Mechanics — Variant 2

A particle of mass $0.6$ kg is projected with a speed of $4$ $\text{m s}^{-1}$ down a line of greatest slope of a smooth plane inclined at $10^\circ$ to the horizontal. Use an energy method to find the speed of the particle after it has moved $15$ m down the plane.

A ring of mass $0.3$ kg is threaded on a horizontal rough rod. The coefficient of friction between the ring and the rod is $0.8$. A force of magnitude $8$ N acts on the ring. This force acts at an angle of $10^\circ$ above the horizontal in the vertical plane containing the rod. Find the time taken for the ring to move, from rest, $0.6$ m along the rod.

A particle of mass $12$ kg is stationary on a rough plane inclined at an angle of $25^\circ$ to the horizontal. A pulling force of magnitude $P$ N acts at an angle of $8^\circ$ above a line of greatest slope of the plane. This force is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is $0.3$. Find the greatest possible value of $P$.

A car of mass $1250$ kg is pulling a caravan of mass $800$ kg along a straight road. The resistances to the motion of the car and caravan are $440$ N and $280$ N respectively. The car and caravan are connected by a light rigid tow-bar.

The car and caravan move along a horizontal part of the road at a constant speed of $30$ $\text{m s}^{-1}$.

Calculate, in kW, the power developed by the engine of the car.

Given that this power is suddenly decreased by $8$ kW, find the instantaneous deceleration of the car and caravan and the tension in the tow-bar.

The car and caravan now travel along a part of the road inclined at $\sin^{-1} 0.06$ to the horizontal. The car and caravan travel up the incline at constant speed with the engine of the car working at $28$ kW.

Find this constant speed.

Find the increase in the potential energy of the caravan in one minute.

A particle $A$ is projected vertically upwards from level ground with an initial speed of $30$ $\text{m s}^{-1}$. At the same instant a particle $B$ is released from rest $15$ m vertically above $A$. The mass of one of the particles is twice the mass of the other particle. During the subsequent motion $A$ and $B$ collide and coalesce to form particle $C$. Find the difference between the two possible times at which $C$ hits the ground.