Mechanics Paper 4 June 2008

A man pulls a crate of weight 20 N along a rough horizontal floor using a string inclined at 60° to the horizontal. The crate is in limiting equilibrium when he pulls with a force of magnitude 4 N. Find the exact normal reaction between the crate and the floor.

Continuing the crate scenario (weight 20 N, pulling force 4 N at 60° to the horizontal, normal reaction $R=20-2\sqrt3$ N, limiting equilibrium), find the exact coefficient of friction between the crate and the floor.

A car travels at 40 ms$^{-1}$ and decelerates uniformly to 25 ms$^{-1}$ over 30 seconds, then continues at the constant speed of 25 ms$^{-1}$ for a further distance of 500 m. Find the total distance travelled by the car.

For the same journey (uniform deceleration from 40 to 25 ms$^{-1}$ over the first 30 s, then constant 25 ms$^{-1}$ for the next 500 m), which statement correctly describes the shape of the displacement-time $(t,x)$ graph?

A stone is thrown from the top of a vertical wall 3 m high with velocity $V$ ms$^{-1}$ at an angle of depression $\theta°$ below the horizontal. It lands 0.5 s later at a point 1.6 m horizontally from the foot of the wall. Taking $g=9.8$ ms$^{-2}$ and neglecting air resistance, find $\theta$.

For the same stone, thrown at an angle of depression $\theta$ from the top of a 3 m wall and landing 0.5 s later 1.6 m from the foot of the wall, with horizontal and vertical velocity components $V\cos\theta=3.2$ ms$^{-1}$ and $V\sin\theta=3.55$ ms$^{-1}$, find $V$.

A toy car P (mass 4 kg) rests on a rough horizontal plane, with coefficient of friction $\frac12$ between P and the plane, and is connected by a light inextensible string over a smooth pulley X to toy car Q (mass 8 kg) held on a smooth plane inclined at 30° to the horizontal. When Q is released, find the acceleration of the system and the tension in the string, in terms of $g$.

For the same system (toy cars P and Q connected over smooth pulley X, tension $T=\dfrac{8g}{3}$ N in the string on both sides of the pulley, one side horizontal to P and the other along the 30° incline to Q), find the magnitude of the force exerted by X on the string, giving the answer in the form $kg\cos\alpha$.