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

Explanation:

To show that it will execute simple harmonic motion:

• During oscillation, the centre of mass does not vary.  Therefore, the driving force can be written as $\mathrm{F}=m g \sin \theta$ where m is the mass of the rectangular plate.  Also, we know that force $F=m a$.  
• Hence, on comparing we get, $a=F m=g \sin \theta$.  For small values of θ, $\sin \theta=\theta$.  
• This implies  $a=g\left(\frac{x}{L}\right)$.  Here, as g and L are constant, a is directly proportional to x.  Therefore, the rectangular plate executes simple harmonic motion.  
• To find Time period: $\mathrm{T}=2 \pi \sqrt{\left(\frac{\text { displacement }}{\text { acceleration }}\right)}=2 \pi \sqrt{\frac{L}{g}}$ where L is length of the string.