Hello dear we know that the particle performing rotational motion can be discussed completely with angular velocity, angular acceleration etc. The motion of particle along circumference of circle is termed as circular motion. The displacement of particle along circumference of circle is considered in terms of angle and radius vector.
Angular displacement
Angle traced by radius vector with the radius vector is known as angular displacement, denoted as θ and has SI unit radian.
If particle completes one revolution in circular motion then,
Angular displacement = θ = 3600 = 2π rad
θ(in rad) = ArcLength/radius = s/r
Instantaneous angular displacement is vector quantity.
Its direction is given by right hand rule and the magnitude in vector from is given as,
Angular velocity
The rate of change of the angular displacement with time is called angular velocity.
S.I. unit is rad/s. Dimensions are [M0L0T-1θ1]
It is vector quantity. Its direction is determined by right hand rule.
Let’s find the relation between linear velocity and angular velocity……………!
Consider a particle performing U.C.M along radius r in anticlockwise of circle of radius ‘r’ in an anticlockwise direction.
Suppose that particle moves from the point A to B. i.e. distance δs in a short time δt. This distance is straight line. The magnitude of instantaneous velocity is given by,
The instantaneous velocity of particle at any instant then can be written as,
V = r (2π/T) = 2πr/T
Also 1/T = n (frequency)
V = 2πrn
Some important terms to be noted…..!
Period: The time taken by a particle performing U.C.M. to complete one revolution is called its period of revolution.
v = distance covered/time = circumference/period
∴ v = 2πr/T
∴ T = 2πr/v
∴ T = 2π/ω
S.I. unit of period is Second (s)
Frequency: The number of revolution per unit time by a particle performing U.C.M. is called frequency.
In one period particle competes one revolution hence, in unit time (1/T) revolutions i.e. frequency is reciprocal of period.
n = 1/T S.I. unit is hertz (Hz).
1 Hz = 1 revolution per second
1 rpm = 1/(60 ) rps = 1/(60 ) Hz.
Points to be remembered……!
- Period of second hand is 60 s.
- Period of minute hand is 1 hour = 3600 s.
- Period of hour hand = 12 X 3600 s.
- Period of earth about own axis = 20 X 3600s.
- Period of earth round the sun T = 365 X 24 X 3600 s.
- Period of moon round the earth = 27.3 days
- Angular speed of second hand= 0.1047 rad/s.
- Angular speed of minute hand= 1.744 X103 rad/s
- Angular speed of hour hand = 1.453 X 104 rad/s.