91Ó°ÊÓ

First, let's list the forces acting on the block: 1. Gravitational force (weight) acting downwards: W = mg 2. Normal force acting from the surface on the block, perpendicular to the surface: N 3. Frictional force acting on the block, parallel to the surface and opposite to the direction of motion: f = μN 4. The external horizontal force acting on the block: F

We can write the equations for the net force acting on the block in the horizontal (x) and vertical (y) directions. Forces in the x-direction: \(F_{net_x} = F - f\) Forces in the y-direction: \(F_{net_y} = N - W\)

The force exerted by the block on the surface (R) is the net force experienced by the block due to all the forces acting on it. The magnitude of R can be found using the Pythagorean theorem: \(R = \sqrt{F_{net_x}^2 + F_{net_y}^2} = \sqrt{(F - f)^2 + (N - W)^2}\) Also, the angle θ that R makes with the vertical can be found using trigonometry: \(\tan(\theta) = \frac{F_{net_x}}{F_{net_y}} = \frac{F - f}{N - W}\)

(A) The magnitude of R will gradually increase. - As the external force F is gradually increased, the frictional force f will increase until it reaches its maximum value (μN). After that, it remains constant, so the net horizontal force (F - f) will increase, which will also lead to an increase in the magnitude of R. Therefore, statement A is correct. (B) \(R \leq mg\sqrt{\mu^2 + 1}\). - We can rewrite the inequality as: \(\sqrt{(F - f)^2 + (N - W)^2} \leq mg\sqrt{\mu^2 + 1}\) This inequality holds true in general, so statement B is also correct. (C) The angle made by R with the vertical will gradually increase. - As the external force F increases, the net horizontal force (F - f) will increase, causing the angle between R and the vertical to increase. Therefore, statement C is correct. (D) The angle made by R with the vertical \(\leq \tan^{-1} \mu\). - For small values of F, the angle between R and the vertical will be less than the maximum angle \(\tan^{-1} \mu\). However, as F increases, the net horizontal force (F - f) will increase, causing the angle between R and the vertical to increase and reach a value equal to \(\tan^{-1} \mu\) for the limiting case. So, statement D is also correct. In conclusion, all the given statements A, B, C, and D are correct.