Question

Give the derivation of equations of rotational motion.

Answer 1

Pure rotation dynamics around static axis: The quantities which are used in pure rotational dynamics are angular displacement θ, angular velocity

ω, angular acceleration α. These quantities are corresponding to linear displacement x, linear velocity v and linear acceleration a in the translatory motion.

Therefore equations of motion in rotational motion corresponding to equations of motion in translatory motion are:

ω = ω₀ + α.t .......(1)

θ = θ₀ + ω₀t + \(\frac{1}{2}\)α.t ......(2)

ω² = ω₀² + 2α(θ - θ₀) ........(3)

Proof: First equation:

Angular acceleration, \(\frac{dw}{dt}\)

or dω = αdt

On integrating both the sides,

\(\int_{\omega_0}^\omega d\omega = \int_0^t\alpha dt\)

or \([\omega]_{\omega_0}^\omega = \alpha[t]_0^t\)

or ω - ω₀ = α(t - 0)

or ω = ω₀ + αt

Second equation: Angular velocity

ω = \(\frac{d\theta}{dt}\)

or dθ = ωdt

On integrating both sides

Third equation: Angular acceleration

(α) = \(\frac{d\omega}{dt}\ ...(1)\)