Mechanics Paper 4 June 2013

A particle of mass 8 kg rests on a rough plane inclined at 30$^{\circ}$ to the horizontal and is on the point of slipping. Take $g = 9.81$ ms$^{-2}$.

Calculate the component of the particle's weight acting down the line of greatest slope, in newtons.

A particle of mass 8 kg rests on a rough plane inclined at 30$^{\circ}$ to the horizontal and is on the point of slipping. Take $g = 9.81$ ms$^{-2}$.

Calculate the coefficient of friction between the plane and the particle, giving your answer as a decimal.

A ball is thrown from a height of 0.9 m above the ground at an angle of 30$^{\circ}$ to the horizontal. It is to clear a wall which stands 10,5 m away horizontally and is 4,5 m high. Air resistance is neglected and $g = 9.81$ ms$^{-2}$.

The trajectory satisfies $y = x\tan\theta - \dfrac{gx^{2}\left(1+\tan^{2}\theta\right)}{2u^{2}}$, where x and y are measured from the point of projection.

Find the least speed of projection, in ms$^{-1}$, for which the ball clears the wall.

Coplanar forces of magnitudes 3 N, 4 N and 8 N act at a point as shown in the diagram. The 3 N force acts along the direction of $\mathbf{j}$, the 4 N force acts along the direction of $\mathbf{i}$, and the 8 N force acts at an angle $\theta$ below the direction of $-\mathbf{i}$.

The resultant force in the $\mathbf{i}$ direction is $4\left(1-\sqrt{3}\right)$ N. Find $\theta$, in degrees.

Coplanar forces of magnitudes 3 N, 4 N and 8 N act at a point as shown in the diagram. The 3 N force acts along the direction of $\mathbf{j}$, the 4 N force acts along the direction of $\mathbf{i}$, and the 8 N force acts at an angle $\theta$ below the direction of $-\mathbf{i}$.

Given that $\theta = 30^{\circ}$, calculate the magnitude of the resultant force, in newtons.

Coplanar forces of magnitudes 3 N, 4 N and 8 N act at a point as shown in the diagram. The 3 N force acts along the direction of $\mathbf{j}$, the 4 N force acts along the direction of $\mathbf{i}$, and the 8 N force acts at an angle $\theta$ below the direction of $-\mathbf{i}$.

Given that $\theta = 30^{\circ}$, the resultant force is $-2.928\mathbf{i}-\mathbf{j}$ newtons. Find the angle, in degrees, between the resultant and the direction of $-\mathbf{i}$.

A particle moves in a straight line. It starts with a velocity of 8 ms$^{-1}$ and is subjected to a retardation of $\dfrac{3}{2}$ ms$^{-2}$ for 6 seconds.

Find the time, in seconds, at which the particle is momentarily at rest.

A particle moves in a straight line. It starts with a velocity of 8 ms$^{-1}$ and is subjected to a retardation of $\dfrac{3}{2}$ ms$^{-2}$ for 6 seconds.

Find the displacement of the particle from its starting point after the 6 seconds, in metres.

A particle moves in a straight line. It starts with a velocity of 8 ms$^{-1}$ and is subjected to a retardation of $\dfrac{3}{2}$ ms$^{-2}$ for 6 seconds.

Find the total distance travelled by the particle in the 6 seconds, in metres.