In figure the coefficient of friction between the floor and the body B is 0.1. The coefficient of friction between the bodies B and A is $0.2 .$ A force F is applied as shown on B. The mass of A is $m / 2$ and of B is m. How many number of the following statements are true?
a) The bodies will move together if F = 0.25 mg
b) The body A will slip with respect to B if F = 0.5 mg
c) The bodies will move together if F = 0.5 mg
d) The bodies will be at rest if F = 0.1 mg
e) The maximum value of F for which the two bodies will move together is 0.45 mg

Question

Accepted Answer

Consider the adjacent diagram. Fictional force on

B (f_{1})

and frictional force on

A (f_{2})

will be shown as
Let A and B are moving together

a_{c o m m o n} = \frac{F - f_{1}}{m_{A} + m_{B}} = \frac{F - f_{1}}{(m / 2) + m} = \frac{2 (F - f_{1})}{3 m}

Pseudo force on

A = (m_{A}) \times a_{c o m m o n} = m_{A} \times \frac{2 (F - f_{1})}{3 m} = \frac{m}{2} \times \frac{2 (F - f_{1})}{3 m} = \times \frac{(F - f_{1})}{3}

The force (F) will be maximum when Pseudo force on A= Frictional force on A

\Rightarrow \frac{F_{\max} - f_{1}}{3} = μ m_{A} g = 0.2 \times \frac{m}{2} \times g = 0.1 m g

\Rightarrow F_{\max} = 0.3 m g + f_{1} = 0.3 m g + (0.1) \frac{3}{2} m g = 0.45 m g

Hence, maximum force upto which bodies will move together is

Detailed Solution