OCR M1 (Mechanics 1) 2007 June

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\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_415_823_264_660} Two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) act at the origin O of rectangular coordinates Oxy (see diagram). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 14 N and 5 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 9 N and 7 N respectively.

1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
2. Hence find the magnitude of this resultant, and the angle the resultant makes with the positive \(x\)-axis.

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\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_714_1048_1231_552} A particle starts from the point A and travels in a straight line. The diagram shows the ( \(\mathrm { t } , \mathrm { v }\) ) graph, consisting of three straight line segments, for the motion of the particle during the interval \(0 \leqslant t \leqslant 290\).

1. Find the value of ther which the distance of the particle from A is greatest.
2. Find the displacement of the particle from A when \(\mathrm { t } = 290\).
3. Find the total distance travelled by the particle during the interval \(0 \leqslant \mathrm { t } \leqslant 290\).

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\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_437_846_274_651} A block of mass 50 kg is in equilibrium on smooth horizontal ground with one end of a light wire attached to its upper surface. The other end of the wire is attached to an object of mass mkg . The wire passes over a small smooth pulley, and the object hangs vertically below the pulley. The part of the wire between the block and the pulley makes an angle of \(72 ^ { \circ }\) with the horizontal. A horizontal force of magnitude X N acts on the block in the vertical plane containing the wire (see diagram). The tension in the wire is T N and the contact force exerted by the ground on the block is R N.

1. By resolving forces on the block vertically, find a relationship between T and R . It is given that the block is on the point of lifting off the ground.
2. Show that \(\mathrm { T } = 515\), correct to 3 significant figures, and hence find the value of m .
3. By resolving forces on the block horizontally, write down a relationship between T and X , and hence find the value of \(X\).

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\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_149_606_1626_772} Two particles of masses 0.18 kg and m kg move on a smooth horizontal plane. They are moving towards each other in the same straight line when they collide. Immediately before the impact the speeds of the particles are \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).

1. Given that the particles are brought to rest by the impact, find m .
2. Given instead that the particles move with equal speeds of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after the impact, find
   (a) the value of m , assuming that the particles move in opposite directions after the impact,
   (b) the two possible values of m , assuming that the particles coalesce.

5 A particle \(P\) is projected vertically upwards, from horizontal ground, with speed \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the greatest height above the ground reached by P is 3.6 m . A particle Q is projected vertically upwards, from a point 2 m above the ground, with speed \(\mathrm { um } \mathrm { s } ^ { - 1 }\). The greatest height abovetheground reached by Q is also 3.6 m .
2. Find the value of \(u\). It is given that P and Q are projected simultaneously.
3. Show that, at the instant when P and Q are at the same height, the particles have the same speed and are moving in opposite directions.

6 A particle starts from rest at the point A and travels in a straight line. The displacement sm of the particle from A at time ts after leaving A is given by $$s = 0.001 t ^ { 4 } - 0.04 t ^ { 3 } + 0.6 t ^ { 2 } , \quad \text { for } 0 \leqslant t \leqslant 10 .$$

1. Show that the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(\mathrm { t } = 10\). The acceleration of the particle for \(t \geqslant 10\) is \(( 0.8 - 0.08 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Show that the velocity of the particle is zero when \(\mathrm { t } = 20\).
3. Find the displacement from A of the particle when \(\mathrm { t } = 20\).

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\includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-5_488_739_269_703} One end of a light inextensible string is attached to a block of mass 1.5 kg . The other end of the string is attached to an object of mass 1.2 kg . The block is held at rest in contact with a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The string is taut and passes over a small smooth pulley at the bottom edge of the plane. The part of the string above the pulley is parallel to a line of greatest slope of the plane and the object hangs freely below the pulley (see diagram). The block is released and the object moves vertically downwards with acceleration \(\mathrm { am } \mathrm { s } ^ { - 2 }\). The tension in the string is TN . The coefficient of friction between the block and the plane is 0.8 .

1. Show that the frictional force acting on the block has magnitude 10.98 N , correct to 2 decimal places.
2. By applying Newton's second law to the block and to the object, find a pair of simultaneous equations in T and a .
3. Hence show that \(\mathrm { a } = 2.24\), correct to 2 decimal places.
4. Given that the object is initially 2 m above a horizontal floor and that the block is 2.8 m from the pulley, find the speed of the block at the instant when
   (a) the object reaches the floor,
   (b) the block reaches the pulley. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
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