8 A rough slope is inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude \(T\) newtons. The diagram shows the slope, the box and the rope.
\includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-008_307_469_500_840}

1. The box is held in equilibrium by the rope.
   1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
   2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
   3. Find the least possible tension in the rope to prevent the box from moving down the slope.
   4. Find the greatest possible tension in the rope.
   5. Show that the mass of the box is approximately 8.16 kg .
2. The rope is now released and the box slides down the slope. Find the acceleration of the box. General Certificate of Education
   June 2006
   Advanced Subsidiary Examination ASSESSMENT and
   REALIFIEATIONS
   ALLIANCE Tuesday 6 June 20061.30 pm to 2.45 pm \section*{For this paper you must have:}
   • an 8-page answer book
   • the blue AQA booklet of formulae and statistical tables
   You may use a graphics calculator. Time allowed: 1 hour 15 minutes \section*{Instructions}
   • Use blue or black ink or ball-point pen. Pencil should only be used for drawing.
   • Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MM1A/W.
   • Answer all questions.
   • Show all necessary working; otherwise marks for method may be lost.
   • The final answer to questions requiring the use of calculators should be given to three significant figures, unless stated otherwise.
   • Take \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), unless stated otherwise.
   \section*{Information}
   • The maximum mark for this paper is 60 .
   • The marks for questions are shown in brackets.
   • Unit Mechanics 1A has a written paper and coursework.
   \section*{Advice}
   • Unless stated otherwise, you may quote formulae, without proof, from the booklet.
   Answer all questions. 1 A small stone is dropped from a high bridge and falls vertically.
3. Find the distance that the stone falls during the first 4 seconds of its motion. (3 marks)
4. Find the speed of the stone when it has been falling for 4 seconds. 2 A car travels along a straight horizontal road. The motion of the car can be modelled as three separate stages. During the first stage, the car accelerates uniformly from rest to a velocity of \(10 \mathrm {~ms} ^ { - 1 }\) in 6 seconds. During the second stage, the car travels with a constant velocity of \(10 \mathrm {~ms} ^ { - 1 }\) for a further 4 seconds. During the third stage of the motion, the car travels with a uniform retardation of magnitude \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest.
5. Show that the time taken for the third stage of the motion is 12.5 seconds.
6. Sketch a velocity-time graph for the car during the three stages of the motion.
7. Find the total distance travelled by the car during the motion. 3 A stone rests in equilibrium on a rough plane inclined at an angle of \(16 ^ { \circ }\) to the horizontal, as shown in the diagram. The mass of the stone is 0.5 kg .
   \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-010_222_528_1978_731}
8. Draw a diagram to show the forces acting on the stone.
9. Show that the magnitude of the frictional force acting on the stone is 1.35 newtons, correct to three significant figures.
10. Find the magnitude of the normal reaction force between the stone and the plane.
11. Hence find an inequality for the value of \(\mu\), the coefficient of friction between the stone and the plane. 4 A block \(P\) is attached to a can \(Q\) by a light inextensible string. The string hangs over a smooth peg so that \(P\) and \(Q\) hang freely, as shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-011_246_259_470_886} The block \(P\) and the can \(Q\) each has mass 0.2 kg . The can \(Q\) contains a small stone of mass 0.1 kg . The system is released from rest and the can \(Q\) and the stone move vertically downwards.
12. By forming two equations of motion, show that the magnitude of the acceleration of \(P\) and \(Q\) is \(1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
13. Find the magnitude of the reaction force between the can and the stone. 5 The points \(A\) and \(B\) have position vectors \(( 3 \mathbf { i } + 2 \mathbf { j } )\) metres and \(( 6 \mathbf { i } - 4 \mathbf { j } )\) metres respectively. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.
14. A particle moves from \(A\) to \(B\) with constant velocity \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Calculate the time that the particle takes to move from \(A\) to \(B\).
15. The particle then moves from \(B\) to a point \(C\) with a constant acceleration of \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It takes 4 seconds to move from \(B\) to \(C\).
    1. Find the position vector of \(C\).
    2. Find the distance \(A C\).