are directed towards the centre of the circular path.
Work Done by Centripetal Force
The centripetal force is directed towards the centre of the circle, whereas the instantaneous displacement of the particle is along the tangent to the circular path. Therefore, the centripetal force is perpendicular to the instantaneous displacement.
\[ W=Fs\cos\theta \]
Here, \(\theta=90^\circ\). Hence,
\[ W=Fs\cos90^\circ \]
\[ \boxed{W=0} \]
Therefore, centripetal force does no work on a particle in uniform circular motion.
What Happens When Centripetal Force Ceases to Act?
If the centripetal force suddenly ceases to act, the body no longer receives the inward force required to maintain its circular path.
Due to its inertia, the body moves along the tangent to the circular path at the point where the centripetal force ceases to act.
Therefore, the body does not move radially outward from the centre. It moves along the tangent to the circular path at that instant.
Nature of Centripetal Force
Centripetal force is a vector quantity because it has both magnitude and direction.
Its magnitude is given by
\[ F_c=\frac{mv^2}{r} \]
and its direction is always towards the centre of the circular path.
SI Unit
The SI unit of centripetal force is newton (N).
\[ 1\,N=1\,kg\,m\,s^{-2} \]
Dimensional Formula
From
\[ F_c=ma_c \]
The dimensions of mass are \([M]\), and the dimensions of acceleration are \([LT^{-2}]\).
\[ [F_c]=[M][LT^{-2}] \]
\[ \boxed{[F_c]=[MLT^{-2}]} \]
Factors Affecting Centripetal Force
From
\[ F_c=\frac{mv^2}{r} \]
Centripetal force is directly proportional to the mass of the body when speed and radius remain constant:
\[ F_c\propto m \]
It is directly proportional to the square of the speed when mass and radius remain constant:
\[ F_c\propto v^2 \]
It is inversely proportional to the radius when mass and speed remain constant:
\[ F_c\propto\frac{1}{r} \]
Examples of Centripetal Force
1. Ball tied to a string: When a ball attached to a string is rotated in a circular path, the tension in the string provides the required centripetal force.
2. Motion of the Moon around the Earth: The gravitational force of attraction between the Earth and the Moon provides the required centripetal force for the Moon’s orbital motion.
3. A car moving around a curved road: When a car moves around a curved road, the frictional force between the tyres and the road provides the required centripetal force.
4. An aeroplane turning in the air: When an aeroplane makes a turn, an appropriate component of the lift force produced by its wings provides the required centripetal force.
5. Charged particle: In situations where a charged particle follows a circular path under the action of an attractive electrostatic force, that force can provide the required centripetal force.
Important Points
- Centripetal force is the net inward force acting on a body moving in a circular path.
- It is always directed towards the centre of the circular path.
- Centripetal force is perpendicular to the instantaneous linear velocity.
- Centripetal force continuously changes the direction of velocity.
- Centripetal force produces centripetal acceleration.
- Centripetal force is not a separate fundamental type of force. It is the role played by the net inward force in maintaining circular motion.
- If the centripetal force ceases to act, the body moves along the tangent to the circular path at that point.
- In uniform circular motion, centripetal force does no work on the particle.
- The magnitude of centripetal force is \[ F_c=\frac{mv^2}{r}=mr\omega^2=mv\omega \]
- The centripetal acceleration is \[ a_c=\frac{v^2}{r}=r\omega^2 \]
- Centripetal force is a vector quantity.
- The SI unit of centripetal force is newton (N).
- The dimensional formula of centripetal force is \[ [MLT^{-2}] \]
Centripetal Acceleration
Centripetal acceleration is the acceleration produced by the continuous change in the direction of velocity of a particle moving in a circular path, and it is directed along the radius towards the centre of the circle.
When a particle moves in a circular path, its velocity changes continuously because the direction of velocity changes from point to point. Therefore, the particle possesses acceleration even when its speed remains constant.
This acceleration is directed towards the centre of the circular path. Hence, it is called centripetal acceleration.
Position Vector of the Particle
Consider a particle moving on the circumference of a circle of radius r with constant angular velocity ω.
At any instant of time t, let its position vector \(\vec{r}\) be represented by \(\overrightarrow{OP}\).
If the particle rotates through an angle \(\theta = \omega t\), its position vector can be resolved into X- and Y-components.
Therefore, the position vector of the particle is
Linear Velocity of the Particle
The linear velocity of the particle is obtained by differentiating its position vector with respect to time.
Linear Acceleration of the Particle
The linear acceleration is obtained by differentiating the velocity with respect to time.
But
Therefore,
Magnitude of Centripetal Acceleration
The magnitude of centripetal acceleration is
Since
we can also write
Also, since
therefore,
Direction of Centripetal Acceleration
The centripetal acceleration is directed towards the centre of the circular path. Hence, it acts along the radius in the inward direction.
It is perpendicular to the instantaneous velocity of the particle. Therefore, in uniform circular motion, centripetal acceleration changes only the direction of velocity and not its magnitude.
Vector Form
Physical Characteristics
| Physical quantity | Centripetal acceleration |
|---|---|
| Symbol | ac |
| SI unit | m s−2 |
| Dimensional formula | [L T−2] |
| Nature of quantity | Vector quantity |
| Direction | Towards the centre of the circular path |
Important Point
Even when the speed of a particle moving in a circular path is constant, its velocity changes continuously because its direction changes continuously. Hence, the particle possesses centripetal acceleration.
Key Formulae
Centrifugal Force
Centrifugal force is an apparent force experienced by an object in a rotating reference frame, which appears to act away from the centre or axis of rotation.
What is Centrifugal Force?
When an object moves in a circular path, it must have an acceleration directed towards the centre of the circle. This inward acceleration is called centripetal acceleration.
From an inertial frame of reference, an object moving in a circular path requires an inward force called centripetal force. There is no actual outward force acting on the object in this frame.
However, when the motion is observed from a rotating frame of reference, the observer finds that the object appears to experience an outward force. This apparent outward force is called centrifugal force.
Why Does Centrifugal Force Appear?
Consider a passenger sitting inside a car that is taking a sharp turn. The car changes its direction, but the passenger’s body tends to continue moving in its original direction because of inertia.
Relative to the turning car, the passenger appears to move towards the outer side of the turn. Therefore, to the passenger inside the turning car, it appears as though a force is pushing the body outward.
This apparent outward force is called centrifugal force.
Mathematical Expression
The magnitude of centrifugal force is equal to the magnitude of the centripetal force when viewed from the corresponding rotating frame.
Since the magnitude of centripetal force is
therefore, the magnitude of centrifugal force is
Using the relation \(v=r\omega\), we obtain
Vector Form
In a rotating frame, centrifugal force is directed radially outward. Therefore, its vector expression is opposite in direction to the centripetal acceleration.
Since
the centrifugal force can be written as
Here, \(\vec{r}\) is taken from the axis of rotation towards the particle, so the centrifugal force is directed outward.
Differences between Centrifugal Force and Centripetal Force
| Centripetal Force | Centrifugal Force |
|---|---|
| It acts towards the centre of the circular path. | It appears to act away from the centre of the circular path. |
| It is a real force acting on the object. | It is an apparent or pseudo force in a rotating frame. |
| It is required to produce centripetal acceleration. | It appears because the observer is using a rotating frame of reference. |
| It can be produced by tension, friction, gravity, normal reaction, etc. | It does not arise from a separate physical interaction. |
| Its direction is towards the centre. | Its direction is away from the centre. |
| Magnitude: \(F_c=\frac{mv^2}{r}\) | Magnitude: \(F_{\mathrm{cf}}=\frac{mv^2}{r}\) |
Why is Centrifugal Force Called a Pseudo Force?
A pseudo force is a force that appears when the motion of an object is described from a non-inertial reference frame, such as a rotating frame.
Centrifugal force does not arise from a physical interaction between the object and another body. It is introduced to describe the motion conveniently from the rotating frame.
Examples of Centrifugal Force
1. Passenger in a Turning Car
When a car takes a sharp turn, the passengers appear to move towards the outer side of the turn. This happens because their bodies tend to maintain their original state of motion due to inertia.
From the rotating frame of the car, this apparent outward tendency is described using centrifugal force.
2. Water in a Rotating Bucket
When a bucket containing water is rotated rapidly in a vertical circle, the water tends to remain pressed against the bottom of the bucket when the bucket is overhead.
From the rotating frame of the bucket, the outward tendency of the water can be described using centrifugal force.
3. Clothes in a Spin Dryer
In a washing machine spin cycle, the drum rotates rapidly. Water in wet clothes tends to move away from the axis of rotation.
The rotating frame of the drum can describe this outward tendency in terms of centrifugal force. The water then passes through the holes in the drum.
Examples of Centrifugal Force
1) A centrifugal governor or Watt’s governor controls the speed of an engine by using masses that respond to centrifugal force provided by the engine. If the engine increases the speed, the masses move and trigger a cut in the throttle.
2) The weight of an object on the surface of the earth increases as it is moved from the equator to the poles. This can be explained by taking the earth to be a non-inertial frame because of its spinning motion. In such a case we have to consider centrifugal force on the bodies. At the equator, the centrifugal force is maximum as the radius of the circle is equal to the radius of the earth. As we move towards the poles the radius of the circle decreases and the centrifugal force decreases. When the object is at the equator the centrifugal force which is maximum is in the opposite direction of gravitational force due to the earth on the body. So, the weight of the object is minimum at the equator. It is maximum at the poles. (The weight of the object decreases further at equator due to the bulging of the earth.)
3) A centrifuge is a device that puts a substance in rotation about a fixed axis for the centrifugal force to separate lighter and heavier substances. A centrifuge is used to separate cream from milk, butter from curd or substances of different masses.
4) When a coin is placed on a rotating gramophone disc or on a rotating circular turntable it remains at rest when the force of static friction balances the centrifugal force in the rotating frame. The weight (\(mg\)) of the coin is equal to the normal reaction (\(N\)). In the limiting condition,
\(N = mg\)
\(f_s = mr\omega^2\)
\[ \mu_s = \frac{f_s}{N} = \frac{mr\omega^2}{mg} = \frac{r\omega^2}{g} \]
Coefficient of static friction:
\[ \boxed{\mu_s = \frac{r\omega^2}{g}} \]
Applications of Centrifugal Effect
The centrifugal effect associated with rotating systems is useful in many practical devices and processes.
Centrifuges: Centrifuges use rapid rotation to separate substances of different densities.
Spin dryers: Washing machines use rapid rotation to remove water from clothes.
Centrifugal separators: Rotating devices can be used to separate particles or substances according to their density.
Centrifugal pumps: Rotating impellers transfer energy to fluids and are widely used for moving liquids.
Physical Characteristics
| Physical quantity | Centrifugal force |
|---|---|
| Symbol | Fcf |
| SI unit | newton (N) |
| Dimensional formula | [M L T−2] |
| Nature of quantity | Vector quantity |
| Direction | Away from the axis of rotation in the rotating frame |
Important Formulae
Key Points
1. Centrifugal force appears to act away from the centre or axis of rotation.
2. It is a pseudo force observed when motion is described from a rotating reference frame.
3. Its magnitude is equal to the magnitude of the centripetal force for the corresponding circular motion.
4. Centripetal force and centrifugal force act in opposite directions in the rotating-frame description.
5. Centripetal force is directed inward, whereas centrifugal force appears outward.
Dynamics of Non-Uniform Circular Motion
When a particle moves in a circular path with varying speed, its motion is called non-uniform circular motion. In such motion, the particle has both centripetal (radial) acceleration and tangential acceleration. Therefore, the net force acting on the particle also has both radial and tangential components.
The radial component of acceleration continuously changes the direction of velocity, while the tangential component changes the magnitude of velocity, that is, the speed of the particle.
Radial and Tangential Acceleration
Since the particle is moving along a circular path with varying speed, its total acceleration has two mutually perpendicular components:
\( \vec{a} = \vec{a}_c + \vec{a}_t \)
Here, \( \vec{a}_c \) is the centripetal or radial acceleration directed towards the centre of the circular path, and \( \vec{a}_t \) is the tangential acceleration directed along the tangent to the circular path.
Centripetal or Radial Force
The centripetal force \( \vec{F}_c \) is directed towards the centre of the circle and is responsible for producing the centripetal acceleration.
\( F_c = ma_c = \frac{mv^2}{r} = mr\omega^2 \)
Thus, the magnitude of the centripetal force depends on the mass of the particle, its speed, and the radius of the circular path.
Tangential Force
The tangential force \( \vec{F}_t \) acts along the tangent to the circular path and is responsible for changing the speed of the particle.
\( F_t = ma_t = m\frac{dv}{dt} = mr\alpha \)
Here, \( \alpha \) is the angular acceleration of the particle. The tangential force changes the magnitude of velocity, whereas the centripetal force changes its direction.
Net Force in Non-Uniform Circular Motion
Since the radial and tangential components of acceleration are perpendicular to each other, the total force acting on the particle is the vector sum of the centripetal force and the tangential force.
\( \vec{F} = \vec{F}_c + \vec{F}_t \)
The magnitude of the net force is therefore obtained using the Pythagorean theorem:
\( F = \sqrt{F_c^2 + F_t^2} \)
\( F = \sqrt{\left(\frac{mv^2}{r}\right)^2 + (mr\alpha)^2} \)
The diagram shows that the centripetal force \( \vec{F}_c \) acts towards the centre of the circle, while the tangential force \( \vec{F}_t \) acts along the tangent. Their vector sum gives the net force \( \vec{F} \).
Direction of the Net Force
The direction of the net force \( \vec{F} \) makes an angle \( \phi \) with the radial component \( \vec{F}_c \).
\( \tan\phi = \frac{F_t}{F_c} \)
Substituting the expressions for \( F_t \) and \( F_c \),
\( \tan\phi = \frac{mr\alpha}{\frac{mv^2}{r}} \)
\( \tan\phi = \frac{r^2\alpha}{v^2} \)
Important Points
1. The centripetal force is directed towards the centre of the circular path.
2. The tangential force acts along the tangent to the circular path.
3. The centripetal force changes the direction of velocity.
4. The tangential force changes the magnitude of velocity, that is, the speed.
5. In non-uniform circular motion, both the magnitude and direction of the velocity may change.
6. The total force acting on the particle is the vector sum of the radial and tangential components.
7. The net force is generally neither purely radial nor purely tangential. Its direction depends on the relative magnitudes of \( F_c \) and \( F_t \).
Special Case: Uniform Circular Motion
If the speed of the particle remains constant, then \( \frac{dv}{dt}=0 \). Therefore, the tangential acceleration and tangential force become zero.
\( a_t = 0 \)
\( F_t = 0 \)
Hence, the net force is purely centripetal and is directed towards the centre of the circle.
Examples of Non-Uniform Circular Motion
1. A car moving around a circular road while accelerating or braking has both centripetal and tangential acceleration.
2. A stone attached to a string and rotated in a vertical circle with changing speed undergoes non-uniform circular motion.
3. A particle moving around a circular path while its angular speed changes continuously has both radial and tangential acceleration.
4. A rotating object whose speed is increased or decreased while it moves along a circular path experiences non-uniform circular motion.