Two blocks of masses $m$ and $M,(M \gt m)$, are placed on a frictionless table as shown in figure. A…
Two blocks of masses $m$ and $M,(M \gt m)$, are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released then ($\mu=$ coefficient of friction between the two blocks) (A) The time period of small oscillation of the two blocks is $\mathrm{T}=2 \pi \sqrt{\frac{(\mathrm{~m}+\mathrm{M})}{\mathrm{k}}}$ (B) The acceleration of the blocks is $\mathrm{a}=\frac{\mathrm{kx}}{\mathrm{M}+\mathrm{m}}$ ($\mathrm{x}=$ displacement of the blocks from the mean position) (C) The magnitude of the frictional force on the upper block is $\frac{m \mu|x|}{M+m}$ (D) The maximum amplitude of the upper block, if it does not slip, is $\frac{\mu(M+m) g}{k}$ (E) Maximum frictional force can be $\mu(\mathrm{M}+\mathrm{m}) \mathrm{g}$. Choose the correct answer from the options given below:
A, B, D Only
B, C, D Only
C, D, E Only
A, B, C Only
Solution
(A) As both blocks moving together so Time period $=2 \pi \sqrt{\frac{\mathrm{~m}}{\mathrm{~K}}} ;$ where $\mathrm{m}=\mathrm{M}+\mathrm{m}$ $\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{M}+\mathrm{m}}{\mathrm{~K}}}$ (B) Let block is displaced by x in $(+\mathrm{ve})$ direction so force on block will be in(-ve) direction $\begin{aligned} & \mathrm{F}=-\mathrm{Kx} \\
& (\mathrm{M}+\mathrm{m}) \mathrm{a}=-\mathrm{Kx} \\
& \mathrm{a}=-\frac{\mathrm{Kx}}{(\mathrm{M}+\mathrm{m})} \end{aligned}$ (C) As upper block is moving due to friction thus $\mathrm{f}=\mathrm{ma}=\frac{\mathrm{mKx}}{(\mathrm{M}+\mathrm{m})}$ (D) This option is like two block problem in friction for maximum amplitude, force on block is also maximum, for which both blocks are moving together. $\begin{aligned} & K A=(M+m) a \\
& a=\frac{K A}{(M+m)} \\
& f=m a=\frac{m K A}{(M+m)} \\
& f_{\max }=f_L=\mu \mathrm{mg} \\
& f=\mu m g \\
& \frac{m K A}{(M+m)}=\mu m g \\
& A=\frac{\mu(M+m) g}{K} \end{aligned}$ (E) Maximum friction can be $\mu \mathrm{mg}$ as force is acting between blocks \& normal force here is mg.
