A block takes time to slide down a plane inclined at to the horizontal. If the surface is made smooth (frictionless)

Question

A block takes $t$ time to slide down a plane inclined at $45 °$ to the horizontal. If the surface is made smooth (frictionless), the block takes time $\frac{t}{2}$ to slide down the plane. The coefficient of friction between the block and the inclined plane is $(\frac{α}{100})$ . The value of $α$ is _________. [2026]

Smart Achievers · Accepted Answer

(75)

$The block takes time t to slide down the inclined plane when friction is present,$

$s = \frac{1}{2} a_{1} t^{2}$ $. . . (1)$

$The block takes time \frac{t}{2} to slide down the inclined plane when friction is absent,$

$s = \frac{1}{2} a_{2} {(\frac{t}{2})}^{2}$ $. . . (2)$

$From (i) and (ii),$

$\frac{1}{2} a_{1} t^{2} = \frac{1}{2} a_{2} \times \frac{t^{2}}{4} \Rightarrow a_{1} = \frac{a_{2}}{4}$

$a_{1} = g \sin 45 ° - μ g \cos 45 ° = \frac{g}{\sqrt{2}} (1 - μ),$ $a_{2} = \frac{g}{\sqrt{2}}$

$\frac{g}{\sqrt{2}} = 4 (\frac{g}{\sqrt{2}} (1 - μ))$ $\Rightarrow μ = 1 - \frac{1}{4} = \frac{3}{4}$

$\frac{α}{100} = \frac{3}{4} \Rightarrow α = 75$

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Q. A block takes t time to slide down a plane inclined at 45° to the horizontal. If the surface is made smooth (frictionless), the block takes time t2 to slide down the plane. The coefficient of friction between the block and the inclined plane is (α100). The value of α is _________. [2026]

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