Question

A square plate of mass 120 g and edge 5⋅0 cm rotates about one of the edges. If it has a uniform angular acceleration of 0⋅2 rad/s², what torque acts on the plate?

Answer 1

Consider a small area element of width dx and length a at a distance x from the axis of rotation.
Mass of element, dm $=\frac{m}{a^2}\times adx$
(m = mass of the square, a = side of the plate)
$$\begin{gathered} l={\int }_0^a{x}^{2}dm \\ l={\int }_0^a\frac{m}{a^2}\times a{x}^{2}dx \\ =\frac{m}{a}{\left[\frac{x^3}{3}\right]}_0^a=\frac{m{a}^2}{3} \end{gathered}$$



Now, torque produced,
$$\begin{gathered} \tau =\left(\frac{m{a}^2}{3}\right)\times \alpha \\ =\left\{\frac{120\times {10}^{-3}\times 5^2\times {10}^{-4}}{3}\right\}\times .02 \\ =40\times 25\times {10}^{-7}\times 0.2 \\ =2\times {10}^{-5}\mathrm{N}-\mathrm{m}. \end{gathered}$$