(A) A mass of 4 Kg rests on a horizontal plane . The plane is gradually inclined until at an $a n g l e θ = 15^{o}$ with the horizontal,the mass just beings to slide, What is the coefficient of static friction between the block and the surface?
(B) Two blocks of masses 1 Kg and 2 Kg are connected by an intextensible string which passes over a frictionless pulley as shown in the figure.calculate the acceleration of these blocks

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Solution

(a) Here, friction force is balanced by weight of the body.
$m g sin 15 = μ m g cos 15$
$μ = tan 15$
$= tan (45 - 30)$
$μ = \frac{(tan 45 - tan 30)}{1 + tan 45 tan 30}$
$μ = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$
(B) For block A
$T - m g = m a$
$T = (10 + a) . . . . . . . . . (1)$
For block B
$m g - T = m a$
$T = m (g - a)$
$T = 2 (10 - a) . . . . . . . (2)$
Solve equation (1) and (2)

$a = \frac{10}{3} m / s^{2}$

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Q. A mass of 4 Kg rest on the horizontal plane. The plane is gradually inclined until at an angle of

θ = 15^{\circ}

with the horizontal,the mass just being to slide what is the coffeficient of static friction between the block & the surface.

Two blocks of masses $m_{1} = 4 k g$ and $m_{2} = 6 k g$ are connected by a string of negligible mass passing over a frictionless pulley as shown in the figure. The coefficient of friction between block $m_{1}$ and the horizontal surface is 0.4. When the system is released, the masses $m_{1} a n d m_{2}$ start accelerating. What additional mass $m$ should be placed over mass $m_{1}$ so that the masses $(m_{1} + m)$ slide with a uniform speed?