Question

The friction coefficient between the two blocks shown in the figure is $μ$ but the floor is smooth.
(a) What maximum horizontal force $F$ can be applied without disturbing the equilibrium of the system?

(b) Suppose the horizontal force applied is double the value found in part (a). Find the accelerations of the two masses.

Accepted Answer

(a) Let the tension in the string be $T$ when the maximum external force is $F .$ Since neither of the two blocks move vertically, the normal reaction on the top block by the bottom block will be equal to its weight. Hence, $N = m g .$ Therefore, the maximum value of the friction force between the two blocks will be $f = μ m g .$ This friction force will be due right on the top block to oppose the maximum external force. It will be due left on the lower block by Newton's third law of motion and will oppose tension on this block. Since both the blocks remain stationary, the force equations for the top and the bottom blocks, respectively, are written as

$F = T + f . . . (1)$

and $T = f . . . (2)$

Solving these two equations and putting the values of all the forces, we get the required maximum external force for the equilibrium of the system, i.e., $F = 2 μ m g .$

(b) The external force is doubled in magnitude. The two blocks being tied to the same string, will move with the same acceleration in opposite directions. Let the acceleration be $a .$ Let the new tension in the strings be $T'$ . The value of the friction force remains unchanged as it is the kinetic friction in this case.

Writing force equations similarly for both the blocks, we get

$2 F - T' - f = m a . . . (3)$

and $T' - f = M a . . . (4)$

Adding these two equations and putting the respective values of all the forces, we get

$2 μ m g = (m + M) a \Rightarrow a = \frac{2 μ m g}{(m + M)}$ .

H C Verma Solutions for Chapter: Friction, Exercise 4: EXERCISES

H C Verma Physics Solutions for Exercise - H C Verma Solutions for Chapter: Friction, Exercise 4: EXERCISES

Questions from H C Verma Solutions for Chapter: Friction, Exercise 4: EXERCISES with Hints & Solutions

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