> Q. A smooth block is released at rest on a 45$^{circ}$ incline and then slides a distance $d$. The time taken to slide is $n$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is – LIVE ANSWER TODAY

Q. A smooth block is released at rest on a 45$^{circ}$ incline and then slides a distance $d$. The time taken to slide is $n$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

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A smooth block is released at rest on a 45$^{circ}$ incline and then slides a distance $d$. The time taken to slide is $n$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

Solution:

When friction is absent

$ , , , , , , , , , , a_1 = g sin ,
theta $

$therefore , , , , , , , , s_1 = frac{1}{2 } a_1t_1^2 , , , , , , , , , , , , $…(i)

Whrn friction is present

$ , , , , , , , , , , , a_2 = g sin ,
theta – mu_kg cos
theta$

$therefore , , , , , , , , , , s_2 = frac{1}{2 a_2 t^2_2} , , , , , , , , , , , $…(ii)

From Eqs. (i) and (ii)

or $ , , , , , , , , , , , , frac{1}{2} a_1t^2_1 = frac{1}{2} a_2t^2_1 , , , , , because (t_2 = nt_1)$

or $, , , , , , , , a_1 = n^2 a_2$

$ , , , , , , , , frac{a_2}{a_1} = frac{g sin theta – mu_k g cos theta}{g sin theta } = frac{1}{n^2}$

or $ , , , , , , , , frac{g sin , 45^circ – mu_k g cos , 45^circ}{g , sin , 45^circ } = frac{1}{n^2}$

or $ , , , , , , , , , , , , , , 1 – mu_k = frac{1}{n^2} $ or $mu_k = 1 – frac{1}{n^2}$

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