Newton's Laws of Motion

A block of mass $12\text{ kg}$ is being pulled by a rope up a rough plane. The plane is inclined at an angle of $20^\circ$ above the horizontal. The rope pulling the block is parallel to a line of greatest slope of the plane. The coefficient of friction between the block and the plane is $0.4$. The acceleration of the block is $2\text{ m s}^{-2}$.

Find the tension in the rope.

A van of mass $4500\text{ kg}$ is towing a trailer of mass $350\text{ kg}$ along a straight horizontal road. The van and trailer are connected by a light rigid tow-bar which is parallel to the road. There are resistance forces of $X\text{ N}$ on the van and $120\text{ N}$ on the trailer. The driving force produced by the van's engine is $2500\text{ N}$. The tension in the tow-bar is $T\text{ N}$, and the acceleration of the van is $0.4\text{ ms}^{-2}$.

Find the value of $X$ and the value of $T$.

A particle $A$ of mass $2.5\text{ kg}$ is released from rest from the top of a smooth plane, which makes an angle of $\sin^{-1} 0.2$ with the horizontal. The particle $A$ collides $2$ seconds later with a particle $B$, of mass $3\text{ kg}$, which is moving up a line of greatest slope of the plane. The speed of $B$ immediately before the collision is $3.5\text{ m s}^{-1}$. Immediately after the collision, $B$ has a velocity of $0.5\text{ m s}^{-1}$ down the plane.

Find the distance $A$ moves up the plane after the collision.

A railway locomotive of mass $240\,000\text{ kg}$ is towing a coach of mass $36\,000\text{ kg}$ down a hill inclined at an angle of $\sin^{-1} 0.04$ to the horizontal. The driving force produced by the locomotive is $450\,000\text{ N}$ and there are resistances to motion of $120\,000\text{ N}$ on the locomotive and $15\,000\text{ N}$ on the coach. The coupling between the locomotive and the coach is light, rigid and parallel to the hill.

Find the acceleration of the locomotive and the tension in the coupling.

A block of mass $5\text{ kg}$ is being pulled straight down a line of greatest slope of a rough plane by a force of magnitude $20\text{ N}$. The plane is inclined at an angle of $10^\circ$ to the horizontal and the $20\text{ N}$ force acts at an angle of $35^\circ$ above the line of greatest slope of the plane (see diagram). The coefficient of friction between the block and the plane is $0.4$. The speed of the block when it passes a point $O$ is $2\text{ m s}^{-1}$.

Find the speed of the block when it has moved $3\text{ m}$ down the plane from $O$.