A block of mass m 1 = m is placed on a long plank of mass m 2 = 2 m . The plank rests on a smooth horizontal surface. The coefficient of static friction between the block and the plank is μ s =

Question

A block of mass m 1 ​ = m is placed on a long plank of mass m 2 ​ = 2 m . The plank rests on a smooth horizontal surface. The coefficient of static friction between the block and the plank is μ s ​ = 0.5 , and the coefficient of kinetic friction is μ k ​ = 0.4 . A horizontal force F is applied to the plank. Match the physical quantities in List-I with their correct values in List-II. (Take g as acceleration due to gravity) List - I List - II (P) Maximum force F that can be applied to the plank such that the block m 1 ​ does not slip relative to the plank (1) 0.4 g (Q) Acceleration of the block m 1 ​ when F = 3 m g (2) 1.5 m g (R) Friction force on m 1 ​ when F = m g (3) m g /3 (S) Acceleration of the plank m 2 ​ when F = 3 m g (4) 1.3 g

TestHub · Accepted Answer

P: For no slipping, the maximum static friction force on $m_{1}$ is $f_{s, ma x} = μ_{s} m_{1} g = 0.5 m g$ . The maximum acceleration of $m_{1}$ without slipping is $a_{ma x} = f_{s, ma x} / m_{1} = 0.5 g$ . If no slip, both blocks move together with acceleration $a$ . $F = (m_{1} + m_{2}) a = (m + 2 m) a = 3 ma$ . For no slip, $a \leq a_{ma x}$ , so $F / (3 m) \leq 0.5 g ⟹ F_{ma x} = 1.5 m g$ .

R: When $F = m g$ , since $m g < F_{ma x} (1.5 m g)$ , there is no slipping. The system accelerates together with $a = F / (m_{1} + m_{2}) = m g / (3 m) = g /3$ . The friction force on $m_{1}$ is $f = m_{1} a = m (g /3) = m g /3$ .

Q and S: When $F = 3 m g$ , since $3 m g > F_{ma x} (1.5 m g)$ , slipping occurs. Kinetic friction acts between the blocks: $f_{k} = μ_{k} m_{1} g = 0.4 m g$ . For block $m_{1}$ : $f_{k} = m_{1} a_{1} ⟹ 0.4 m g = m a_{1} ⟹ a_{1} = 0.4 g$ . So, Q is $0.4 g$ . For plank $m_{2}$ : $F - f_{k} = m_{2} a_{2} ⟹ 3 m g - 0.4 m g = 2 m a_{2} ⟹ 2.6 m g = 2 m a_{2} ⟹ a_{2} = 1.3 g$ . So, S is $1.3 g$ .

List - I	List - II
(P) Maximum force $F$ that can be applied to the plank such that the block $m_{1}$ does not slip relative to the plank	(1) $0.4 g$
(Q) Acceleration of the block $m_{1}$ when $F = 3 m g$	(2) $1.5 m g$
(R) Friction force on $m_{1}$ when $F = m g$	(3) $m g /3$
(S) Acceleration of the plank $m_{2}$ when $F = 3 m g$	(4) $1.3 g$

Physics - NLM Question with Solution | TestHub

Options:

Doubts & Discussion