From Mechanics

\displaystyle \large {\underline{ {\textbf {Mechanics Test 2011 (1 hour)}}}}

This test accounts for 50% of your final mechanics mark.

1. A ball is projected vertically upwards at 24.5 \displaystyle \text{ m}{{\text{s}}^{-1}} from ground level. Assume that the ball is a particle and that no resistance forces act on it as it moves.

(a) Find the time that it takes the ball to reach its maximum height.

(3 marks)

(b) Find the maximum height of the ball.

(3 marks)

(c) State the total distance travelled by the ball.

(1 marks)

2. A crane lifts a crate of mass 500 kg using a single vertical cable.

(a) Draw a diagram to show the forces acting on the crate.

(1 marks)

(b) Find the tension in the cable if rising at a constant speed.

(1 marks)

(c) Find the tension in the cable if the crate is accelerating upwards at 0.6 \displaystyle \text{ m}{{\text{s}}^{-2} }.

(3 marks)

(d) If the tension in the cable is 4800 N. Describe fully the acceleration of the crate.

(4 marks)

3. A block of mass 5 kg is placed on a rough inclined plane which is inclined at an angle of 42\displaystyle {}^\circ to the horizontal. When released the block slides down the slope. The coefficient of friction between the block and the surface is 0.4.

(a) Calculate the magnitude of the normal reaction force on the block.

(2 marks)

(b) Calculate the magnitude of the friction force acting on the block.

(2 marks)

(c) Find the acceleration of the block as it slides down the slope.

(3 marks)

(d) Find the time that it would take the block to travel 2 metres.

(3 marks)

4. A particle, of mass 8 kg, is held at rest on an inclined plane by a horizontal force of magnitude Q N. The slope is inclined at 40\displaystyle {}^\circ to the horizontal.

(a) Find Q if the slope is smooth.

(2 marks)

(b) Find Q if the coefficient of friction between the slope and the particle is 0.3

(2 marks)

5. A boat is initially at the origin, O, where it has velocity \displaystyle (\mathbf{i}+\mathbf{j})\text{ m}{{\text{s}}^{\text{-1}}}. It moves with constant acceleration of \displaystyle (0\textrm{.}05\mathbf{i}-0\textrm{.}1\mathbf{j})\text{ m}{{\text{s}}^{\text{-2}}}. The unit vectors \displaystyle \mathbf{i} and \displaystyle \mathbf{j} are directed east and north respectively.

(a) Write down an expression for the velocity of the boat t seconds after it has left the origin.

(2 marks)

(b) Find the time it takes for the boat to move to the point where it is travelling south east

(4 marks)

40 Marks