For uniform circular motion Angular speed ω, Linear speed t, kinetic energy, Angular momentum (L) are constant. The angular acceleration α and the tangential acceleration a_T are zero.

Problems Based on Hands of Clock:

1. The second hand of a clock takes 60 seconds to complete one rotation. Its angular speed is 0.105 rad /s.
2. The minute hand of a clock takes 60 minutes = 60 x 60 seconds to complete one rotation. Its angular speed is 1.746 x 10^-3 rad /s.
3. The hour hand of a clock takes 12 hours = 12 x 60 x 60 seconds to complete one rotation. Its angular speed is 1.455 x 10^-4 rad /s.
4. The ratio of angular speeds of the second hand of a clock and the minute hand of a clock is 60:1.
5. The ratio of angular speeds of the minute hand of a clock and an hour hand of a clock is 12:1.
6. The ratio of angular speeds of the second hand of a clock and the hour hand of a clock is 720:1.

Example – 01:

Calculate the angular speed of the second hand, minute hand and hour hand of a clock.

Given: For second Hand T_S = 60 sec, For minute hand T_M = 60 min = 60 x 60 sec, For hour hand T_H = 12 hr = 12 x 60 x 60 sec.

To Find: Angular speed ω_S=? ω_M=? ω_H=?

Solution: 

The tips of the second hand, the minute hand, and the hour hand perform the uniform circular motion.

Ans: Angular speed of second hand = 0.105 rad/s,

Angular speed of minute hand =1.746 x 10^-3 rad/s,

Angular speed of hour hand =1.454 x 10^-4 rad/s.

Example – 02:

What is the angular velocity of the minute hand of a clock? What is the angular displacement of the minute hand in 20 minutes? If the minute hand is 5 cm long, what is the linear velocity of its tip?

Given: For minute hand  T_M = 60 min = 60 x 60 sec, t = 20 min = 20 x 60 sec, r = 5 cm = 5 x 10^-2 m

To Find: Angular speed ω_M=?, Angular displacement  θ =?, Linear velocity = v =?

Solution: 

The tip of the minute hand performs the uniform circular motion.

Ans: The angular speed of minute hand =1.746 x 10^-3 rad/s,

The angular displacement of minute hand in 20 minutes =2.095 rad,

The linear speed of the tip of minute hand = 8.73 x 10^-5 m/s.

Example – 03:

What is the angular displacement of the minute hand of a clock in 25 minutes?

Given: For minute hand  T_M = 60 min = 60 x 60 sec, t = 25 min = 25 x 60 sec,

To Find:  Angular displacement  θ =?

Solution: 

The tip of the minute hand performs the uniform circular motion.

Ans: Angular displacement of minute hand = 2.618 rad

Example – 04:

What is the angular velocity of the second hand of a clock? If the second hand is 10 cm long find the linear velocity of its tip.

Given: For second hand  T_s = 60  sec, r = 10 cm = 10 x 10^-2 m,

To Find:  Linear speed = v =?

Solution: 

The tip of the second hand performs the uniform circular motion.

Now v = r ω = 10 x 10^-2 x 0.105 = 1.05 x 10^-2 m/s = 1.05 cm/s

Ans: Angular velocity of second hand = 0.105 rad/s, Linear speed of tip of second hand = 1.05 cm/s

Example – 05:

The second hand of a watch is 1.5 cm long. Find the linear speed of a point on the second hand at a distance of 0.5 cm from the tip.

Given: For second hand  T_s = 60  sec, r = 1.5 cm – 0.5 cm = 1 cm = 1 x 10^-2 m,

To Find:  Linear speed = v =?

Solution:

The tip of the second hand performs the uniform circular motion.

Now v = r ω = 1 x 10^-2 x 0.105 = 1.05 x 10^-3 m/s = 1.05 mm/s

Ans: Linear speed of the point on the second hand = 1.05 mm/s

Example – 06:

The extremity of the hour hand of a clock moves 1/20th as fast as the minute hand. What is the length of the hour hand if the minute hand is 10 cm long?

Given: v_H = 1/20 v_M , r_M = 10 cm

To Find: r_H =?

Solution:

Ans: Length of hour hand = 6 cm

Example – 07:

The length of an hour hand of a wristwatch is 1.5 cm. Find the magnitudes of following w.r.t. tip of the hour hand a) angular velocity b) linear velocity c) angular acceleration d) radial acceleration e) tangential acceleration f) linear acceleration

Given: r = 1.5 cm = 1.5 x 10^-2 m, For hour hand T_H = 12 hr = 12 x 60 x 60 sec

To Find: Angular velocity = ω = ?, linear velocity = v = ?, angular acceleration = α = ?, radial acceleration = a_r = ?, tangential acceleration = a_T =?, linear acceleration a = ?

Solution:

Now v = r ω = 1.5 x 10^-2 x 1.454 x 10^-4 = 2.181 x 10^-6 m/s

Tip of hour hand performs uniform circular motion

∴  Angular acceleration = α = 0 and tangential acceleartion = a_T = 0

Radial acceleration is given by

a_r = v² /r = ( 2.181 x 10^-6 ) ² / (1.5 x 10^-2) = 3.171 x 10^-10 m/s²

a² = a_r ² + a_T ²

as a_T = 0, a = a_r = 3.171 x 10^-10 m/s²

Ans: Angular speed = 1.454x 10^-4 rad/s, Linear speed =2.181 x 10^-6 m/s,
Angular acceleration = 0, Radial acceleration = 3.171 x 10^-10 m/s².
Tangential acceleration = 0, Linear acceleartion = 3.171 x 10^-10 m/s².

Problems Based on Earth:

The earth takes 24 hours to complete one rotation about its axis. The angular speed of the earth of its rotation about its axis is 7.273 x 10^-5 rad /s.

Example – 08:

Calculate the angular velocity of the earth due to its spin motion.

Given: For the earth = T = 24 hr = 24 x 60 x 60 sec

To Find: Angular velocity = ω = ?

Solution:

Ans: Angular speed of the earth due to its spin motion is 7.273 x 10^-5 rad/s.

Problems Based on Rotating Discs:

Example – 09:

A turntable rotates at 100 rev/sec. Calculate its angular speed in rad/s and degrees/s.

Given: n = 100 r.p.s.

To Find: Angular speed =?

Solution:

Ans: Angular speed = 10.47 rad/s, Angular speed = 600 degrees/s

Example – 10:

Propeller blades of an aeroplane are 2 m long. When the propeller is rotating at 1800 rev/min, compute the tangential velocity of the tip of the blade. Also, find the tangential velocity at a point on the blade midway between tips and axis.

Given: r = 2 m, Angular speed = N = 1800 r.p.m.

To Find: v_Tip =? v_Mid =?

Solution:

Ans: The angular speed of the tip of blade = 376.8 rad/s, AThe angular speed of point midway = 188.4 rad/s

Example – 11:

A turntable has a constant angular speed of 45 r.p.m. Express this in rad per second and degrees per second. If the radius of the turntable is 0.5 m, what is the linear speed of a point on the rim?

Given: N = 45 r.p.m., r = 0.5 m

To Find: angular speed in rad/s and degrees/s, linear velocity = v = ?

Solution:

Ans: Angular speed = 2.355 rad/s = 270 degrees/s Linear speed on point on rim = 2.355 m/s.

Example – 12:

The linear velocity of a point on the rotating disc is 3 times greater than at a point on the at a distance of 8 cm from it. What is the diameter of the disc?

Given: v_T = 3 v_P, Let r_T = r, r_P = (r – 8) cm

To Find: Diameter of the disc =?

Solution:

Angular velocity for both the points is the same.
Ans: Diameter of disc = 24 cm.

Example – 13:

A disc has a diameter of one metre and rotates about an axis passing through its centre and at right angles to its plane at the rate of 120 rev/min. What is the angular and linear speed of a point on the rim and at a point halfway to the centre.

Given: d = 1m , r_T = r = 0.5 m, N = 120 r.p.m., for r_P = r/2 = 0.25 m

To Find: v_T = ?, V_P = ?

Solution:

Angular velocity for both the points is the same.

The linear speed of point on the rim

v_T =  r_Tω  = 0.5 x 12.56 = 6.28 m/s

v_P =  r_Pω  = 0.25 x 12.56 = 3.14 m/s

Ans: Angular speed of point on rim = 12.57 rad/s, Linear peed of point on rim = 6.28 m/s,
Angular speed of point on halfway =12.57m/s, Linear speed of point on halfway = 3.14 m/s.

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Angular Displacement:

For a particle performing a circular motion the angle, traced by the radius vector at the centre of the circular path in a given time is called the angular displacement of the particle at that time. It is denoted by ‘θ’. Its S.I. unit is radian (rad). It is a dimensionless quantity. [MºLºTº]

The direction of angular displacement: 

For smaller magnitude (infinitesimal) angular displacement is a vector quantity and its direction is given by the right-hand thumb rule.

Right-Hand Thumb Rule:

If we curl the fingers of our right hand and hold the axis of rotation with fingers pointing in the direction of motion then the outstretched thumb gives the direction of the angular displacement vector.

Sign Convention: 

An angular displacement in counter clock-wise direction is considered positive and that in the clockwise direction is considered as negative.

Vector relation between linear and angular displacement is

Characteristics of Angular Displacement:

• It is the angle, traced by the radius vector at the centre of the circular path in a given time is called the angular displacement of the particle at that time.
• For smaller magnitude (infinitesimal) angular displacement is a vector quantity and its direction is given by the right-hand thumb rule.
• Finite angular displacement is a vector quantity.
• Instantaneous angular velocity is a vector quantity.
• The direction of angular displacement in an anticlockwise sense is considered as positive, while the direction of angular displacement in a clockwise sense is considered as negative.
• The angular displacement of the particle performing uniform circular motion in equal time is equal.
• It is denoted by ‘θ’. Its S.I. unit is radian (rad). It is a dimensionless quantity. [MºLºTº].

Larger angular displacement is not treated as a vector quantity.

If a quantity has both the direction and magnitude then it seems to be vector quantity but it can only be treated as vector quantity if its satisfies laws of vector addition. Consider the following two cases of angular displacement

The commutative law of vector addition which states that if we add two vectors, the order in which we add them does not matter.

We can see that if the order is interchanged the final outcome is different. Thus the angular displacement fails to obey the law of vector addition. Hence larger angular displacement is not a vector quantity.

Angular Velocity:

The rate of change of angular displacement with respect to time is called the angular velocity of the particle. It is denoted by the letter ‘ω’. Its S.I. unit is radians per second (rad s^-1). Its dimensions are [MºLºT ^-1].

Mathematically,

For uniform circular motion, the magnitude of angular velocity is given by

Where ω = Angular speed, T = Period
N = Angular speed in r.p.m., n = Angular speed in r.ps. or Hz.
θ = Angular displacement, t = time taken

The Direction of Angular Velocity: 

For smaller magnitude (infinitesimal) the angular velocity is the vector quantity. Its direction is given by the right-hand thumb rule. It states that “If we curl the fingers of our right hand and hold the axis of rotation with fingers pointing in the direction of motion then the outstretched thumb gives the direction of the angular velocity vector”. Thus, the direction of angular velocity is the same as that of angular displacement.

By this rule, the direction of the angular velocity of the second hand, the minute hand, and the hour hand is perpendicular to the dial and directed inwards.

Angular Speed:

The angle traced by radius vector in unit time is called the angular speed or The magnitude of angular velocity is known an angular speed.

Uniform motion is that motion in which both the magnitude and direction of velocity remain constant. In UCM the magnitude of velocity is constant but its direction changes continuously. Hence UCM is not uniform motion. For uniform circular motion, the angular velocity is constant.

For uniform circular motion, the magnitude of velocity at P = magnitude of velocity at Q = magnitude of velocity at R and the direction of velocity at P ≠ direction of velocity at Q ≠ direction of velocity at R. In uniform circular motion a body moves in a circle describes equal angles in equal interval of time. Thus for a body performing UCM has uniform speed.

For non-uniform circular motion, The magnitude of velocity at P ≠ magnitude of velocity at Q ≠ magnitude of velocity at R and the direction of velocity at P ≠ direction of velocity at Q ≠ direction of velocity at R.  In non-uniform circular motion a body moves in a circle describes unequal angles in equal interval of time.

Characteristics of Angular Velocity:

• The rate of change of angular displacement with respect to time is called the angular velocity of the particle.
• Its direction is given by the right-hand thumb rule.
• The direction of angular velocity is the same as that of angular displacement.
• For uniform circular motion, the magnitude of angular velocity is constant.
• The magnitude of angular velocity (ω) is related to the magnitude of linear velocity (v) by the relation v = rω.
• It is denoted by the letter ‘ω’. Its S.I. unit is radians per second (rad s-1). Its dimensions are [MºLºT ^-1].

Example – 1:

The graph shows angular positions of a rotating disc at different instants. What is the sign of angular displacement and angular acceleration?

The angular velocity at any instant is given by ω = dθ/dt,

At t = 1 second the graph is rising up, thus the slope (dθ/dt) of the tangent at t = 1 second is positive. Hence angular velocity is positive.

At t = 2 seconds the graph reaches the topmost point, thus the slope (dθ/dt) of the tangent at t = 2 seconds is zero. Hence angular velocity is zero.

At t = 3 seconds the graph is going down, thus the slope (dθ/dt) of the tangent at t = 3 seconds is negative. Hence angular velocity is negative.

We can see the change in angular velocity as positive → zero → negative. Thus angular velocity is decreasing. Hence angular acceleration is negative.

Angular Acceleration: 

The average angular acceleration is defined as the time rate of change of angular velocity. It is denoted by the letter ‘α’. Its S.I. unit is radians per second square (rad /s²). Its dimensions are [MºLºT ^-2]. Mathematically,

If the initial angular velocity of the particle changes from initial angular velocity ω₁  to final ω₂ angular velocity in time ‘t’ then

The direction of Angular Acceleration:

The direction of angular acceleration is given by right-hand thumb rule. If the angular velocity is increasing then the angular acceleration has the same direction as that of the angular velocity.

If the angular velocity is decreasing then the angular acceleration has the opposite direction as that of the angular velocity.

For uniform circular motion angular acceleration is zero.

Characteristics of Angular Acceleration:

• The average angular acceleration is defined as the time rate of change of angular velocity.
• If the angular velocity is increasing angular acceleration is positive (e.g. the angular acceleration of the tip of a fan just switched on). If the angular velocity is decreasing angular acceleration is negative (e.g. the angular acceleration of the tip of a fan just switched off)
• If the angular velocity is increasing then the angular acceleration has the same direction as that of the angular velocity. If the angular velocity is decreasing then the angular acceleration has the opposite direction as that of the angular velocity.
• For uniform circular motion angular acceleration is zero.
• The magnitude of angular acceleration (α) is related to the magnitude of linear acceleration (a) by the relation a = rα.
• It is denoted by the letter ‘α’. Its S.I. unit is radians per second square (rad /s²). Its dimensions are [M⁰L⁰T^-2].

Right Handed Screw Rule:

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.

Relation Between Linear Velocity and Angular Velocity:

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 the magnitude of linear velocity is given by

Thus the linear velocity of a particle performing uniform circular motion is radius times its angular velocity. In vector form above equation can be written as

The linear velocity can be expressed as the vector product of angular velocity and radius vector.

The following figure shows relative positions of the linear velocity vector, angular velocity vector, and radius or position vector.

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 ( δθ). Let the corresponding linear displacement (arc length) be ( δs). By geometry

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

Period of Revolution:

Let us consider particle performing a uniform circular motion. Let ‘T’ be its period of revolution. During the periodic time (T), particle covers a distance equal to the circumference 2pr of the circle with linear velocity v.

This is an expression for the period of revolution for particle performing the uniform circular motion.

The Expression for 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 as 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.

If speed is increasing linear acceleration is in the same direction as that of linear velocity. If speed is decreasing linear acceleration is in the opposite direction to that of linear velocity. It is also referred as tangential acceleration. For uniform circular motion α = 0.

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Concept of Circular Motion:

The motion of a particle along the circumference of a circle is called circular motion. It is a translational motion along a curved path.

Examples:

• The motion of the earth around the sun.
• The motion of a satellite around the planet.
• The motion of an electron around the nucleus.
• The motion of a tip of a blade of a fan. (Note it is the motion of the tip of a blade of the fan and not the motion of the fan. The motion of fan is a rotational motion)

Characteristics of Circular Motion:

• In a circular motion, the particle moves along the circumference of a circle.
• It is a translational motion along a curved path.
• The magnitude of the radius vector or position vector is constant and equals to the radius of the circular path.
• The direction of the radius vector or position vector changes continuously.
• If the magnitude of the velocity (speed) of the particle performing constant, the particle is said to perform the uniform circular motion.
• If the magnitude of the velocity (speed) of the particle performing changes continuously, the particle is said to perform the non- uniform circular motion.

Radius Vector:

A vector drawn from the centre of a circular path to the position of the particle at any instant is called a radius vector at that instant. It is also called as a position vector. In the figure at position P,  r or OP is a position vector. The magnitude of the position vector is equal to the radius of the circular path. Hence for a circular motion, the magnitude of the radius vector is constant but its direction changes continuously.

Characteristics of Radius or Position Vector:

• It is always directed along the radius of the circular path.
• Its direction is from the centre of the circular path to the position of the particle at that instant i.e. is directed radially outward.
• In a circular motion, the magnitude of the radius vector or position vector is constant and equals to the radius of the circular path.
• The direction of the radius vector or position vector in circular motion changes continuously.
• Its direction is opposite to that of centripetal acceleration and the centripetal force.
• It is denoted by r. Its dimensions are [MºL¹Tº]. Its S.I. unit is metre (m) and c.g.s. unit is centimetre (cm).

Instantaneous Velocity (v):

A linear velocity of a particle performing a circular motion, which is directed along the tangent to the circular path at a given point on the circular path at that instant is called instantaneous velocity. It is also called as tangential velocity.

For uniform circular motion the magnitude of instantaneous velocity is always constant but direction changes continuously. For non-uniform circular motion, the magnitude and direction of the instantaneous velocity change continuously.

The tangential velocity is directed perpendicular to the direction of the radius vector.

If a stone is tied to one end of a string and whirled in a horizontal circle at the other en, necessary centripetal force is provided by the tension in the string. If the speed of rotation is increased gradually, the tension in the string increases, a stage is reached when the tension in string becomes larger than the breaking tension of the string, the string breaks and the stone flies off tangentially.

Characteristics of Instantaneous Velocity or Tangential Velocity:

• It is always directed along the tangent to the circular path at a given point on the circular path at that instant.
• It is always perpendicular to the direction of the radius vector at the point represented by the radius vector on the circular path.
• For uniform circular motion the magnitude of instantaneous velocity is always constant but direction changes continuously.
• For non-uniform circular motion, the magnitude and direction of the instantaneous velocity change continuously.
• It is denoted by v. Its dimensions are  [MºL¹T^-1]. Its S.I. unit is metre per second  (m s^-1) and c.g.s. unit is centimetre per second (cm s^-1).

Axis of Rotation:

The normal drawn to the plane of the circular path through the centre of the circular path is called the axis of rotation.

Note: In rotational motion, the particles on the axis of rotation are stationary, while all other particles perform circular motion about the axis of rotation.

Uniform Circular Motion:

The motion of a particle along the circumference of a circle with a constant speed is called uniform circular motion (U.C.M.).

Examples: The motion of the earth around the sun, The motion of an electron around the nucleus.

Characteristics of Uniform Circular Motion:

• The magnitude of the velocity (speed) of the particle performing U.C.M. is constant.
• The magnitude of the instantaneous velocity of the particle performing U.C.M.  remains constant but its direction changes continuously. Hence U.C.M. is accelerated motion.
• It is a periodic motion with a definite period and frequency.
• in U.C.M., the magnitude of the centripetal force acting on the body is constant
• in U.C.M., the linear speed, angular speed, radial (centripetal) acceleration, kinetic energy, angular momentum and magnitude of the linear momentum of the body remain constant.
• in U.C.M., the angular acceleration, tangential acceleration is zero.

The Period of Revolution:

The time taken by a particle performing uniform circular motion to complete one revolution is called the period of revolution or periodic time or simply period (T).

It is denoted by ‘T’. The S. I. Unit of the period is second (s). Its dimensions are [MºLºT ¹].

Characteristics of Time Period of U.C.M.:

• The time taken by a particle performing uniform circular motion to complete one revolution is called the period of revolution or periodic time or simply period (T).
• For U.C.M. it is constant.
• It is denoted by ‘T’. The S. I. Unit of the period is second (s). Its dimensions are [MºLºT ¹].

The Frequency of Revolution:

The number of revolutions by the particle performing uniform circular motion in unit time is called as frequency (n) of revolution.

The frequency is denoted by letter ‘n’ or ‘f’. The S. I. Unit of frequency is hertz (Hz). Its dimensions are [MºLºT^-1].

In time T the particles complete one revolution. Thus the particle completes 1/T revolutions in unit time. Thus n = 1/T.

Characteristics of Frequency of U.C.M.:

• The number of revolutions by the particle performing uniform circular motion in unit time is called as frequency (n) of revolution.
• For U.C.M. it is constant.
• The frequency is denoted by letter ‘n’ or ‘f’. The S. I. Unit of frequency is hertz (Hz). Its dimensions are [MºLºT-1].
• In time T the particles complete one revolution. Thus the particle completes 1/T revolutions in unit time. Thus n = 1/T.

Next Topic: angular Displacement, Angular Velocity, and Angular Acceleration

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