Question

A uniform rod is placed on two spinning wheels as shown in the figure. The axes of the wheels are separated by $l$. The coefficient of friction between the rod and the wheel is $\mu$. Show that rod performs SHM and find the period of small oscillations.

• A. $2\pi \sqrt{\frac{l}{8g\mu }}$
• B. $2\pi \sqrt{\frac{l}{4g\mu }}$
• C. $2\pi \sqrt{\frac{l}{g\mu }}$
• D. $2\pi \sqrt{\frac{l}{2g\mu }}$

Answer 1

The correct option is D $2\pi \sqrt{\frac{l}{2g\mu }}$

$N_1+N_2=mg$ .......(1)
$\mu N_1-\mu N_2=ma$ ......(2)
$N_1(\frac{l}{2}+x)=N_2(\frac{l}{2}-x)$
Solving (1)(2) and (3), we get
$a=-\frac{\mu 2gx}{l}$
or $T=2\pi \sqrt{\frac{l}{2g\mu }}$