Incline

A sled is held on an inclined plane by a cord pulling up the plane. The sled is on the verge of moving up the plane. The magnitude $F$ of the cord's force is a linear function of the coefficient of static friction $\mu_s$. The data shows that the force is $F_1$ when $\mu_s = 0$, and the force is $F_2$ when $\mu_s = \mu_{s2}$.

A block of mass $m_A$ on an inclined plane at angle $\theta$ is connected by a rope over a frictionless, massless pulley to a hanging block of mass $m_B$. The coefficients of static and kinetic friction between block A and the incline are $\mu_s$ and $\mu_k$. The positive direction is defined as up the incline, represented by the unit vector $\hat{i}$.

Two blocks are connected over a frictionless, massless pulley as shown in the figure. The mass of block A is $m_A$. The coefficient of kinetic friction between block A and the incline is $\mu_k$. The angle of the incline is $\theta$. Block A slides down the incline at a constant speed. The connecting rope has negligible mass.

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.

A block C of mass $m = 2$ kg is placed on a smooth inclined plane with an angle of inclination $\alpha = 30^\circ$. It is attached to a light spring with an unstretched length of $l_0 = 0.2$ m. The other end of the spring is fixed at the top of the incline. When the system is at rest, the spring stretches to a length of $l_1 = 0.25$ m. The entire system (incline and block) rotates about a vertical axis AB as shown. (Use $g = 9.8$ m/s²)

A block of mass $m_1$ is on a frictionless plane inclined at an angle $\theta$. It is connected by a massless cord over a massless, frictionless pulley to a second, hanging block of mass $m_2$.

A box of mass $m_1$ rests on a frictionless plane inclined at an angle $\theta_1$. It is connected by a massless cord over a massless, frictionless pulley to a second box of mass $m_2$ on a frictionless plane inclined at an angle $\theta_2$.

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.

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 $\mu$.

In Figure below, a 5.0 kg block is sent sliding up a plane inclined at $\theta = 37^\circ$ while a horizontal force $\vec{F}$ of magnitude 50 N acts on it. The coefficient of kinetic friction between block and plane is 0.30. The block's initial speed is 4.0 m/s.