Find the dimensions of

(a) angular speed ω,

(b) angular acceleration *α*,

(c) torque τ and

(d) moment of interia *I*.

Some of the equations involving these quantities are \[\omega = \frac{\theta_2 - \theta_1}{t_2 - t_1}, \alpha = \frac{\omega_2 - \omega_1}{t_2 - t_1}, \tau = F . r \text{ and }I = m r^2\].

The symbols have standard meanings.

#### Solution

(a) Dimensions of angular speed,

\[\omega = \frac{\theta}{t} = \left[ M^0 L^0 T^{- 1} \right]\]

(b) Angular acceleration,

\[\alpha = \frac{\omega}{t}\]

Here, ω = [M^{0}L^{0}T^{−1}] and* **t *= [T]

So, dimensions of angular acceleration = [M^{0}L^{0}T^{−2}]

(c) Torque,* τ *=*Fr*sinθ

Here,* F* = [MLT^{−2}] and* r* = [L]

So, dimensions of torque = [ML^{2}T^{−2}]

(d) Moment of inertia = *m**r*^{2}

Here, *m* = [M] and *r*^{2} = [L^{2}]

So, dimensions of moment of inertia = [ML^{2}T^{0}]