Questions Further Mechanics A AS (52 questions)

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OCR MEI Further Mechanics A AS 2024 June Q1
1 Two horizontal forces of magnitudes 7 N and 15 N act at a point O .
The 15 N force acts an angle of \(\theta ^ { \circ }\) above the positive \(x\)-axis.
The 7 N force acts at an angle of \(70 ^ { \circ }\) below the negative \(x\)-axis (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-2_606_773_402_239} The resultant of the two forces acts only in the positive \(x\)-direction.
  1. Calculate the value of \(\theta\).
  2. Calculate the magnitude of the resultant of the two forces.
OCR MEI Further Mechanics A AS 2024 June Q2
2
  1. Find the dimensions of energy. The moment of inertia, \(I\), of a rigid body rotating about a fixed axis is measured in \(\mathrm { kg } \mathrm { m } ^ { 2 }\).
  2. State the dimensions of \(I\). The kinetic energy, \(E\), of a rigid body rotating about a fixed axis is given by the formula
    \(\mathrm { E } = \frac { 1 } { 2 } \mathrm { I } \omega ^ { 2 }\),
    where \(\omega\) is the angular velocity (angle per unit time) of the rigid body.
  3. Show that the formula for \(E\) is dimensionally consistent. When a rigid body is pivoted from one of its end points and allowed to swing freely, it forms a pendulum. The period, \(t\), of the pendulum is the time taken for it to complete one oscillation. A student conjectures the formula
    \(\mathrm { t } = \left. \mathrm { k } ( \mathrm { mg } ) ^ { \alpha } \mathrm { r } ^ { \beta } \right| ^ { \gamma }\),
    where
    • \(k\) is a dimensionless constant,
    • \(m\) is the mass of the rigid body,
    • \(g\) is the acceleration due to gravity,
    • \(r\) is the distance between the pivot point and the rigid body’s centre of mass.
    • Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\).
    The moment of inertia of a thin uniform rigid rod of mass 1.5 kg and length 0.8 m , rotating about one of its endpoints, is \(0.32 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The student suspends such a rod from one of its endpoints and allows it to swing freely. The student measures the period of this pendulum and finds that it is 1.47 seconds.
  4. Using the formula conjectured by the student, determine the value of \(k\).
OCR MEI Further Mechanics A AS 2024 June Q3
3 The diagram shows a uniform beam AB , of weight 80 N and length 7 m , resting in equilibrium in a vertical plane. The end A is in contact with a rough vertical wall, and the angle between the beam and the upward vertical is \(60 ^ { \circ }\). The beam is supported by a smooth peg at a point C , where \(\mathrm { AC } = 2 \mathrm {~m}\).
\includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-4_474_709_445_244}
  1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on the beam.
    1. Show that the magnitude of the frictional force exerted on the beam by the wall is 25 N .
    2. Hence determine the magnitude of the total contact force exerted on the beam by the wall.
  3. Determine the direction of the total contact force exerted on the beam by the wall. The coefficient of friction between the beam and the wall is \(\mu\).
  4. Find the range of possible values for \(\mu\).
  5. Explain how your answer to part (b)(ii) would change if the peg were situated closer to A but the angle between the beam and the upward vertical remained at \(60 ^ { \circ }\).
OCR MEI Further Mechanics A AS 2024 June Q4
4 Three spheres A, B, and C, of equal radius are in the same straight line on a smooth horizontal surface. The masses of \(\mathrm { A } , \mathrm { B }\) and C are \(2 \mathrm {~kg} , 4 \mathrm {~kg}\) and 1 kg respectively. Initially the three spheres are at rest.
Spheres \(A\) and \(C\) are each given impulses so that \(A\) moves towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-5_325_1591_603_239} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 5 }\).
It is given that the first collision occurs between A and B .
  1. State how you can tell from the information given above that kinetic energy is lost when A collides with B .
  2. Show that the combined kinetic energy of A and B decreases by \(24 \%\) during their collision. Sphere B next collides with C. The coefficient of restitution between B and C is \(\frac { 2 } { 3 }\).
  3. Given that a third collision occurs, determine the range of possible values for \(u\).
  4. State one limitation of the model used in this question.
OCR MEI Further Mechanics A AS 2024 June Q5
5 In the diagram below, points \(\mathrm { A } , \mathrm { B }\) and C lie in the same vertical plane. The slope AB is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(\mathrm { AB } = 5 \mathrm {~m}\). The point B is a vertical distance of 6.5 m above horizontal ground. The point C lies on the horizontal ground.
\includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-6_601_1285_395_244} Starting at A , a particle P , of mass \(m \mathrm {~kg}\), moves along the slope towards B , under the action of a constant force \(\mathbf { F }\). The force \(\mathbf { F }\) has a magnitude of 50 N and acts at an angle of \(\theta ^ { \circ }\) to AB in the same vertical plane as A and B . When P reaches \(\mathrm { B } , \mathbf { F }\) is removed, and P moves under gravity landing at C . It is given that
  • the speed of P at A is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  • the speed of P at B is \(6 \mathrm {~ms} ^ { - 1 }\),
  • the speed of P at C is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  • 58 J of work is done against non-gravitational resistances as P moves from A to B ,
  • 42 J of work is done against non-gravitational resistances as P moves from B to C .
    1. By considering the motion from B to C, show that \(m = 4.33\) correct to 3 significant figures.
    2. By considering the motion from A to B , determine the value of \(\theta\).
    3. Calculate the power of \(\mathbf { F }\) at the instant that P reaches B .
OCR MEI Further Mechanics A AS 2024 June Q6
6 A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points \(A , B\) and \(C\) have coordinates \(( 12,0 ) , ( 12 + p , q )\) and \(( 0 , q )\) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-7_536_917_349_239}
  1. Determine, in terms of \(p\) and \(q\), the coordinates of the centre of mass of OABC . The point D has coordinates \(( 7.6 , q )\). When OABC is suspended from D , the lamina hangs in equilibrium with BC horizontal.
  2. Determine the value of \(p\). When OABC is suspended from C, the lamina hangs in equilibrium with BC at an angle of \(35 ^ { \circ }\) to the downward vertical.
  3. Determine the value of \(q\), giving your answer correct to \(\mathbf { 3 }\) significant figures.
OCR MEI Further Mechanics A AS 2020 November Q1
1 Brent is riding his bicycle along a straight horizontal road.
While riding along this road Brent can attain a maximum speed of \(6.25 \mathrm {~ms} ^ { - 1 }\) and the wind resistance acting on Brent and his bicycle is constant and equal to 19.2 N . Brent and his bicycle have a combined mass of 72 kg . Brent later begins to ride up a hill which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal.
Given that the wind resistance and the maximum power developed by the bicycle is unchanged, determine Brent's maximum speed up the hill.
OCR MEI Further Mechanics A AS 2020 November Q2
2 George is investigating the time it takes for a ball to reach a certain height when projected vertically upwards. George believes that the time, \(t\), for the ball to reach a certain height, \(h\), depends on
  • the ball's mass \(m\),
  • the projection speed \(u\), and
  • the height \(h\).
George suggests the following formula to model this situation
\(t = k m ^ { \alpha } u ^ { \beta } h ^ { \gamma }\),
where \(k\) is a dimensionless constant.
  1. Use dimensional analysis to show that \(t = \frac { k h } { u }\).
  2. Hence explain why George’s formula is unrealistic. Mandy argues that any model of this situation must consider the acceleration due to gravity, \(g\). She suggests the alternative formula
    \(t = \frac { u - \sqrt { u ^ { 2 } + g h } } { g }\).
  3. Show that Mandy's formula is dimensionally consistent.
  4. Explain why Mandy’s formula is incorrect.
OCR MEI Further Mechanics A AS 2020 November Q3
3 Fig. 3 shows a light square lamina ABCD , of side length 0.75 m , suspended vertically by wires attached to A and B so that AB is horizontal. A particle P of mass \(m \mathrm {~kg}\) is attached to the edge DC . The lamina hangs in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-3_586_702_404_251} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} The tension in the wire attached to A is 14 N and the tension in the wire attached to B is \(T \mathrm {~N}\). The wire at A makes an angle of \(25 ^ { \circ }\) with the horizontal and the wire at B makes an angle of \(60 ^ { \circ }\) with the horizontal.
  1. Determine the value of \(T\).
  2. Determine
    1. the value of \(m\),
    2. the distance of P from D . P is moved to the midpoint of CD . A couple is applied to the lamina so that it remains in equilibrium with AB horizontal and the tension in both wires unchanged.
  3. Determine
    • the magnitude of the couple,
    • the direction of the couple.
OCR MEI Further Mechanics A AS 2020 November Q4
4 Fig. 4 shows a uniform beam of length \(2 a\) and weight \(W\) leaning against a block of weight \(2 W\) which is on a rough horizontal plane. The beam is freely hinged to the plane at O and makes an angle \(\theta\) with the horizontal. The contact between the beam and the block is smooth. The beam and block are in equilibrium, and it may be assumed that the block does not topple. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-4_350_830_461_246} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Let
  • \(S\) be the normal contact force between the beam and the block,
  • \(R\) be the normal contact force between the plane and the block,
  • \(F\) be the frictional force between the plane and the block.
Partially complete force diagrams showing the beam and the block separately are given in the Printed Answer Booklet.
  1. Add the forces listed above to these diagrams. It is given that \(\theta = 30 ^ { \circ }\).
  2. Determine the minimum possible value of the coefficient of friction between the block and the plane.
  3. In each case explain, with justification, how your answer to part (b) would change (assuming the rest of the system remained unchanged) if
    1. \(\theta < 30 ^ { \circ }\),
    2. the contact between the beam and the block were rough.
OCR MEI Further Mechanics A AS 2020 November Q5
5 Throughout this question it may be assumed that there are no resistances to motion.
Model trucks A and B, with masses 5 kg and 3 kg respectively, rest on a set of straight, horizontal rails. Truck A is given an impulse of 3.8 Ns towards B .
  1. Calculate the initial speed of A. Truck A collides directly with B. After the collision, B moves with a speed of \(0.6 \mathrm {~ms} ^ { - 1 }\).
  2. Determine
    1. the velocity of A after the collision,
    2. the kinetic energy lost due to the collision.
  3. B continues to move with a speed of \(0.6 \mathrm {~ms} ^ { - 1 }\) and collides with a model truck C, of mass 4 kg , which is travelling at a speed of \(0.2 \mathrm {~ms} ^ { - 1 }\) towards B on the same set of rails. After the collision between B and C , the speeds of B and C are in the ratio 1 to 2 . Determine the two possible values of the coefficient of restitution between B and C .
OCR MEI Further Mechanics A AS 2020 November Q6
6 Fig. 6.1 shows a solid uniform prism OABCDEFG . The \(\mathrm { Ox } , \mathrm { Oy }\) and Oz axes are also shown. The cross-section of the prism is a trapezium. Fig. 6.2 shows the face OABC . The dimensions shown in the figures are in centimetres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-6_528_672_571_274} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-6_524_538_571_1242} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
\end{figure} The centre of mass of the prism has coordinates \(( \bar { x } , \bar { y } , \bar { z } )\).
  1. Determine the values of \(\bar { x } , \bar { y }\) and \(\bar { z }\).
  2. By considering triangle PBA, where P has coordinates ( \(\bar { x } , 0 , \bar { z }\) ), determine whether it is possible for the prism to rest with the face ABEF in contact with a horizontal plane without toppling.
OCR MEI Further Mechanics A AS 2020 November Q7
7 Fig. 7.1 shows one end of a light inextensible string attached to a block A of mass 4.4 kg . The other end of the string is attached to a block B of mass 5.2 kg . Block A is in contact with a smooth horizontal plane. The string is taut and passes over a small smooth pulley at the end of the plane. Block B is inside a hollow vertical tube and the vertical sides of B are in contact with the tube. Initially B is 1.6 m above the horizontal base of the tube. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-7_641_771_559_264} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
\end{figure} The blocks are released from rest. It may be assumed that in the subsequent motion A does not reach the pulley and the string remains taut. Block B reaches the base of the tube with speed \(3.5 \mathrm {~ms} ^ { - 1 }\).
  1. Given that the frictional force exerted by the tube on B is constant, use an energy method to show that the magnitude of this force is 14.21 N . Blocks A and B remain attached to the opposite ends of a light inextensible string, but A is now in contact with a rough plane inclined at \(\theta ^ { \circ }\) to the horizontal, as shown in Fig. 7.2. The string connecting A and B is taut and passes over a small smooth pulley at the top of the plane. Block B is inside the same hollow vertical tube as before with the vertical sides of B in contact with the tube. It may be assumed that the frictional force exerted by the tube on B remains unchanged. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-8_623_723_552_260} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure} The coefficient of friction between block A and the plane is \(\frac { 3 } { 11 }\).
    The blocks are released from rest, with block B 1.6 m above the base of the tube. It may be assumed that in the subsequent motion A does not reach the pulley and the string remains taut.
  2. Given that block B reaches the base of the tube with speed \(0.7 \mathrm {~ms} ^ { - 1 }\), show that \(\theta\) satisfies the equation
    \(3 \cos \theta + 11 \sin \theta = k\),
    where \(k\) is a constant to be determined. \section*{END OF QUESTION PAPER} \section*{}
OCR MEI Further Mechanics A AS 2021 November Q1
1 The specific energy of a substance has SI unit \(\mathrm { J } \mathrm { kg } ^ { - 1 }\) (joule per kilogram).
  1. Determine the dimensions of specific energy. A particular brand of protein powder contains approximately 345 Calories (Cal) per 4 ounce (oz) serving. An athlete is recommended to take 40 grams of the powder each day. You are given that \(1 \mathrm { oz } = 28.35\) grams and \(1 \mathrm { Cal } = 4184 \mathrm {~J}\).
  2. Determine, in joules, the amount of energy in the athlete's recommended daily serving of the protein powder.
OCR MEI Further Mechanics A AS 2021 November Q2
2 The vertices of a triangular lamina, which is in the \(x - y\) plane, are at the origin O and the points \(\mathrm { A } ( 4,0 )\) and \(\mathrm { B } ( 0,3 )\). Forces, of magnitude \(T _ { 1 } \mathrm {~N} , T _ { 2 } \mathrm {~N}\) and 10 N , whose lines of action are in the \(x - y\) plane, are applied to the lamina at \(\mathrm { O } , \mathrm { A }\) and B respectively, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-2_814_922_1135_246}
    1. Show that \(\sin \alpha = 0.6\).
    2. Write down the value of \(\cos \alpha\). The lamina is in equilibrium.
  2. Determine the values of \(T _ { 1 } , T _ { 2 }\) and \(\theta\).
OCR MEI Further Mechanics A AS 2021 November Q3
3 Three small uniform spheres A, B and C have masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively. The spheres move in the same straight line on a smooth horizontal table, with B between A and C . Sphere A moves towards B with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 } , \mathrm {~B}\) is at rest and C moves towards B with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-3_181_1291_461_251} Spheres A and B collide. Collisions between A and B can be modelled as perfectly elastic.
  1. Determine the magnitude of the impulse of A on B in this collision.
  2. Use this collision to verify that in a perfectly elastic collision no kinetic energy is lost. After the collision between A and B, sphere B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 4 }\).
  3. Show that, after the collision between B and C , B has a speed of \(( 1.225 - 0.78125 \mathrm { u } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) towards C.
  4. Determine the range of values for \(u\) for there to be a second collision between A and B .
OCR MEI Further Mechanics A AS 2021 November Q4
4 The diagram shows the path of a particle P of mass 2 kg as it moves from the origin O to C via A and B . The lengths of the sections \(\mathrm { OA } , \mathrm { AB }\) and BC are given in the diagram. The units of the axes are metres.
\includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-4_670_1322_404_246} P , starting from O , moves along the path indicated in the diagram to C under the action of a constant force of magnitude \(T \mathrm {~N}\) acting in the positive \(x\)-direction. As P moves, it does \(R \mathrm {~J}\) of work for every metre travelled against resistances to motion. It is given that
  • the speed of P at O is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  • the speed of P at A is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  • the speed of P at C is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
You should assume that both \(x\) - and \(y\)-axes lie in a horizontal plane.
  1. By considering the entire path of P from O to C , show that $$20 \mathrm {~T} - 30 \mathrm { R } = 108 .$$
  2. By formulating a second equation, determine the values of \(T\) and \(R\).
  3. It is now given that the \(x\)-axis is horizontal, and the \(y\)-axis is directed vertically upwards. By considering the kinetic energy of P at B , show that the motion as described above is impossible.
OCR MEI Further Mechanics A AS 2021 November Q5
5 A car of mass 1600 kg is travelling uphill along a straight road inclined at \(4.7 ^ { \circ }\) to the horizontal. The power developed by the car is constant and equal to 120 kW . The car is towing a caravan and together they have a maximum speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) uphill. In this question you may model any resistances to motion as negligible.
  1. Determine the mass of the caravan. The caravan is now detached from the car. Continuing up the same road, the car passes a point A at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car later passes through a point \(B\) on the same road such that \(A B = 80 \mathrm {~m}\) and the car takes 3.54 seconds to travel from A to B . The power developed by the car while travelling from A to B is constant and equal to 80 kW .
  2. Determine the speed of the car at B .
  3. State one possible refinement to the model used in parts (a) and (b).
OCR MEI Further Mechanics A AS 2021 November Q6
6 Fig. 6.1 shows a cross-section through a block of mass 5 kg which is on top of a trolley of mass 11 kg . The trolley is on top of a smooth horizontal surface. The coefficient of friction between the block and the trolley is 0.3 . Throughout this question you may assume that there are no other resistances to motion on either the block or the trolley. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-6_339_1317_552_294} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure} Initially, both the block and trolley are at rest. A constant force of magnitude 50 N is now applied horizontally to the trolley, as shown in Fig. 6.1.
  1. Show that in the subsequent motion the block will slide.
  2. Find the acceleration of
    1. the block,
    2. the trolley. The same block and trolley are again at rest. An obstruction, in the form of a fixed horizontal pole, is placed in front of the block, the pole is 91 cm above the trolley and the width of the block is 56 cm as shown in Fig. 6.2, as well as the forward direction of motion. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-6_426_1324_1793_269} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
      \end{figure} It is given that the block is uniform and that the contact between the pole and the block is smooth. A small horizontal force is now applied to the trolley in the forward direction of motion and gradually increased.
  3. Determine whether the block will topple or slide.
OCR MEI Further Mechanics A AS 2021 November Q7
7 The vertices of a uniform triangular lamina, which is in the \(x - y\) plane, are at the origin and the points \(( 20,60 )\) and \(( 100,0 )\).
  1. Determine the coordinates of the lamina's centre of mass. Fig. 7.1 shows a uniform lamina consisting of a triangular section and two identical rectangular sections. The coordinates of some of the vertices of the lamina are given in Fig. 7.1. The rectangular sections are then folded at right-angles to the triangular section, to give the rigid three-dimensional object illustrated in Fig. 7.2. Two of the edges, \(E _ { 1 }\) and \(E _ { 2 }\), are marked on both figures. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-7_933_739_799_164} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-7_924_725_808_1133} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
    \end{figure}
  2. Show that the \(x\)-coordinate of the centre of mass of the folded object is 43.6, and determine the \(y\) - and \(z\)-coordinates.
  3. Determine whether it is possible for the folded object to rest in equilibrium with edges \(E _ { 1 }\) and \(E _ { 2 }\) in contact with a horizontal surface.
OCR MEI Further Mechanics A AS Specimen Q1
1 A clock is driven by a 5 kg sphere falling once through a vertical distance of 120 cm over 2 days. Calculate, in watts, the average power developed by the falling sphere.
OCR MEI Further Mechanics A AS Specimen Q2
2 A triangular lamina, ABC , is cut from a piece of thin uniform plane sheet metal. The dimensions of ABC are shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-2_410_572_689_792} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} This piece of metal is freely suspended from a string attached to C and hangs in equilibrium. Calculate the angle of BC with the downward vertical, giving your answer in degrees.
OCR MEI Further Mechanics A AS Specimen Q3
3 Solid toy aeroplane nose cones of various sizes are made in the shape shown in Fig. 3.1, where OA is its line of symmetry. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-3_364_432_395_845} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure} The air resistance against the nose cone as the aeroplane flies through the air is initially modelled by \(R = k r v \eta\), where \(R\) is the air resistance, \(r\) is the radius of the circular flat end of the nose cone, \(v\) is the velocity of the nose cone, \(\eta\) is the viscosity of the air and \(k\) is a dimensionless constant.
  1. Use dimensional analysis to show that the dimensions of \(\eta\) are \(\mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 1 }\). In an experiment conducted on a particular nose cone, measurements of air resistance are taken for different velocities. The viscosity of the air does not vary during the experiment. The graph in Fig. 3.2 shows the results. Measurements are given using the appropriate S.I. units. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-3_794_1166_1411_427} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  2. Comment on whether the results of this experiment are consistent with the initial model. It is now suggested that a better model for the air resistance is \(R = K r v \left( \frac { \rho r v } { \eta } \right) ^ { \alpha }\), where \(\rho\) is the density of the air, \(K\) is a dimensionless constant and \(R , r , v\) and \(\eta\) are as before.
  3. (A) Find the dimensions of \(\frac { \rho r v } { \eta }\).
    (B) Explain why you cannot use dimensional analysis to find the value of \(\alpha\).
OCR MEI Further Mechanics A AS Specimen Q4
4 Fig. 4 shows a thin rigid non-uniform rod PQ of length 0.5 m . End P rests on a rough circular peg. A force of \(T \mathrm {~N}\) acts at the end Q at \(60 ^ { \circ }\) to QP . The weight of the rod is 40 N and its centre of mass is 0.3 m from P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-4_506_960_977_605} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The rod does not slip on the peg and is in equilibrium with PQ horizontal.
  1. Show that the vertical component of \(T\) is 24 N .
  2. \(F\) is the contact force at P between the rod and the peg. Find
    • the vertical component of \(F\),
    • the horizontal component of \(F\).
    • Given that the rod is about to slip on the peg, find the coefficient of friction between the rod and the peg.
OCR MEI Further Mechanics A AS Specimen Q5
5 In this question, all coordinates refer to the axes shown in Fig. 5.1. Fig. 5.1 shows a system of four particles with masses \(4 m , 3 m , m\) and \(2 m\) at the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . These points have coordinates \(( - 3,4 ) , ( 0,0 ) , ( 2,0 )\) and \(( 5,4 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-5_436_817_513_639} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
\end{figure}
  1. Calculate the coordinates of the centre of mass of the system of particles. A thin uniform rigid wire of mass \(12 m\) connects the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D with straight line sections, as shown in Fig. 5.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-5_460_903_1338_573} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure}
  2. Calculate the coordinates of the centre of mass of the wire. The particles at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are now fixed to the wire to form a rigid object, \(R\).
  3. Calculate the \(x\)-coordinate of the centre of mass of \(R\).