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#### 1.1 Definition of Angular Displacement:

- The angular velocity is defined as the angle described by radius vector in a given time at the centre of the circle.

#### 1.2 Definition of Angular Velocity:

- The angular velocity of a particle performing circular motion is defined as the time rate of change of limiting angular displacement.

#### 1.3 Definition of Angular Acceleration:

- The average angular acceleration is defined as the time rate of change of angular velocity.

#### 2.1 Right-Hand Thumb Rule to Find the Direction of Definition of Angular Displacement:

- Imagine the axis of rotation to be held in the right hand with the fingers curled around it and the thumb outstretched. If the curled fingers give the direction of motion of a particle performing the circular motion, then the direction of outstretched thumb gives the direction of the angular displacement vector.

#### 2.2 Right-Hand Screw Rule to Find the Direction of Definition of Angular Displacement:

- When a right-handed screw is rotated in the sense of revolution of the particle, then the direction of the advance of the screw gives the direction of the angular displacement vector.

**3. 1 Proof of With All Symbols With Their Usual Meanings:**

- For smaller magnitudes angular displacement, angular velocity are vector quantities. Let ( r) be the position vector of the particle at some instant. Let the angular displacement in small time δt be (dq). Let the corresponding linear displacement (arc length) be ( ds). By geometry

Dividing both sides of the equation by δt and taking the limit

#### 4.1 Definition of Uniform Circular Motion:

- Uniform circular motion (U.C.M.) is defined as the motion of the particle along the circumference of a circle with constant speed OR Uniform circular motion (U.C.M.) is defined as the periodic motion of a particle moving along the circumference of a circle with constant angular speed.
- Examples: The motion of the earth around the sun. the motion of the moon around the earth.

#### 4.2 To show that the linear speed of a particle performing circular motion is the product of radius of the circle and angular speed of the particle.

- Consider a particle performing uniform circular motion, along the circumference of the circle of radius ‘r’ with constant linear velocity ‘v’ and constant angular speed ‘ω’ moving in the anticlockwise sense as shown in the figure.
- Suppose the particle moves from point P to point Q through a distance ‘δx’along the circumference of the circular path and subtends the angle ‘δθ’ at the centre O of the circle in a small interval of time ‘δt’. By geometry

δx = r . δθ

- If the time interval is very very small then arc PQ can be considered to be almost a straight line. Therefore magnitude of linear velocity is given by

- Thus the linear speed of a particle performing circular motion is the product of radius of the circle and angular speed of the particle.

#### 5.1 Definition of Period of U.C.M.:

- The time taken by a particle performing uniform circular motion (U.C.M.) to complete one revolution is called a periodic time or period. OR the time taken by a particle performing uniform circular motion (U.C.M.)to travel the distance equal to the circumference of a circle is called periodic time or period.
- It is denoted by ‘T’. The S. I. Unit of the period is second (s). Its dimensions are[MºLºT
^{1}].

#### 5.2 Definition of Frequency of U.C.M.:

- The frequency of revolution is defined as the number of revolutions performed by particle performing the uniform circular motion.
- The frequency is denoted by letter ‘n’ or ‘f’. The S. I. Unit of frequency is hertz (Hz). Its dimensions are [MºLºT
^{-1}].

#### 6.1 Relation Between Magnitude of Linear Acceleration and Angular Acceleration:

- When a body is performing a non-uniform circular motion, its angular velocity changes. Hence the body possesses angular acceleration.

The rate of change of angular velocity w.r.t. time is called the angular acceleration.

- We know that acceleration is the rate of change of velocity with respect to time.

r = radius of circular path = constant.

ω = angular velocity of the particle performing a circular motion

Where ‘α’ is angular acceleration.

Hence, linear acceleration = radius x angular acceleration.

#### 7.1 The Expression for Acceleration for Particle Performing U.C.M.:

- The magnitude of the velocity of a particle performing uniform circular motion is constant but its direction changes constantly in the direction. Hence the particle in circular motion has linear acceleration.
- Let us consider a particle performing uniform circular motion with a linear velocity of magnitude v and angular velocity of magnitude ω along a circle of radius r with centre O in an anticlockwise sense (moving from initial position A to final position B)as shown in the figure.

AP = velocity of the particle at position A

BQ = velocity of the particle at position B

**Construction:**Draw BR || AP and BR =| AP and join RQ

Consider velocity triangle QBR. By triangle law of vector addition

Now by definition,

The triangles AOB and RBQ are similar. Hence ∠ QBR = δθ

For smaller angular displacement δθ,

Substituting in equation (1), we get

This is the expression for acceleration of particle performing the uniform circular motion.

- This acceleration is directed towards the centre of circular path along the radius. This acceleration is called as radial acceleration or centripetal acceleration. In vector form, centripetal acceleration can be given as

- The negative sign indicates that centripetal acceleration is oppositely directed to that of radius vector i.e. directed towards the centre of the circle along the radius.

#### 8.1 Definition of Centripetal Force:

- Centripetal force is a force which is acting on a body performing circular motion and is acting along the radius of the circular path and directed towards the centre of the circle.

#### 8.2 Examples of Centripetal Force:

- When a stone tied to one end of a string is whirled horizontally, there is an inward force exerted by the string on the stone called tension. This force provides necessary centripetal force for circular motion.
- The moon or a Satellite revolves around the earth in circular orbit. Necessary centripetal force is provided by the gravitational force of attraction between satellite and earth.
- When a vehicle moves around a horizontal circular road, the centripetal force for circular motion is provided by the frictional force between the road and the wheels.
- In an atom, an electron moves around the nucleus in an orbit. The centripetal force required for the motion of the electron is provided by the electrostatic force of attraction between the negatively charged electron and positively protons.

#### 9.1 Definition of Centrifugal Force:

- The imaginary (pseudo) force which acts on the particle performing a circular motion in the direction away from the centre along the radius of the circular path having the same magnitude as that of centripetal force is called as centrifugal force.

#### 9.2 Examples of Centrifugal Force:

- When moving car along a horizontal curved road takes a turn, persons in the car experience a force in an outward direction. This force is centrifugal force.
- When the horizontal merry go round rotates about the vertical axis the chairs are pulled out due to centrifugal force.
- When a stone is whirled in a circle, we feel that stone is pulling our hand because of centrifugal force.
- The earth is flattened at the poles and bulged at the equator because the centrifugal force acting on the particles on the equator is maximum.
- The drier of washing machine acts on the principle of centrifugal force. Water particles from wet clothes are thrown outward due to centrifugal force acting on them. Drier in washing machine consists of a cylindrical vessel with perforated walls. As the cylindrical vessel is rotated fast, centrifugal force acts on the water particles of wet clothes. Under the action of this centrifugal force, water particles are forced out of the perforations, thereby drying of the clothes.
- A coin kept slightly away from the centre of rotating gramophone disc slips off towards the edge of the disc at a particular speed. This is due to centrifugal force acting on the coin.

#### 10.1 Distinguishing Between Centripetal Force and Centrifugal Force:

Centripetal Force |
Centrifugal Force |

Centripetal force is a force which is acting on a body performing circular motion and is acting along the radius of the circular path and directed towards the centre of the circle. | The imaginary (pseudo) force which acts on the particle performing a circular motion in the direction away from the centre along the radius of the circular path having the same magnitude as that of centripetal force is called as centrifugal force. |

It is a real force. | It is an imaginary force or a pseudo force. |

It arises in an inertial frame of reference. | It is experienced in non – inertial frame of reference. |

It is always directed towards the centre of the circular path. | It is always directed away from the centre of the circle along the radius. |

Without it, the circular motion is not possible. | Centrifugal force doesn’t have an independent existence. |

Example: The moon or a Satellite revolves around the earth in circular orbit. Necessary centripetal force is provided by the gravitational force of attraction between satellite and earth. | Example: When moving car takes a turn along a horizontal curved road, persons in the car experience a force in the outward direction. This force is centrifugal force |

#### 11.1 The Expression for Maximum Safe Speed Along Curved Horizontal Road and its Significance:

- The necessary centripetal force required to negotiate a turn by a vehicle moving along a horizontal unbanked curved road is provided by the force of friction between the wheels (tyres) and the surface of the road.
- Let us consider a vehicle of a mass ‘m’ is moving along a horizontal curved road of radius ‘r’ with speed ‘v’.

Let μ be the coefficient of friction between the road surface and the wheels then

- This expression gives the maximum speed with which a vehicle can be moved safely along a horizontal curved road. If speed is more than this velocity, then there is a danger that the vehicle will get thrown (skid) off the road.

#### 12.1 Definition of Banking of Road:

- The process of raising outer edge of a road over the inner edge through certain angle is known as banking of road.

#### 12.2 Definition of Angle of Banking

- The angle made by the surface of a road with the horizontal surface of the road is called angle of banking.

#### 12.3 The Necessity of Banking of Road:

- As the speed of vehicle increases, the centripetal force (friction between the road and tyres) needed for the circular motion of vehicle also increases. But there is a maximum limit for frictional force, which depends on the coefficient of friction between the wheels and road.
- For this, we may increase the force of friction making the road rough. However, this results in the wear and tear of the tyres of the vehicle.
- The force of friction is not always reliable because it changes when roads are oily or wet due to rains etc.
- Due to banking of the road, the necessary centripetal force is provided by the component of the normal reaction.

#### 13.1 and 14.1 The Expression for Safe Velocity of Vehicle on Curved Banked Road and To show that the angle of banking is independent of the mass of a vehicle:

- Consider a vehicle of mass ‘m’ is moving with speed ‘v’ on a banked road of radius ‘r’ as shown in the figure. Let ‘θ’ be the angle of banking. The weight “mg” of the vehicle acts vertically downwards through its centre of gravity G, and N is the normal reaction exerted on the vehicle by banked road AC. N is perpendicular to road AC. Let ‘f’ be the frictional force between the road and the tyres of the vehicle.
- Now, the normal reaction is resolved into two components (N cosθ) along the vertical (acting vertically upward) and (N sinθ) along the horizontal (towards left) as shown in the figure. Similarly, the frictional force ‘f’ can be resolved into two components (f sinθ) along the vertical (acting vertically downward) and( f cosθ) along the horizontal (towards left) as shown in the figure.

The free body diagram of a car is as follows

Considering equilibrium

Total upward force = Total downward force

∴ N cosθ = mg + f sinθ

∴ mg = N cosθ – f sinθ ….. (1)

The horizontal components N sinθ and f cosθ provides necessary centripetal force

mv^{2}/r = N sinθ + f cosθ …… (2)

Dividing equation (2) by (1)

Now, frictional force f = μ_{s}N

Where μ_{s} = coefficient of friction between road and tyres

This is an expression for the velocity of a vehicle on a curved banked road.

When the frictional force between the road and tyres of the vehicle is negligible μ_{s} = 0.

This is an expression for the angle of banking of a road.

When friction is not mentioned in the problem, use this expression to solve the problems.

- The expression for safe velocity on the banked road is

- The expression of the angle of banking does not contain the term m representing mass, thus the angle of banking is independent of mass ‘m’ of the vehicle. Thus the angle of banking is the same for heavy and light vehicles.

#### 15.1 Definition of Period of Conical Pendulum:

- The time taken by the bob of a conical pendulum to complete one horizontal circle is called time period of the conical pendulum

#### 15.2 The Expression for Time Period of Conical Pendulum:

But v = rω

Where ω is angular speed and T is the period of the pendulum.

From figure tan θ = r/h

Substituting in equation (4)

This is an expression for the time period of a conical pendulum.

#### 16.1 Definition of Conical Pendulum:

- A conical pendulum is a simple pendulum, which is given such a motion that bob describes a horizontal circle and the string describes the cone.

#### 16.2 The Expression for Angle Made by String of Conical Pendulum with Vertical and the Velocity of the Bob of Pendulum:

- Let us consider a conical pendulum consists of a bob of mass ‘m’ revolving in a horizontal circle with constant speed ‘v’ at the end of a string of length ‘l’. Let the string makes a constant angle ‘θ’ with the vertical. let ‘h’ be the depth of the bob below the support.

- The tension ‘F’ in the string can be resolved into two components. Horizontal ‘Fsin θ’ and vertical ‘Fcos θ’.

The vertical component (F cos θ) balances the weight mg of the vehicle.

F cosθ = mg ………….. (1)

The horizontal component (F sin θ) provides the necessary centripetal force.

F sin θ = mv^{2}/r ………… (2)

Dividing equation (2) by (1) we get,

The angle made by the string of conical pendulum with vertical is θ = tan^{-1}(v^{2}/rg)

The velocity of the bob of a conical pendulum is v = √r g tanθ

#### 17.1 The Expression for Velocity of Particle Moving in a Vertical at Different Positions:

- Consider a small body of mass ‘m’ attached to one end of a string and whirled in a vertical circle of radius ‘r’. In this case, the acceleration of the body increases as it goes down the vertical circle and decreases when goes up the vertical circle. Hence the speed of the body changes continuously. It is maximum at the bottommost position and minimum at the uppermost position of the vertical circle. Hence the motion of the body is not uniform circular motion. Irrespective of the position of the particle on the circle, the weight ‘mg’ always acts vertically downward.

- Let ‘v’ be the velocity of the body at any point P on the vertical circle. Let L be the lowest point of the vertical circle. Let ‘h’ be the height of point P above point L. let ‘u’ be the velocity of the body at L. By the law of conservation of energy

Energy at point P = Energy at point L

This is an expression for the velocity of a particle at any point performing a circular motion in a vertical circle.

Consider the centripetal force at point P

Substituting in equation (2)

This is the expression for the tension in the string.

Velocities of Body at Different Positions When Looping a Loop in a Vertical circle

Lowest Point L (h = 0):

Highest Point H (h = 2r):

When String is Horizontal (h = r):

#### 18.1 Definition of Vertical Circular Motion:

- A body revolving in a vertical circle such that its motion at different points is different then the motion of the body is said to be vertical circular motion.

#### 18.2 Vertical Circular Motion is Not Uniform Circular Motion:

- When studying the motion of a body in a vertical circle we have to consider the effect of gravity.
- Due to the influence of the earth’s gravitational field, the magnitudes of the velocity of the body and tension in the string change continuously. It is maximum at the lowest point and minimum at the highest point.
- Hence the motion of the body in a vertical circle is not uniform circular motion.

#### 19.1 Difference Between Uniform Circular Motion and Non-Uniform Circular Motion:

Uniform Circular Motion |
Non-Uniform Circular Motion |

The magnitude of the velocity (speed) of the body is constant. | the magnitude of the velocity (speed) of the body changes continuously. |

The magnitude of the centripetal force acting on the body is constant | The magnitude of the centripetal force acting on the body changes continuously |

linear speed, angular speed, radial (centripetal) acceleration, kinetic energy, angular momentum and magnitude of the linear momentum of the body remain constant. Angular acceleration, tangential acceleration is zero. | linear speed, angular speed, radial (centripetal) acceleration, kinetic energy, angular momentum and magnitude of the linear momentum of the body do not remain constant. Angular acceleration, tangential acceleration is not zero. |

Example: The motion of the moon around the Earth. | Example: The motion of a body in a vertical circle. |

#### 20.1 The Difference in Tension at the Highest Point and the Lowest Point:

The tension in the string for a body moving in a vertical circle is given by

When the body is at the lowermost position i.e. body is at L) (h = 0)

When the body is at the uppermost position i.e. body is at H (h = 2r)

Thus the difference in tensions at the two positions

- Thus the tension in the string at the lowest point L is greater than the tension at the highest point H by six times the weight of the body.

#### 21.1 The Expression for Tension in a String for Body Moving in a Vertical Circle at Different Positions:

- Consider a small body of mass ‘m’ attached to one end of a string and whirled in a vertical circle of radius ‘r’. In this case, the acceleration of the body increases as it goes down the vertical circle and decreases when goes up the vertical circle. Hence the speed of the body changes continuously. It is maximum at the bottommost position and minimum at the uppermost position of the vertical circle. Hence the motion of the body is not uniform circular motion. Irrespective of the position of the particle on the circle, the weight ‘mg’ always acts vertically downward.

- Let ‘v’ be the velocity of the body at any point P on the vertical circle. Let L be the lowest point of the vertical circle. Let ‘h’ be the height of point P above point L. let ‘u’ be the velocity of the body at L. By the law of conservation of energy

Energy at point P = Energy at point L

This is an expression for the velocity of a particle at any point performing a circular motion in a vertical circle.

Consider the centripetal force at point P

Substituting in equation (2)

This is the expression for the tension in the string.

Case – I: When the body is at the lowermost position i.e. body is at L) (h = 0)

Case – II: When the body is at the uppermost position i.e. body is at H (h = 2r)

Case – III: When the string is horizontal (midway position), i.e. the body is at M (h = r)

#### 22.1 The Expression for Energy of a Body Moving in a Vertical Circle at Different Positions:

- The energy of the body has two components a) kinetic energy (E
_{K}) and b) potential energy (E_{P}). Sum of the two energies is total energy (E_{T})

At Lowest Point L

At Highest Point H

When String is Horizontal (Midway)

#### 23.1 Kinematical Equations for Circular Motion in Analogy With Linear Motion:

- The quantities linear displacement (s), linear velocity (v) and linear acceleration (a) of linear motion are analogous to angular displacement (θ), angular velocity (ω) and angular acceleration (α) of angular motion.
- The equations of motion for the linear motion are

v = u + at

s = ut + ½at^{2}

v^{2} = u^{2} + 2as

- By analogy, the equations of motion for the angular motion are

ω = ω_{o} + αt

θ = ω_{o}t + ½αt^{2}

ω^{2} = ω_{o}^{2} + 2αθ

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