The angular frequency of a fan increases from 30 rpm to 60 rpm in 3.142 s. A dust particle is present on the blades of the fan at a distance of 20 cm from the axis of rotation. Find i) the average angular acceleration ii) the radial acceleration of the dust particle at the end of 3.142 s, iii) the tangential acceleration of the dust particle at the end of 3.142 s and iv) the net acceleration of the dust particle at the end of 3.142 s.
-
i)
Average angular acceleration
= (Change in angular velocity) / time
= (60 - 30) * (2π/60) / (3.142) rad/s^2
≈ 1 rad/s^2
ii)
radial acceleration at 3.142 sec
= ω^2 R, where ω = angular velocity at 3.142 sec
= (60 * 2π/60)^2 * (0.20) m/s^2
≈ 7.90 m/s^2
(iii)
Tangential acceleration
= (average angular acceleration) * R
= 1 * (0.20)
= 0.20 m/s^2
(iv)
Net acceleration
= √[(7.9)^2 + (0.20)^2]
≈ 7.9 m/s^2
Average angular acceleration
= (Change in angular velocity) / time
= (60 - 30) * (2π/60) / (3.142) rad/s^2
≈ 1 rad/s^2
ii)
radial acceleration at 3.142 sec
= ω^2 R, where ω = angular velocity at 3.142 sec
= (60 * 2π/60)^2 * (0.20) m/s^2
≈ 7.90 m/s^2
(iii)
Tangential acceleration
= (average angular acceleration) * R
= 1 * (0.20)
= 0.20 m/s^2
(iv)
Net acceleration
= √[(7.9)^2 + (0.20)^2]
≈ 7.9 m/s^2
-
The angular frequency of a fan increases from 30 rpm to 60 rpm in 3.142 s. A dust particle is present on the blades of the fan at a distance of 20 cm from the axis of rotation.
Angular velocity is measured in radians per second. One revolution = 2 * π radians
30 rpm = 30 * 2 * π = 60 * π rad/s
60 rpm = 60 * 2 * π = 120 * π rad/s
Average angular acceleration = ∆ angular velocity ÷ time = (60 * π) ÷ 3.142 = 59.999 ≈ 60 rad/s^2
Tangential velocity = angular velocity * radius
30 rpm = 60 * π * 0.20 = 12 * π
60 rpm = 120 * π * 0.20 = 24 * π
Radial acceleration = (tangential velocity)^2 ÷ radius
Radial acceleration at 3.142 s = (24 * π) ÷ 0.20 = 377 m/s^2
This is the same as the centripetal acceleration!
So, the radial acceleration is directed toward the center of the circle.
Tangential acceleration = ∆ tangential velocity ÷ time
Tangential acceleration = (24 * π – 12 * π) ÷ 3.142 = 11.998 ≈ 12 m/s^2
The tangential acceleration is perpendicular to the radial acceleration.
Net acceleration is the sum of the 2 vectors which are 90° apart.
Magnitude = (377^2 + 12^2)^0.5 = 377.19 m/s^2
The tangent of the angle = 377 / 12
angle = tan^-1 (377/12) = 88.2° from the tangent line toward the radius.
Angular velocity is measured in radians per second. One revolution = 2 * π radians
30 rpm = 30 * 2 * π = 60 * π rad/s
60 rpm = 60 * 2 * π = 120 * π rad/s
Average angular acceleration = ∆ angular velocity ÷ time = (60 * π) ÷ 3.142 = 59.999 ≈ 60 rad/s^2
Tangential velocity = angular velocity * radius
30 rpm = 60 * π * 0.20 = 12 * π
60 rpm = 120 * π * 0.20 = 24 * π
Radial acceleration = (tangential velocity)^2 ÷ radius
Radial acceleration at 3.142 s = (24 * π) ÷ 0.20 = 377 m/s^2
This is the same as the centripetal acceleration!
So, the radial acceleration is directed toward the center of the circle.
Tangential acceleration = ∆ tangential velocity ÷ time
Tangential acceleration = (24 * π – 12 * π) ÷ 3.142 = 11.998 ≈ 12 m/s^2
The tangential acceleration is perpendicular to the radial acceleration.
Net acceleration is the sum of the 2 vectors which are 90° apart.
Magnitude = (377^2 + 12^2)^0.5 = 377.19 m/s^2
The tangent of the angle = 377 / 12
angle = tan^-1 (377/12) = 88.2° from the tangent line toward the radius.