Incline

A 3.0 m long wooden plank is used to push a cargo box with a mass of $m = 200$ kg up onto a platform that is 1.3 m high. The coefficient of kinetic friction between the box and the plank is 0.3. The box is pushed at a constant speed.

As shown in the figure, a right-angled triangle ABC is situated in a vertical plane, with side BC being horizontal and side AB being vertical. The angle between the hypotenuse AC and the horizontal side BC is $\alpha$. A point mass starts from rest at point A and travels to point C under the influence of gravity, constrained to move along specified paths.

Path 1 involves the particle moving along the sides AB and then BC. The time taken for the segment AB is $t_1$, and the time for the segment BC is $t_2$. Path 2 involves the particle moving along the hypotenuse AC. The time taken is $t_3$.

It is assumed that the particle turns at corner B instantaneously and that the magnitude of its velocity is conserved during the turn. The paths are smooth, so there is no friction.

As shown in Figure, a small block is placed on a fixed inclined plane with an inclination angle of $\theta$. The coefficient of friction between the block and the plane is given as $\mu = \tan\varphi$, where $\varphi$ is a fixed angle. A force $F$ is applied to the block at an angle $\beta$ with respect to the inclined plane, pulling it upwards along the plane.

As shown in Figure, a wooden wedge ABC with mass $M=10$ kg is stationary on a rough horizontal ground. A block with mass $m=1.0$ kg starts to slide down from rest on the inclined surface of the wedge, which has an angle $\theta = 30^\circ$. When the block has slid a distance $s=1.4$ m, its velocity is $v=1.4$ m/s. Throughout this process, the wedge does not move. The gravitational acceleration is $g=10$ m/s². The coefficient of kinetic friction between the wedge and the ground is given as $\mu=0.02$, though this information might not be necessary for the primary calculation.

A small object is initially at the bottom of a slope inclined at an angle $\alpha$ with the horizontal. It is projected upward along the slope with an initial velocity $v_0$, and reaches the maximum height. It then slides downward and returns to the initial position with final velocity $v_1$. The coefficient of sliding friction between the object and the slope surface is $\mu$.

A triangular wooden block ABC rests on a rough horizontal surface. On its two rough inclined surfaces, two blocks with masses $m_1$ and $m_2$ ($m_1 > m_2$) are placed, as shown in Figure. The angles of inclination are $\theta_1$ and $\theta_2$. The entire system, consisting of the triangular block and the two smaller blocks, is at rest.

An inclined plane has a length $L=5$ m and height $H=3$ m. At the bottom, there is an object A with mass $m=5$ kg. The coefficient of kinetic friction between A and the plane is $\mu=0.3$. A horizontal force $F=100$ N pushes A, causing it to move up the incline from rest. After A has moved a distance $s_0=2$ m along the incline, the force F is removed. Assume $g=10$ m/s².

Two blocks, A and B, with masses $m_A$ and $m_B$ and coefficients of kinetic friction $\mu_A$ and $\mu_B$ respectively, are connected by a light rod. They slide down an inclined plane with an angle of inclination $\theta$ together, as shown in the figure. It is assumed that the condition for sliding down is met, i.e., $\tan\theta > \mu_A$ and $\tan\theta > \mu_B$.

An object of mass $m$ starts from rest at the top of a fixed inclined plane of height $h$ and angle of inclination $\theta$. The coefficient of kinetic friction between the object and the plane is $\mu$.