Question

A rectangle plate of sides a and b is suspended from a ceiling by two parallel string of length L each (figure 12−E11). The separation between the string is d. The plate is displaced slightly in its plane keeping the strings tight. Show that it will execute simple harmonic motion. Find the time period.

Figure

Answer 1

Let m is the mass of rectangular plate and x is the displacement of the rectangular plate.
During the oscillation, the centre of mass does not change.
Driving force$\left(F\right)$ is given as,
F = mgsin θ
Comparing the above equation with F = ma, we get:
$a=\frac{F}{m}=g\sin \theta$
For small values of θ, sinθ can be taken as equal to θ.
Thus, the above equation reduces to:
$a=g\mathrm{\theta }=g\left(\frac{x}{L}\right)\left[\mathrm{Where}g\mathrm{and}\mathit{}L\mathrm{are}\mathrm{constant}.\right]$
It can be seen from the above equation that, a α x.
Hence, the motion is simple harmonic.
Time period of simple harmonic motion $\left(T\right)$ is given by,
$$\begin{gathered} \mathit{}T=2\mathrm{\pi }\sqrt{\frac{\mathrm{displacement}}{\mathrm{Acceleration}}} \\ =2\mathrm{\pi }\sqrt{\frac{x}{gx/L}}=2\mathrm{\pi }\sqrt{\frac{L}{g}} \end{gathered}$$