Questions M2 (1537 questions)

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AQA M2 2012 January Q7
11 marks Standard +0.3
  1. Show that \(v ^ { 2 } = u ^ { 2 } - 4 a g\).
  2. The ratio of the tensions in the string when the bead is at the two points \(A\) and \(B\) is \(2 : 5\).
    1. Find \(u\) in terms of \(g\) and \(a\).
    2. Find the ratio \(u : v\).
AQA M2 2010 June Q7
12 marks Standard +0.8
  1. Draw a diagram to show the forces acting on the rod.
  2. Find the magnitude of the normal reaction force between the rod and the ground.
    1. Find the normal reaction acting on the rod at \(C\).
    2. Find the friction force acting on the rod at \(C\).
  4. In this position, the rod is on the point of slipping. Calculate the coefficient of friction between the rod and the peg.
    \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-15_2484_1709_223_153}
OCR M2 2008 January Q6
11 marks Standard +0.3
  1. Show that the tension in the string is 4.16 N , correct to 3 significant figures.
  2. Calculate \(\omega\).
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_510_417_1238_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lower part of the string is now attached to a point \(R\), vertically below \(P\). \(P B\) makes an angle \(30 ^ { \circ }\) with the vertical and \(R B\) makes an angle \(60 ^ { \circ }\) with the vertical. The bead \(B\) now moves in a horizontal circle of radius 1.5 m with constant speed \(v _ { \mathrm { m } } \mathrm { m } ^ { - 1 }\) (see Fig. 2).
    1. Calculate the tension in the string.
    2. Calculate \(v\).
OCR M2 2006 June Q6
11 marks Standard +0.3
  1. Calculate the tension in the string and hence find the angular speed of \(Q\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The particle \(Q\) on the plane is now fixed to a point 0.2 m from the hole at \(A\) and the particle \(P\) rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
  2. Calculate the tension in the string.
  3. Calculate the speed of \(P\).
OCR M2 2008 June Q5
8 marks Standard +0.3
  1. Show that the distance from the ball to the centre of mass of the toy is 10.7 cm , correct to 1 decimal place.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_312_1051_1509_587} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The toy lies on horizontal ground in a position such that the ball is touching the ground (see Fig. 2). Determine whether the toy is lying in equilibrium or whether it will move to a position where the rod is vertical.
OCR M2 2009 June Q5
11 marks Standard +0.3
  1. Fig. 1 Fig. 1 shows a uniform lamina \(B C D\) in the shape of a quarter circle of radius 6 cm . Show that the distance of the centre of mass of the lamina from \(B\) is 3.60 cm , correct to 3 significant figures. A uniform rectangular lamina \(A B D E\) has dimensions \(A B = 12 \mathrm {~cm}\) and \(A E = 6 \mathrm {~cm}\). A single plane object is formed by attaching the rectangular lamina to the lamina \(B C D\) along \(B D\) (see Fig. 2). The mass of \(A B D E\) is 3 kg and the mass of \(B C D\) is 2 kg . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-3_959_447_1123_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the object. The object is freely suspended at \(C\) and rests in equilibrium.
  3. Calculate the angle that \(A C\) makes with the vertical.
OCR M2 2015 June Q7
11 marks Standard +0.8
  1. Show that \(\mu = \frac { 2 } { 3 }\). A small object of weight \(a W \mathrm {~N}\) is placed on the ladder at its mid-point and the object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at its lowest point \(A\).
  2. Given that the system is in equilibrium, find the set of possible values of \(a\).
OCR MEI M2 2010 June Q2
18 marks Standard +0.3
  1. Calculate the coordinates of the centre of mass of the stand. A small object of mass 5 kg is fixed to the rod AB at a distance of 40 cm from A .
  2. Show that the coordinates of the centre of mass of the stand with the object are ( 22,68 ). The stand is tilted about the edge PQ until it is on the point of toppling. The angle through which the stand is tilted is called 'the angle of tilt'. This procedure is repeated about the edges QR and RS.
  3. Making your method clear, determine which edge requires the smallest angle of tilt for the stand to topple. The small object is removed. A light string is attached to the stand at A and pulled at an angle of \(50 ^ { \circ }\) to the downward vertical in the plane \(\mathrm { O } x y\) in an attempt to tip the stand about the edge RS.
  4. Assuming that the stand does not slide, find the tension in the string when the stand is about to turn about the edge RS.
OCR MEI M2 2016 June Q3
18 marks Standard +0.3
  1. Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
  2. Calculate the power of the tension in the string when the block has a speed of \(7 \mathrm {~ms} ^ { - 1 }\). Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is \(M \mathrm {~kg}\). The sides OP and PQ have thin rigid uniform reinforcements attached with masses \(0.6 M \mathrm {~kg}\) and \(0.4 M \mathrm {~kg}\), respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.
  1. Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces \(Y _ { \mathrm { Q } } \mathrm { N }\) and \(Y _ { \mathrm { R } } \mathrm { N }\) act at Q and R respectively.
  2. Calculate the value of \(Y _ { \mathrm { Q } }\) and show that \(Y _ { \mathrm { R } } = 120\). The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CA }\) and CD freely pin-jointed together at \(\mathrm { A } , \mathrm { B }\) and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces \(X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }\) acting at A and \(X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }\) acting at D . You are given that \(\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
    \end{figure}
  3. Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , including those internal to the rods, when the inn sign is hanging from the framework.
  4. Show that \(X _ { \mathrm { D } } = 261\).
  5. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of \(Y _ { \mathrm { D } }\) and \(Y _ { \mathrm { A } }\). [Your working in this part should correspond to your diagram in part (iii).]
Edexcel M2 Q3
11 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-02_650_1043_367_317} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 2.
  1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
  2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
  3. find the value of \(k\),
  4. find the coordinates of \(D\).
Edexcel M2 Q4
5 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-03_725_560_310_571} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Fig. 4. Given that the capacity of a carton has to be \(1030 \mathrm {~cm} ^ { 3 }\),
  1. express \(h\) in terms of \(x\),
  2. show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$
Edexcel M2 Q16
13 marks Moderate -0.8
16. \section*{Figure 3}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-08_581_575_395_609}
The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 3.
  1. Find the coordinates of the centre of the circle.
  2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
  4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.
Edexcel M2 Q20
14 marks Moderate -0.5
20. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$$ and that \(C\) passes through the point \(P ( 2,6 )\),
  1. find \(y\) in terms of \(x\).
  2. Verify that \(C\) passes through the point ( 4,0 ).
  3. Find an equation of the tangent to \(C\) at \(P\). The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
  4. Calculate the exact \(x\)-coordinate of \(Q\).
    21. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find \(\int y \mathrm {~d} x\).
22.
  1. Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
  2. Solve the simultaneous equations $$\begin{gathered} x = 2 y - 2 \\ x ^ { 2 } = y ^ { 2 } + 7 \end{gathered}$$
    1. The straight line \(l _ { 1 }\) with equation \(y = \frac { 3 } { 2 } x - 2\) crosses the \(y\)-axis at the point \(P\). The point \(Q\) has coordinates \(( 5 , - 3 )\).
    The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(Q\).
  1. Calculate the coordinates of the mid-point of \(P Q\).
  2. Find an equation for \(l _ { 2 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integer constants. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(R\).
  3. Calculate the exact coordinates of \(R\).
    24. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$
  1. Use integration to find \(y\) in terms of \(x\).
  2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).
    25. Find the set of values for \(x\) for which
  1. \(6 x - 7 < 2 x + 3\),
  2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
  3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).
    [0pt] [P1 June 2003 Question 2]
    26. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \(( 280 + x )\) phones will be sold in the second month, \(( 280 + 2 x )\) in the third month, and so on. Using this model with \(x = 5\), calculate
    1. the number of phones sold in the 36th month,
    2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
      Using the same model,
  2. find the least value of \(x\) required to achieve this target.
    [0pt] [P1 June 2003 Question 3]
    27. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
  1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integers. The straight line \(l _ { 2 }\) passes through the origin and has gradient - 4 .
  2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
  3. Find the exact coordinates of the mid-point of \(A C\).
    28. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
  2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
    [0pt] [P1 June 2003 Question 8*]
    29. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$
  1. Write down the first two terms of the series.
  2. Find the common difference of the series. Given that \(n = 50\),
  3. find the sum of the series.
30.
  1. Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c$$ where \(c\) is a constant.
  2. Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).
    31. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 1 = 0 \\ & x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{aligned}$$
    1. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.
    Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$ 33. \section*{Figure 1}
    \includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-15_668_748_358_699}
    The points \(A\) and \(B\) have coordinates \(( 2 , - 3 )\) and \(( 8,5 )\) respectively, and \(A B\) is a chord of a circle with centre \(C\), as shown in Fig. 1.
  1. Find the gradient of \(A B\). The point \(M\) is the mid-point of \(A B\).
  2. Find an equation for the line through \(C\) and \(M\). Given that the \(x\)-coordinate of \(C\) is 4 ,
  3. find the \(y\)-coordinate of \(C\),
  4. show that the radius of the circle is \(\frac { 5 \sqrt { } 17 } { 4 }\).
    34. The first three terms of an arithmetic series are \(p , 5 p - 8\), and \(3 p + 8\) respectively.
  1. Show that \(p = 4\).
  2. Find the value of the 40th term of this series.
  3. Prove that the sum of the first \(n\) terms of the series is a perfect square.
    35. $$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$
  1. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has no real solutions. Given that \(k = 4\),
  2. express \(\mathrm { f } ( x )\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found,
    36. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$
  1. Show that, when \(x = 8\), the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(9 \sqrt { } 2\). The curve \(C\) passes through the point \(( 4,30 )\).
  2. Using integration, find \(\mathrm { f } ( x )\).
    37. \section*{Figure 2}
    \includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-17_687_1074_351_539}
    Figure 2 shows the curve with equation \(y ^ { 2 } = 4 ( x - 2 )\) and the line with equation \(2 x - 3 y = 12\).
    The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).
  1. Write down the coordinates of \(A\).
  2. Find, using algebra, the coordinates of \(P\) and \(Q\).
  3. Show that \(\angle P A Q\) is a right angle.
    38. A sequence is defined by the recurrence relation $$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where \(a\) is a constant.
  1. Given that \(a = 20\) and \(u _ { 1 } = 3\), find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\), giving your answers to 2 decimal places.
  2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
    1. calculate the value of \(a\),
    2. write down the value of \(u _ { 5 }\).
      [0pt] [P2 January 2004 Question 2]
      39. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\).
  2. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).
    40. Giving your answers in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are rational numbers, find
  1. \(( 3 - \sqrt { } 8 ) ^ { 2 }\),
  2. \(\frac { 1 } { 4 - \sqrt { 8 } }\).
    41. The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,
  1. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
  2. form a quadratic inequality in \(x\).
  3. by solving your inequalities, find the set of possible values of \(x\).
    42. The curve \(C\) has equation \(y = x ^ { 2 } - 4\) and the straight line \(l\) has equation \(y + 3 x = 0\).
  1. In the space below, sketch \(C\) and \(l\) on the same axes.
  2. Write down the coordinates of the points at which \(C\) meets the coordinate axes.
  3. Using algebra, find the coordinates of the points at which \(l\) intersects \(C\).
    43. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$
  1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
  2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
AQA M2 Q1
Moderate -0.3
1 A uniform beam, \(A B\), has mass 20 kg and length 7 metres. A rope is attached to the beam at \(A\). A second rope is attached to the beam at the point \(C\), which is 2 metres from \(B\). Both of the ropes are vertical. The beam is in equilibrium in a horizontal position, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_298_906_756_552} Find the tensions in the two ropes.
AQA M2 Q2
Moderate -0.8
2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_346_340_1580_842}
  1. Show that the tension in the string is 22.6 N , correct to three significant figures.
  2. Find the speed of the particle.
AQA M2 Q3
Moderate -0.8
3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. State the range of values of the acceleration of the particle.
  2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).
    (4 marks)
AQA M2 Q4
Standard +0.3
4 A car has a maximum speed of \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is moving on a horizontal road. When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons.
  1. Show that the maximum power of the car is 52920 W .
  2. The car has mass 1200 kg . It travels, from rest, up a slope inclined at \(5 ^ { \circ }\) to the horizontal.
    1. Show that, when the car is travelling at its maximum speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) up the slope, $$V ^ { 2 } + 392 \sin 5 ^ { \circ } V - 1764 = 0$$
    2. Hence find \(V\).
AQA M2 Q5
Standard +0.3
5 A car, of mass 1600 kg , is travelling along a straight horizontal road at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the driving force is removed. The car then freewheels and experiences a resistance force. The resistance force has magnitude \(40 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car after it has been freewheeling for \(t\) seconds. Find an expression for \(v\) in terms of \(t\).
AQA M2 Q6
Standard +0.3
6 A particle \(P\), of mass \(m \mathrm {~kg}\), is placed at the point \(Q\) on the top of a smooth upturned hemisphere of radius 3 metres and centre \(O\). The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of \(2 \mathrm {~ms} ^ { - 1 }\). When the particle is on the surface of the hemisphere, the angle between \(O P\) and \(O Q\) is \(\theta\) and the particle has speed \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-005_419_1013_607_511}
  1. Show that \(v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )\).
  2. Find the value of \(\theta\) when the particle leaves the hemisphere.
AQA M2 Q7
Standard +0.3
7 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).
  1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
  2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 J .
  3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
    1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
    2. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.
AQA M2 Q8
Standard +0.3
8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).
  1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-017_385_239_669_881} Find the extension of the spring.
  2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
  3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
    1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
    2. Find the value of \(x\).
AQA M2 2007 January Q1
8 marks Moderate -0.8
1 A child, of mass 35 kg , slides down a slide in a water park. The child, starting from rest, slides from the point \(A\) to the point \(B\), which is 10 metres vertically below the level of \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_259_595_685_705}
  1. In a simple model, all resistance forces are ignored. Use an energy method to find the speed of the child at \(B\).
  2. State one resistance force that has been ignored in answering part (a).
  3. In fact, when the child slides down the slide, she reaches \(B\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the slide is 20 metres long and the sum of the resistance forces has a constant magnitude of \(F\) newtons, use an energy method to find the value of \(F\).
    (4 marks)
AQA M2 2007 January Q2
6 marks Moderate -0.8
2 A hotel sign consists of a uniform rectangular lamina of weight \(W\). The sign is suspended in equilibrium in a vertical plane by two vertical light chains attached to the sign at the points \(A\) and \(B\), as shown in the diagram. The edge containing \(A\) and \(B\) is horizontal. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_289_529_1859_726} The tensions in the chains attached at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively.
  1. Draw a diagram to show the forces acting on the sign.
  2. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(W\).
  3. Explain how you have used the fact that the lamina is uniform in answering part (b).
AQA M2 2007 January Q3
6 marks Moderate -0.3
3 A light inextensible string has length \(2 a\). One end of the string is attached to a fixed point \(O\) and a particle of mass \(m\) is attached to the other end. Initially, the particle is held at the point \(A\) with the string taut and horizontal. The particle is then released from rest and moves in a circular path. Subsequently, it passes through the point \(B\), which is directly below \(O\). The points \(O , A\) and \(B\) are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-3_426_437_575_772}
  1. Show that the speed of the particle at \(B\) is \(2 \sqrt { a g }\).
  2. Find the tension in the string as the particle passes through \(B\). Give your answer in terms of \(m\) and \(g\).
AQA M2 2007 January Q4
9 marks Standard +0.3
4 A uniform T-shaped lamina is formed by rigidly joining two rectangles \(A B C H\) and \(D E F G\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_748_652_456_644}
  1. Show that the centre of mass of the lamina is 26 cm from the edge \(A B\).
  2. Explain why the centre of mass of the lamina is 5 cm from the edge \(G F\).
  3. The point \(X\) is on the edge \(A B\) and is 7 cm from \(A\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_697_534_1576_753} The lamina is freely suspended from \(X\) and hangs in equilibrium.
    Find the angle between the edge \(A B\) and the vertical, giving your answer to the nearest degree.
    (4 marks)