1. **Problem statement:** A body weighing 10 N is placed on a rough inclined plane at angle $\theta$ and is about to move under its own weight. Another body weighing 20 N, made of the same material, is placed on the same plane. We need to determine the state of the second body. 2. **Key concepts and formula:** - The body is on the verge of moving, so the frictional force is at its maximum (limiting friction). - The limiting friction $f_{lim} = \mu N$, where $\mu$ is the coefficient of friction and $N$ is the normal reaction. - For a body about to move down, the component of weight down the plane equals limiting friction plus any opposing forces. - The angle of inclination $\theta$ satisfies $\tan \theta = \mu$ when the body is just about to move. 3. **Analysis for the first body (10 N):** - Weight $W_1 = 10$ N. - Since it is about to move, $\tan \theta = \mu$. 4. **For the second body (20 N):** - Weight $W_2 = 20$ N. - The coefficient of friction $\mu$ is the same (same material and surface). - The frictional force $f_{lim} = \mu N = \mu W_2 \cos \theta$. - The component of weight down the plane is $W_2 \sin \theta$. 5. **Compare forces for the second body:** - Since $\tan \theta = \mu$, then $\mu = \frac{W_1 \sin \theta}{W_1 \cos \theta} = \tan \theta$. - For the second body, the ratio $\frac{W_2 \sin \theta}{W_2 \cos \theta} = \tan \theta = \mu$. - So the second body is also at the verge of moving. 6. **Conclusion:** - The second body will be about to move down the plane. --- 7. **Second question:** If a body is about to slide on a rough inclined plane, then $\tan$ of the friction angle equals all of the following except one. - (a) the coefficient of friction $\mu$. - (b) the ratio between the normal reaction and the magnitude of the resultant reaction. - (c) tangent of the inclination angle between the plane and the horizontal. - (d) the ratio between the limiting friction and the magnitude of normal reaction. 8. **Explanation:** - $\tan$ of the friction angle equals the coefficient of friction $\mu$. - It also equals the ratio of limiting friction to normal reaction. - It equals the tangent of the angle of inclination $\theta$ when the body is just about to move. - However, the ratio between the normal reaction and the magnitude of the resultant reaction is not equal to $\tan$ of the friction angle. **Answer:** (b) is the exception. --- **Final answers:** - For the first problem: (a) be about to move down. - For the second problem: (b) the ratio between the normal reaction and the magnitude of the resultant reaction.