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  4. A block P of mass m is placed on a horizontal frictionless plane. A second block of same mass m is placed on it and is connected to a spring of spring constant k, the two blocks are pulled by a distance A. Block Q oscillates without slipping. What is the maximum value of frictional force between the two blocks?

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Q. A block P of mass m is placed on a horizontal
frictionless plane. A second block of same mass m is
placed on it and is connected to a spring of spring
constant k, the two blocks are pulled by a distance A.
Block Q oscillates without slipping. What is the
maximum value of frictional force between the two
blocks?Physics Question Image

IIT JEEIIT JEE 2004Oscillations

Solution:

Angular frequency of the system, $\omega = \sqrt{\frac{k}{m+m}} = \sqrt{\frac{k}{2m}}$
Maximum acceleration of the system will be ${\omega}^2 A \, \, or \, \, \frac{kA}{2m}$
This acceleration to the lower block is provided by
friction.
Hence , $ f_{max } = ma_{max} = m {\omega}^2 A = m\bigg( \frac{ kA}{2m}\bigg) = \frac{kA}{2}$