Rotary Motion - Engineering Mathematics, Lecture notes of Engineering Mathematics

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ROTATION OF RIGID BODIES

(Kinematics of Rotation)

Kinematics of Rotational Motion

s = r θ θ =s / r

v = rω ω = v / r

a

T

= rα α = a

T

/ r

f = 1 / T

Review : Linear to Angular and vice versa

a

R

= rω

2

Kinematics of Rotational Motion

θ is in radians

θ 1 @ Time = t 1 θ 1 θ 2 @ Time = t 2 θ 1 θ 2 Δθ Δt = t 2 - t 1

Angular Velocity – rate of angular displacement per unit time

ω = Δθ /Δt

Average Angular Velocity

ω(t) = dθ /dt

Instantaneous Angular Velocity

ω is in rad/s

Kinematics of Rotational Motion

θ is in radians

ω 1 @ Time = t 1 θ 1 ω 2 @ Time = t 2 θ 1 θ 2 Δω = ω 2 − ω 1 Δt = t 2 - t 1

Angular Acceleration – rate of change in angular velocity per unit

time

α = Δω /Δt

Average Angular Velocity

α(t) = dω /dt

Instantaneous Angular Velocity

α is in rad/s

2

1. A bicycle wheel turns with constant angular acceleration of 0. 64 rad/s^2 : a) How much time does it take to reach an angular velocity of 8 rad/sec starting from rest? b) How many revolutions does the wheel turn in this time interval?

Sample Problems : Kinematics of Rotational Motion

Answers : a) t = 12.5 sec b) ≈ 8 revolutions

2. A flywheel of radius 30 cm starts from rest and accelerates with constant α = 0. 5 rad/s 2 Compute the tangential acceleration, the radial acceleration and resultant of a point on its rim. a) @ the start b) after it has turned through 120 ° c) after it has turned through 240 °

Sample Problems : Kinematics of Rotational Motion

Answers : a) aT = 15 cm/s 2 , aR = 0 , a = 15 cm/s 2 b) aT = 15 cm/s^2 , aR = 62.83 cm/s^2 , a = 64.6 cm/s^2 c) aT = 15 cm/s 2 , aR = 125.66 cm/s 2 , a = 126.56 cm/s 2

1. A merry-go-round is being pushed by a child. The angle the merry-go-round has turned through varies with time according to :

θ(t) = (3 rad/s)t + (0.05 rad/s

3

)t

3 a) Calculate the average angular velocity f or the time intervals : a. 1 ) t 1 = 0 to t 2 = 2 sec a. 2 ) t 1 = 0 to t 2 = 5 sec b) Calculate the instantaneous angular velocity for t = 2 sec and t = 5 sec

SW : Kinematics of Rotational Motion

Answers : a.1 ) ω = 3.2 rad/s a.2) ω = 4.25 rad/s b.1) ω(2) = 3.6 rad/s b.2) ω(5) = 6.75 rad/s