Question

A disc rotates about a fixed axis. It's angular velocity W varies with time according to eq w=at+b .initalliy at t=0 it's angula velocity is 0.1rad/s and angular position 2rad at the instant t=2s angula velocity is 5.0rad/s determine angular position and angular acceleration when t=4s

Answer 1

answer : angular position = 22 rad

And angular acceleration = 2.45 rad/s²

explanation : angular velocity, ω varies with time according to equation, ω = at + b

at t = 0, angular velocity = 0.1 rad/s

i.e., 0.1 = a × 0 + b

or, b = 0.1

so, ω = at + 0.1.....(1)

and angular position ,θ = 2rad

we know, dθ/dt = ω

or, ∫dθ = ∫ω.dt

or, $\int\limits^{\theta}_2\,d\theta=\int\limits^t_0{(at+0.1)}\,dt$

or, θ - 2 = at²/2 + 0.1t

or, θ = at²/2 + 0.1t + 2 ..........(2)

now at t = 2, angular velocity = 5 rad/s

so, ω = at + 0.1 ⇒5 = a × 2 + 0.1

or, a = 4.9/2 = 2.45

then, ω = (2.45)t + (0.1)

angular acceleration, α = dω/dt = d{2.45t + 0.1}/dt = 2.45 rad/s²

now angular position at t = 4s, θ = (2.45) × 4²/2 + 0.1 × 4 + 2

= 19.6 + 0.4 + 2

= 22 rad