Incline

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 wedge of inclination angle $\theta$ and mass $m$ is placed on a horizontal table. An object, also of mass $m$, is placed on the inclined surface of the wedge. The coefficient of friction between the object and the wedge, and between the wedge and the table, is $\mu$.

As shown in the figure, a block of mass $m$ is placed on the smooth inclined surface of a wedge. It is held in place by a thin string, with the other end of the string attached to a fixed backstop at the top of the wedge. Assume the angle of inclination is $\theta = 30^\circ$. The string will break when its tension reaches $2mg$. The entire system accelerates to the right with acceleration $a$.

A wedge ABC of mass $M$ and height $h$ has an inclined surface AC with an angle $\theta$. A small object of mass $m$ is placed at the top A and slides down from rest. All contact surfaces are frictionless. Find:

As shown in the figure, the masses of the three objects A, B, and C are $m_A = \frac{2\sqrt{3}}{3}m$, $m_B = 2m$, and $m_C = m$, respectively. Object C is connected to sliders A and B by two ropes that pass over two light, smooth pulleys, $O_1$ and $O_2$, at the same height. Sliders A and B are on fixed, smooth inclined planes with angles 60° and 30°, respectively. The system is initially in equilibrium. An object D with mass $m_D=m$ is gently hung on a hook below object C.