The Ladder from the IPhOO

First, draw a free body diagram . We know that $F_1 = \mu N_1$ . Since the system is at translational equilibrium, the net horizontal force is 0 and so $N_2 = F_1$ . Knowing that $N_2$ is the normal force of the wall on the ladder, $F_2 = \mu N_2 = \mu^2 N_1$ . Again, since there is translational equilibrium, the vertical forces balance out and so $N_1 + F_2 = 2mg$ : $N_1 = 2mg - F_2 = 2mg - \mu^2 N_1$ $\mu^2 N_1 + N_1 = N_1(\mu^2 + 1) = 2mg$ $N_1 = \frac{2mg}{\mu^2+1}$ Thus, $N_2 = \frac{2\mu mg}{\mu^2 + 1}$ and $F_2 = \frac{2\mu^2 mg}{\mu^2+1}$ . Now we can use the fact that the system is at rotational equilibrium as well, so $\Sigma \tau = 0$ . Using the base of the ladder as the pivot point: $\Sigma \tau = -mg\cos\theta(\frac{L}{2}) - mg\cos\theta(\frac{2L}{3}) + F_2 L\cos\theta + N_2 L\sin\theta = 0$ $\frac{-mg\cos\theta}{2} - \frac{2mg\cos\theta}{3} + \frac{2\mu^2 mg\cos\theta}{\mu^2+1} + \frac{2\mu mg\sin\theta}{\mu^2+1} = 0$ $\frac{2\mu^2\cos\theta + 2\mu\sin\theta}{\mu^2+1} = \frac{\cos\theta}{2}+\frac{2\cos\theta}{3}=\frac{7\cos\theta}{6}$ $2\mu^2\cos\theta+2\mu\sin\theta = \frac{7\cos\theta}{6}\mu^2 + \frac{7\cos\theta}{6}$ $\mu^2(2\cos\theta - \frac{7\cos\theta}{6}) + 2\mu\sin\theta - \frac{7\cos\theta}{6} = \mu^2(\frac{5\cos\theta}{6}) + 2\mu\sin\theta - \frac{7\cos\theta}{6} = 0$ Plugging into the quadratic formula: $\mu = \frac{-2\sin\theta\pm\sqrt{4\sin^2\theta - 4(\frac{5\cos\theta}{6})(\frac{-7\cos\theta}{6})}}{\frac{7\cos\theta}{3}}$ Setting $\theta = 30^\circ$ and discounting the negative answer (-2.06), we find that $\mu \approx 0.678$ and so $1000\mu \approx$ * 678 *.

just balance the forces in the x and y components and take sum of torques 0 about the end of ladder.write everything in terms of mg. and then cancel out the m and the l from the torque equations.ladder has uniform mass density so com lies at l/2 and is conc. with mass m.we also know theta = 30.we solve for mu and get 0.678. so floor function on mu gives 678.

(p.s ahaan rungta i solved iphoo and got 6/10 [P-1,4,5,7,9,10].(including this one)what rank do i have because topper has 9/10 and rank 2 and 3 are tied at 8/10,i submitted solutions too but no reply from your side).