The quantities of circular motion include angular displacement, angular velocity, and angular acceleration.
1. Angular displacement (θ)
Displacement in circular motion is called angular displacement. Angular displacement including vector quantities, therefore, has magnitude and directions. The direction of angular displacement is usually expressed in a clockwise direction (clockwise or counterclockwise).
There are three units of angular displacement. First, degree (o). One circumference of the circle is equal to 360o. Second, revolution. One circumference of the circle is equal to one revolution. Third, radian. Observe the figure below. If an object moves in a circle then r = the radius of the circle, x = the length of the circular path that the object passes = circumference of the circle.
One circumference of the circle is equal to 2π radians.
Sample problem 1:
1 revolution = 360o . ½ revolution = …. rad?
1 revolution = 360o = 2 π rad = 2(3.14) rad = 6.28 rad
1⁄2 revolution = 180o = 1⁄2 (6.28 rad) = 3.14 rad
Sample problem 2 :
1 rad = …… o ? 1o = … rad ?
180o = π rad = 3.14 rad
2. Angular Speed (ω)
a. Average angular Speed
Sample problem 3:
A wheel rotates clockwise, rotates an angle of 180o for 2 seconds and 90o for 1 second. What is the magnitude and direction of the average angular velocity?
The direction of the average angular velocity = clockwise direction.
9 o/s = … rad/s ?
Sample problem 4:
A wheel rotates clockwise, rotates at an angle of 360o for 4 seconds. The wheel then rotates counterclockwise, rotates at an angle of 180o for 2 seconds. What is the average angular velocity?
The direction of the average angular velocity = clockwise direction.
3 o/s = … rad/s?
b. Instantaneous angular speed
The instantaneous angular velocity is often called the angular velocity. If only the angular velocity is mentioned then what is meant is instantaneous angular velocity. The magnitude of the instantaneous angular velocity = magnitude of the angular velocity during a very short time interval. Mathematically:
If in the linear motion, we can replace the magnitude of the velocity with speed so that in the circular motion we can replace the magnitude of the angular velocity with the angular speed.
3. Angular acceleration (α)
a. Average angular acceleration
Sample problem 5 :
A windmill was initially at rest, blown by the wind so it turned clockwise. After 2 seconds, the angular velocity becomes 90 o/s. What is the average angular speed?
On average, the angular speed of the windmill changes 45 o/second every 1 second = … rad / s every 1 second?
b. Instantaneous angular acceleration
Instantaneous angular acceleration is often abbreviated as angular acceleration. The instantaneous angular acceleration is a change in the angular velocity during a very short time interval. Mathematically:
Linear magnitudes in the circular motion
Linear quantities are quantities in linear motion, such as displacement (distance), velocity (speed), and acceleration. On the other hand, the quantities of circular motion can be referred to as angular quantities.
Observe an object that is spinning, like a wheel or fan or windmill or clock, etc. When an object like a fan rotates, all parts of the fan rotate together. If the fan takes one rotation (360o) then all parts of the fan, both located on the edge and near the axis also take one revolution or 360o. One revolution or 360o is a magnitude of the angular displacement done by all parts of the fan, both on the edge and near the axis.
When the fan takes one revolution, the fan part that is on the edge and the fan part near the axis of rotation moves as far as one circle (2). If the radius of the fan is 20 cm, the distance between the edge of the fan and the axis of rotation is 20 cm. For example the distance between the axis and one part of the fan that is near the axis = 1 cm. When the fan does one revolution, the edge of the fan moves circularly as far as (2)(3.14)(20 cm) = 125.6 cm, while the point located near the axis moves circularly as far as (2)(3.14)( 1 cm) = 6.28 cm. 125.6 cm is the magnitude of displacement done by the edge of the fan, while 6.28 cm is a magnitude of the displacement done by a point located near the axis of rotation. The smaller r, the smaller the displacement. The relationship between displacement (d) and angular displacement (θ) is expressed by the equation:
θ = d / r
d = r θ
d = displacement (meter), r = radius or distance from the axis of rotation (meter), θ = angular displacement (radian)
Sample problem 6:
A CD of 5 cm in radius rotates through an angle of 90o. What is the displacement of a point 2 cm from the axis of rotation?
r = distance from the axis of rotation = 2 cm = 0.02 m
θ = 90o = 1.57 rad (must be stated in radians)
d = (0.02 m)(1.57 rad) = 0.03 m.
Angular displacement does not have an international system unit and has no dimension (its dimension is 1), therefore in the calculation as above, just eliminate the radians unit from the calculation results.
When rotated, a fan or any rotating object of course needs a certain time interval. If the fan rotates clockwise and takes one rotation (360o) for 1 second then the angular velocity of all parts of the fan is 1 rev/s = 360 o/s = 6.28 rad/s and the direction of angular velocity is the same as the direction of the hour needle.
If the fan radius is 20 cm then the edge of the fan moves circularly with a speed of 2(3.14)(20 cm) / 1 second = 125.6 cm/s = 1.256 m/s. The point which is 1 cm (0.01 m) from the axis of rotation with a speed of 2 (3.14) (1 cm)/1 second = 6.28 cm/s = 0.0628 m/s. The smaller r, the smaller the speed. In a circular motion, speed is often called tangential speed.
The relationship between velocity (v) and angular velocity (ω) is expressed by the equation:
v = speed (m/s), r = radius or distance from the axis of rotation (m), ω = angular velocity (rad/s)
Sample problem 7 :
The speed of the second needle is 6.28 rad/minute = 6.28 rad / 60 second = 0.1 rad/s. What is the speed of a point that is 2 cm (0.02 m) from the axis of rotation?
ω = 0.1 rad/s, r = 0.02 m
v = r
ω = (0.02 m)(0.1 rad/s) = 0.002 m/s
Eliminate radians from the calculation results.
A circular moving point accelerates if the magnitude and/or direction of the velocity change. Therefore there are two types of acceleration in a circular motion. First, centripetal acceleration (ac) or also called radial acceleration (ar). Centripetal acceleration occurs due to changes in velocity direction. The direction of the centripetal acceleration always goes to the center of the circle.
The magnitude of the centripetal acceleration is:
ac = centripetal acceleration (m/s2), r = radius or distance from the axis of rotation (m), v = speed (m/s), ω = angular velocity (rad/s)
Second, tangential acceleration (at). Tangential acceleration occurs due to changes in the magnitude of the velocity. Review a point on the edge of the rotating fan. If at first, the fan is rest then the point on the edge of the fan is also rest (v = 0). If one second later, the fan rotates with an angular speed of 1 rev/s = 360 o/s = 6.28 rad/s so that the point on the edge of the fan moves circularly with a speed of 1.2 m/s then the point on the edge of the fan wind experiences a tangential acceleration of 1.2 m/s per second = 1.2 m/s2.
The relationship between the tangential acceleration (at) and the angular acceleration (α) is expressed by the equation:
at = tangential acceleration (m/s2), r = radius or distance from the axis of rotation (m), α = angular acceleration (rad/s2)
Relationship between period (T) and frequency (f) with speed and angular speed
The period states the time interval for an object to do one revolution while the frequency states the number of revolutions for one second. The relationship between period and frequency is expressed through the equation: f = 1/T
The International System Unit for the period is second, the International System unit for frequency is 1 / second (= hertz). Speed (v) and angular velocity (ω) of circular moving particles can be expressed in periods or frequencies.
The velocity (v) of the circular moving particle is expressed by the equation:
The magnitude of the angular velocity (ω) of the circular moving particles is expressed by the equation: