Questions — Edexcel (10514 questions)

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Edexcel M2 Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f5449ec3-ead0-464f-9d03-f225cd21bca6-3_390_725_191_575} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a uniform rod \(A B\) of mass 2 kg and length \(2 a\). The end \(A\) is attached by a smooth hinge to a fixed point on a vertical wall so that the rod can rotate freely in a vertical plane. A mass of 6 kg is placed at \(B\) and the rod is held in a horizontal position by a light string joining the midpoint of the rod to a point \(C\) on the wall, vertically above \(A\). The string is inclined at an angle of \(60 ^ { \circ }\) to the wall.
  1. Show that the tension in the string is \(28 g\).
  2. Find the magnitude and direction of the force exerted by the hinge on the rod, giving your answers correct to 3 significant figures.
Edexcel M2 Q5
13 marks Standard +0.3
5. A particle \(P\) moves in a straight line with an acceleration of \(( 6 t - 10 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) at time \(t\) seconds. Initially \(P\) is at \(O\), a fixed point on the line, and has velocity \(3 \mathrm {~ms} ^ { - 1 }\).
  1. Find the values of \(t\) for which the velocity of \(P\) is zero.
  2. Show that, during the first two seconds, \(P\) travels a distance of \(6 \frac { 26 } { 27 } \mathrm {~m}\).
Edexcel M2 Q6
14 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f5449ec3-ead0-464f-9d03-f225cd21bca6-4_412_770_198_507} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A football player strikes a ball giving it an initial speed of \(14 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal as shown in Figure 2. At the instant he strikes the ball it is 0.6 m vertically above the point \(P\) on the ground. The trajectory of the ball is in a vertical plane containing \(P\) and \(M\), the middle of the goal-line. The distance between \(P\) and \(M\) is 12 m and the ground is horizontal. Given that the ball passes over the point \(M\) without bouncing,
  1. find, to the nearest degree, the minimum value of \(\alpha\). Given that the crossbar of the goal is 2.4 m above \(M\) and that \(\tan \alpha = \frac { 4 } { 3 }\),
  2. show that the ball passes 4.2 m vertically above the crossbar.
Edexcel M2 Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f5449ec3-ead0-464f-9d03-f225cd21bca6-5_536_848_191_397} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a hotel 'key' consisting of a rectangle \(O A B D\), where \(O A = 8 \mathrm {~cm}\) and \(O D = 4 \mathrm {~cm}\), joined to a semicircle whose diameter \(B C\) is 4 cm long. The thickness of the key is negligible and the same material is used throughout. The key is modelled as a uniform lamina.
Using this model,
  1. find, correct to 3 significant figures, the distance of the centre of mass from
    1. OD ,
    2. \(O A\). A small circular hole of negligible diameter is made at the mid-point of \(B C\) so that the key can be hung on a smooth peg. When the key is freely suspended from the peg,
  2. find, correct to 3 significant figures, the acute angle made by \(O A\) with the vertical.
Edexcel M2 Q1
4 marks Moderate -0.8
  1. A ball of mass 0.6 kg bounces against a wall and is given an impulse of \(( 12 \mathbf { i } - 9 \mathbf { j } ) \mathrm { Ns }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors. The velocity of the particle after the impact is \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the velocity of the particle before the impact.
(4 marks)
Edexcel M2 Q2
6 marks Moderate -0.3
2. A particle \(P\) moves along the \(x\)-axis such that its displacement, \(x\) metres, from the origin \(O\) at time \(t\) seconds is given by $$x = 2 + t - \frac { 1 } { 10 } \mathrm { e } ^ { t }$$
  1. Find the distance of \(P\) from \(O\) when \(t = 0\).
  2. Find, correct to 1 decimal place, the value of \(t\) when the velocity of \(P\) is zero.
    (4 marks)
Edexcel M2 Q3
11 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-2_421_474_1080_664} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a ladder of mass 20 kg and length 6 m leaning against a rough vertical wall with its lower end on smooth horizontal ground. The ladder is prevented from slipping along the ground by a light rope which is attached to the ladder 2 m from its bottom end and fastened to the wall so that the rope is horizontal and perpendicular to the wall. The ladder is at an angle \(\theta\) to the horizontal where \(\tan \theta = \frac { 5 } { 2 }\) and the coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\).
  1. Draw a diagram showing all the forces acting on the ladder.
  2. Show that the magnitude of the tension in the rope is \(5 g\). A man wishes to use the ladder but fears the rope will snap as he climbs the ladder.
  3. Suggest, giving a reason for your answer, a more suitable position for the rope.
    (2 marks)
Edexcel M2 Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-3_222_350_242_788} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows an earring consisting of a uniform wire \(A B C D\) of length \(6 a\) bent to form right angles at \(B\) and \(C\) such that \(A B\) and \(C D\) are of length \(2 a\) and \(a\) respectively.
  1. Find, in terms of \(a\), the distance of the centre of mass from
    1. \(\quad A B\),
    2. \(B C\). The earring is to be worn such that it hangs in equilibrium suspended from the point \(A\).
  2. Find, to the nearest degree, the angle made by \(A B\) with the downward vertical.
    (4 marks)
Edexcel M2 Q5
13 marks Moderate -0.3
5. A lorry of mass 40 tonnes moves up a straight road inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 20 }\). The lorry moves at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In a model of the motion of the lorry, the non-gravitational resistance to motion is assumed to be constant and of magnitude 4400 N .
  1. Show that the engine of the lorry is working at a rate of 480 kW . The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion is assumed to remain unchanged.
  2. Find the initial acceleration of the lorry.
  3. Find, correct to 3 significant figures, the maximum speed of the lorry.
  4. Using your answer to part (c), comment on the suitability of the modelling assumption.
Edexcel M2 Q6
13 marks Moderate -0.3
6. Particle \(S\) of mass \(2 M\) is moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it collides directly with a particle \(T\) of mass \(5 M\) which is lying at rest on the plane. The coefficient of restitution between \(S\) and \(T\) is \(\frac { 3 } { 4 }\). Given that the speed of \(T\) after the collision is \(4 \mathrm {~ms} ^ { - 1 }\),
  1. find \(U\). As a result of the collision, \(T\) is projected horizontally from the top of a building of height 19.6 m and falls freely under gravity. \(T\) strikes the ground at the point \(X\) as shown in Figure 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-4_663_928_740_523} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. Find the time taken for \(T\) to reach \(X\).
  3. Show that the angle between the horizontal and the direction of motion of \(T\), just before it strikes the ground at \(X\), is \(78.5 ^ { \circ }\) correct to 3 significant figures.
    (4 marks)
Edexcel M2 Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-5_495_604_214_580} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Figure 4 shows a particle \(P\) projected from the point \(A\) up the line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 4 } { 5 } . P\) is projected with speed \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction between \(P\) and the plane is \(\frac { 4 } { 7 }\). Given that \(P\) first comes to rest at point \(B\),
  1. use the Work-Energy principle to show that the distance \(A B\) is 1.4 m . The particle then slides back down the plane.
  2. Find, correct to 2 significant figures, the speed of \(P\) when it returns to \(A\).
Edexcel M2 Q1
5 marks Moderate -0.3
  1. An ice hockey puck of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors, when it is struck by a stick. After the impact, the puck travels with velocity \(( 13 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the magnitude of the impulse exerted by the stick on the puck.
(5 marks)
Edexcel M2 Q2
8 marks Moderate -0.3
2. A car of mass 1 tonne is climbing a hill inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 7 }\). When the car passes a point \(X\) on the hill, it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the car passes the point \(Y , 200 \mathrm {~m}\) further up the hill, it has speed \(10 \mathrm {~ms} ^ { - 1 }\). In a preliminary model of the situation, the car engine is assumed only to be doing work against gravity. Using this model,
  1. find the change in the total mechanical energy of the car as it moves from \(X\) to \(Y\).
    (6 marks)
    In a more sophisticated model, the car engine is also assumed to work against other forces.
  2. Write down two other forces which this model might include.
    (2 marks)
Edexcel M2 Q3
8 marks Moderate -0.8
3. A particle moves along a straight horizontal track such that its displacement, \(s\) metres, from a fixed point \(O\) on the line after \(t\) seconds is given by $$s = 2 t ^ { 3 } - 13 t ^ { 2 } + 20 t$$
  1. Find the values of \(t\) for which the particle is at \(O\).
  2. Find the values of \(t\) at which the particle comes instantaneously to rest.
Edexcel M2 Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_378_730_196_609} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a uniform rod \(A B\) of length 2 m and mass 6 kg inclined at an angle of \(30 ^ { \circ }\) to the horizontal with \(A\) on smooth horizontal ground and \(B\) supported by a rough peg. The rod is in limiting equilibrium and the coefficient of friction between \(B\) and the peg is \(\mu\).
  1. Find, in terms of \(g\), the magnitude of the reactions at \(A\) and \(B\).
  2. Show that \(\mu = \frac { 1 } { \sqrt { 3 } }\).
Edexcel M2 Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_405_718_1169_555} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} During a cricket match, a batsman hits the ball giving it an initial velocity of \(22 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 7 } { 8 }\). When the batsman strikes the ball it is 1.6 metres above the ground, as shown in Figure 2, and it subsequently moves freely under gravity.
  1. Find, correct to 3 significant figures, the maximum height above the ground reached by the ball. The ball is caught by a fielder when it is 0.2 metres above the ground.
  2. Find the length of time for which the ball is in the air. Assuming that the fielder who caught the ball ran at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  3. find, correct to 3 significant figures, the maximum distance that the fielder could have been from the ball when it was struck.
Edexcel M2 Q6
16 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-4_433_282_196_726} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a uniform rectangular lamina \(A B C D\) of mass \(8 m\) in which the sides \(A B\) and \(B C\) are of length \(a\) and \(2 a\) respectively. Particles of mass \(2 m , 6 m\) and \(4 m\) are fixed to the lamina at the points \(A , B\) and \(D\) respectively.
  1. Write down the distance of the centre of mass from \(A D\).
  2. Show that the distance of the centre of mass from \(A B\) is \(\frac { 4 } { 5 } a\). Another particle of mass \(k m\) is attached to the lamina at the point \(B\).
  3. Show that the distance of the centre of mass from \(A D\) is now given by \(\frac { ( 10 + k ) a } { 20 + k }\).
    (4 marks)
    Given that when the lamina is suspended freely from the point \(A\) the side \(A B\) makes an angle of \(45 ^ { \circ }\) with the vertical,
  4. find the value of \(k\).
Edexcel M2 Q7
17 marks Standard +0.3
7. Particle \(A\) of mass 7 kg is moving with speed \(u _ { 1 }\) on a smooth horizontal surface when it collides directly with particle \(B\) of mass 4 kg moving in the same direction as \(A\) with speed \(u _ { 2 }\). After the impact, \(A\) continues to move in the same direction but its speed has been halved. Given that the coefficient of restitution between the particles is \(e\),
  1. show that \(8 u _ { 2 } ( e + 1 ) = u _ { 1 } ( 8 e - 3 )\). Given also that \(u _ { 1 } = 14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { 2 } = 3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  2. find \(e\),
  3. show that the percentage of the kinetic energy of the particles lost as a result of the impact is \(9.6 \%\), correct to 2 significant figures.
Edexcel M3 Q1
7 marks Standard +0.8
  1. One end of a light inextensible string of length \(2 r \mathrm {~m}\) is attached to a fixed point \(O\). A particle of mass \(m \mathrm {~kg}\) is attached to the other end \(Q\) of the string, so that it can move in a vertical plane. The string is held taut and horizontal and the particle is projected vertically downwards with a speed \(\sqrt { } ( g r ) \mathrm { ms } ^ { - 1 }\). When the string is vertical it begins to wrap round a small, smooth peg \(X\) at a distance \(r \mathrm {~m}\) vertically below \(O\). The particle continues to move.
    1. Find the speed of the particle when it reaches \(O\), in terms of \(g\) and \(r\).
    2. Show that, when \(Q X\) is horizontal, the tension in the string is 3 mgN .
    3. A particle moving along the \(x\)-axis describes simple harmonic motion about the origin \(O\). The period of its motion is \(\frac { \pi } { 2 }\) seconds. When it is at a distance 1 m from \(O\), its speed is \(3 \mathrm {~ms} ^ { - 1 }\). Calculate
    1. the amplitude of its motion,
    2. the maximum acceleration of the particle,
    3. the least time that it takes to move from \(O\) to a point 0.25 m from \(O\).
    4. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length \(8 l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The two ends of the string are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(A B = 81 \mathrm {~m} . P\) is released from rest at the mid-point of \(A B\).
    1. If \(P\) comes to instantaneous rest at a depth \(3 / \mathrm { m }\) below \(A B\), find an expression for \(\lambda\) in terms of \(m\) and \(g\).
    2. Using this value of \(\lambda\), show that the speed \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) when it passes through the point \(2 l \mathrm {~m}\) below \(A B\) is given by \(v ^ { 2 } = 4 ( 24 \sqrt { 5 } - 53 ) g l\).
    3. A particle \(P\) of mass 0.8 kg moves along a straight line \(O L\) and is acted on by a resistive force of magnitude \(R \mathrm {~N}\) directed towards the fixed point \(O\). When the displacement of \(P\) from \(O\) is \(x \mathrm {~m} , R = \frac { 0 \cdot 8 x v ^ { 2 } } { 1 + x ^ { 2 } }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) at that instant.
    1. Write down a differential equation for the motion of \(P\).
    Given that \(v = 2\) when \(x = 0\),
  2. find the speed with which \(P\) passes through the point \(A\), where \(O A = 1 \mathrm {~m}\). \section*{MECHANICS 3 (A) TEST PAPER 3 Page 2}
Edexcel M3 Q5
10 marks Standard +0.3
  1. The diagram shows a uniform solid right circular cone of mass \(m \mathrm {~kg}\), height \(h \mathrm {~m}\) and base radius \(r \mathrm {~m}\) suspended by two vertical strings attached to the points \(P\) and \(Q\) on the circumference of the base. The vertex \(O\) of the cone is vertically below \(P\).
    1. Show that the tension in the string attached at \(Q\) is \(\frac { 3 m g } { 8 } \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_296_277_269_1668}
    2. Find, in terms of \(m\) and \(g\), the tension in the other string.
    3. Two identical particles \(P\) and \(Q\) are connected by a light inextensible string passing through a small smooth-edged hole in a smooth table, as shown. \(P\) moves on the table in a horizontal circle of radius 0.2 m and \(Q\) hangs at rest. \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_309_430_859_1476}
    1. Calculate the number of revolutions made per minute by \(P\).
      (5 marks) \(Q\) is now also made to move in a horizontal circle of radius 0.2 m below the table. The part of the string between \(Q\) and the table makes an angle of \(45 ^ { \circ }\) with the vertical.
    2. Show that the numbers of revolutions per minute made by \(P\) and \(Q\) respectively are in the ratio \(2 ^ { 1 / 4 } : 1\). \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_293_428_1213_1499}
    3. A particle \(P\) of mass \(m \mathrm {~kg}\) is fixed to one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(k m g \mathrm {~N}\). The other end of the string is fixed to a point \(X\) on a horizontal plane. \(P\) rests at \(O\), where \(O X = l \mathrm {~m}\), with the string just taut. It is then pulled away from \(X\) through a distance \(\frac { 3 l } { 4 } \mathrm {~m}\) and released from rest. On this side of \(O\), the plane is smooth.
    1. Show that, as long as the string is taut, \(P\) performs simple harmonic motion.
    2. Given that \(P\) first returns to \(O\) with speed \(\sqrt { } ( g l ) \mathrm { ms } ^ { - 1 }\), find the value of \(k\).
    3. On the other side of \(O\) the plane is rough, the coefficient of friction between \(P\) and the plane being \(\mu\). If \(P\) does not reach \(X\) in the subsequent motion, show that \(\mu > \frac { 1 } { 2 }\). ( 4 marks)
    4. If, further, \(\mu = \frac { 3 } { 4 }\), show that the time which elapses after \(P\) is released and before it comes to rest is \(\frac { 1 } { 24 } ( 9 \pi + 32 ) \sqrt { \frac { l } { g } }\) s.
      (6 marks)
Edexcel M3 Q1
7 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-2_341_652_217_621} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle of mass 0.6 kg is attached to one end of a light elastic spring of natural length 1 m and modulus of elasticity 30 N . The other end of the spring is fixed to a point \(O\) which lies on a smooth plane inclined at an angle \(\alpha\) to the horizontal where \(\tan \alpha = \frac { 3 } { 4 }\) as shown in Figure 1. The particle is held at rest on the slope at a point 1.2 m from \(O\) down the line of greatest slope of the plane.
  1. Find the tension in the spring.
  2. Find the initial acceleration of the particle.
Edexcel M3 Q2
7 marks Standard +0.3
2. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis under the action of a single force directed away from the origin \(O\). When \(P\) is \(x\) metres from \(O\), the magnitude of the force is \(3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}\) and \(P\) has a speed of \(v \mathrm {~ms} ^ { - 1 }\). Given that when \(x = 1 , P\) is moving away from \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),
  1. find an expression for \(v ^ { 2 }\) in terms of \(x\),
  2. show that when \(x = 4 , P\) has a speed of \(7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 1 decimal place.
Edexcel M3 Q3
10 marks Standard +0.3
3. A particle is performing simple harmonic motion along a straight line between the points \(A\) and \(B\) where \(A B = 8 \mathrm {~m}\). The period of the motion is 12 seconds.
  1. Find the maximum speed of the particle in terms of \(\pi\). The points \(P\) and \(Q\) are on the line \(A B\) at distances of 3 m and 6 m respectively from \(A\).
  2. Find, correct to 3 significant figures, the time it takes for the particle to travel directly from \(P\) to \(Q\).
    (6 marks)
Edexcel M3 Q4
11 marks Standard +0.3
4. Whilst in free-fall a parachutist falls vertically such that his velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when he is \(x\) metres below his initial position is given by $$v ^ { 2 } = k g \left( 1 - \mathrm { e } ^ { - \frac { 2 x } { k } } \right) ,$$ where \(k\) is a constant.
Given that he experiences an acceleration of \(\mathrm { f } \mathrm { m } \mathrm { s } ^ { - 2 }\),
  1. show that \(f = g \mathrm { e } ^ { - \frac { 2 x } { k } }\). After falling a large distance, his velocity is constant at \(49 \mathrm {~ms} ^ { - 1 }\).
  2. Find the value of \(k\).
  3. Hence, express \(f\) in the form ( \(\lambda - \mu v ^ { 2 }\) ) where \(\lambda\) and \(\mu\) are constants which you should find.
    (4 marks)
Edexcel M3 Q5
13 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-3_588_291_1126_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A firework is modelled as a uniform solid formed by joining the plane surface of a right circular cone of height \(2 r\) and base radius \(r\), to one of the plane surfaces of a cylinder of height \(h\) and base radius \(r\) as shown in Figure 2. Using this model,
  1. show that the distance of the centre of mass of the firework from its plane base is $$\frac { 3 h ^ { 2 } + 4 h r + 2 r ^ { 2 } } { 2 ( 3 h + 2 r ) }$$ The firework is to be launched from rough ground inclined at an angle \(\alpha\) to the horizontal. Given that the firework does not slip or topple and that \(h = 4 r\),
  2. Find, correct to the nearest degree, the maximum value of \(\alpha\).