Questions M3 (796 questions)

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Edexcel M3 2007 January Q2
5 marks Challenging +1.3
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-03_513_399_303_785}
\end{figure} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac { 1 } { 3 }\). The cone is freely suspended by a string attached at a point \(A\) on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta ^ { \circ }\) with the upward vertical, as shown in Figure 1. Find, to one decimal place, the value of \(\theta\).
Edexcel M3 2007 January Q3
9 marks Standard +0.8
3. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity 3.6 mg . The other end of the string is fixed at a point \(O\) on a rough horizontal table. The particle is projected along the surface of the table from \(O\) with speed \(\sqrt { } ( 2 a g )\). At its furthest point from \(O\), the particle is at the point \(A\), where \(O A = \frac { 4 } { 3 } a\).
  1. Find, in terms of \(m , g\) and \(a\), the elastic energy stored in the string when \(P\) is at \(A\).
  2. Using the work-energy principle, or otherwise, find the coefficient of friction between \(P\) and the table.
Edexcel M3 2007 January Q4
13 marks Standard +0.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-05_574_510_324_726}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a point \(O\). The point \(A\) is vertically below \(O\), and \(O A = a\). The particle is projected horizontally from \(A\) with speed \(\sqrt { } ( 3 a g )\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) and the string is still taut, the tension in the string is \(T\) and the speed of \(P\) is \(v\), as shown in Figure 2.
  1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
  2. Show that \(T = ( 1 - 3 \cos \theta ) m g\). The string becomes slack when \(P\) is at the point \(B\).
  3. Find, in terms of \(a\), the vertical height of \(B\) above \(A\). After the string becomes slack, the highest point reached by \(P\) is \(C\).
  4. Find, in terms of \(a\), the vertical height of \(C\) above \(B\).
Edexcel M3 2007 January Q5
13 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-07_531_691_299_657}
\end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.
  1. Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
  2. Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\). Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
  3. find, in terms of \(m\) and \(g\), the tension in the string.
Edexcel M3 2007 January Q6
13 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-09_515_1015_319_477}
\end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\), as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid \(S\) has the shape made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Show that the centre of mass of \(S\) is \(\frac { 2 } { 7 } \mathrm {~m}\) from its larger plane face. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-09_616_431_1420_778}
    \end{figure} A sporting trophy \(T\) is a uniform solid hemisphere \(H\) joined to the solid \(S\). The hemisphere has radius \(\frac { 1 } { 2 } \mathrm {~m}\) and its plane face coincides with the larger plane face of \(S\), as shown in Figure 5. Both \(H\) and \(S\) are made of the same material.
  2. Find the distance of the centre of mass of \(T\) from its plane face.
Edexcel M3 2007 January Q7
16 marks Challenging +1.2
  1. A particle \(P\) of mass 0.25 kg is attached to one end of a light elastic string. The string has natural length 0.8 m and modulus of elasticity \(\lambda \mathrm { N }\). The other end of the string is attached to a fixed point \(A\). In its equilibrium position, \(P\) is 0.85 m vertically below \(A\).
    1. Show that \(\lambda = 39.2\).
    The particle is now displaced to a point \(B , 0.95 \mathrm {~m}\) vertically below \(A\), and released from rest.
  2. Prove that, while the string remains stretched, \(P\) moves with simple harmonic motion of period \(\frac { \pi } { 7 } \mathrm {~s}\).
  3. Calculate the speed of \(P\) at the instant when the string first becomes slack. The particle first comes to instantaneous rest at the point \(C\).
  4. Find, to 3 significant figures, the time taken for \(P\) to move from \(B\) to \(C\).
Edexcel M3 2008 January Q1
6 marks Moderate -0.3
  1. A light elastic string of natural length 0.4 m has one end \(A\) attached to a fixed point. The other end of the string is attached to a particle \(P\) of mass 2 kg . When \(P\) hangs in equilibrium vertically below \(A\), the length of the string is 0.56 m .
    1. Find the modulus of elasticity of the string.
    A horizontal force is applied to \(P\) so that it is held in equilibrium with the string making an angle \(\theta\) with the downward vertical. The length of the string is now 0.72 m .
  2. Find the angle \(\theta\).
Edexcel M3 2008 January Q2
8 marks Standard +0.8
2. A particle \(P\) of mass 0.1 kg moves in a straight line on a smooth horizontal table. When \(P\) is a distance \(x\) metres from a fixed point \(O\) on the line, it experiences a force of magnitude \(\frac { 16 } { 5 x ^ { 2 } } \mathrm {~N}\) away from \(O\) in the direction \(O P\). Initially \(P\) is at a point 2 m from \(O\) and is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when \(P\) first comes to rest.
Edexcel M3 2008 January Q3
8 marks Standard +0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{39c2d25a-a39b-4eb9-a17b-6e741ab5ae98-04_519_709_315_603}
\end{figure} A uniform solid \(S\) is formed by taking a uniform solid right circular cone, of base radius \(2 r\) and height \(2 h\), and removing the cone, with base radius \(r\) and height \(h\), which has the same vertex as the original cone, as shown in Figure 1.
  1. Show that the distance of the centre of mass of \(S\) from its larger plane face is \(\frac { 11 } { 28 } h\). The solid \(S\) lies with its larger plane face on a rough table which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The table is sufficiently rough to prevent \(S\) from slipping. Given that \(h = 2 r\),
  2. find the greatest value of \(\theta\) for which \(S\) does not topple.
Edexcel M3 2008 January Q4
10 marks Standard +0.8
  1. A particle \(P\) of mass \(m\) lies on a smooth plane inclined at an angle \(30 ^ { \circ }\) to the horizontal. The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m g\). The other end of the string is attached to a fixed point \(O\) on the plane. The particle \(P\) is in equilibrium at the point \(A\) on the plane and the extension of the string is \(\frac { 1 } { 4 } a\). The particle \(P\) is now projected from \(A\) down a line of greatest slope of the plane with speed \(V\). It comes to instantaneous rest after moving a distance \(\frac { 1 } { 2 } a\).
By using the principle of conservation of energy,
  1. find \(V\) in terms of \(a\) and \(g\),
  2. find, in terms of \(a\) and \(g\), the speed of \(P\) when the string first becomes slack.
Edexcel M3 2008 January Q5
12 marks Standard +0.8
5. A car of mass \(m\) moves in a circular path of radius 75 m round a bend in a road. The maximum speed at which it can move without slipping sideways on the road is \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that this section of the road is horizontal,
  1. show that the coefficient of friction between the car and the road is 0.6 . The car comes to another bend in the road. The car's path now forms an arc of a horizontal circle of radius 44 m . The road is banked at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between the car and the road is again 0.6. The car moves at its maximum speed without slipping sideways.
  2. Find, as a multiple of \(m g\), the normal reaction between the car and road as the car moves round this bend.
  3. Find the speed of the car as it goes round this bend.
Edexcel M3 2008 January Q6
15 marks Challenging +1.2
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{39c2d25a-a39b-4eb9-a17b-6e741ab5ae98-09_357_606_315_717}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). At time \(t = 0 , P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 5 } { 2 } g a \right)\) from a point \(A\) which is at the same level as \(O\) and a distance \(a\) from \(O\). When the string has turned through an angle \(\theta\) and the string is still taut, the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 2.
  1. Show that \(v ^ { 2 } = \frac { g a } { 2 } ( 5 + 4 \sin \theta )\).
  2. Find \(T\) in terms of \(m , g\) and \(\theta\). The string becomes slack when \(\theta = \alpha\).
  3. Find the value of \(\alpha\). The particle is projected again from \(A\) with the same velocity as before. When \(P\) is at the same level as \(O\) for the first time after leaving \(A\), the string meets a small smooth peg \(B\) which has been fixed at a distance \(\frac { 1 } { 2 } a\) from \(O\). The particle now moves on an arc of a circle centre \(B\). Given that the particle reaches the point \(C\), which is \(\frac { 1 } { 2 } a\) vertically above the point \(B\), without the string going slack,
  4. find the tension in the string when \(P\) is at the point \(C\).
Edexcel M3 2008 January Q7
16 marks Challenging +1.3
7. A particle \(P\) of mass 2 kg is attached to one end of a light elastic string, of natural length 1 m and modulus of elasticity 98 N . The other end of the string is attached to a fixed point \(A\). When \(P\) hangs freely below \(A\) in equilibrium, \(P\) is at the point \(E , 1.2 \mathrm {~m}\) below \(A\). The particle is now pulled down to a point \(B\) which is 0.4 m vertically below \(E\) and released from rest.
  1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion about \(E\) with period \(\frac { 2 \pi } { 7 } \mathrm {~s}\).
  2. Find the greatest magnitude of the acceleration of \(P\) while the string is taut.
  3. Find the speed of \(P\) when the string first becomes slack.
  4. Find, to 3 significant figures, the time taken, from release, for \(P\) to return to \(B\) for the first time.
Edexcel M3 2009 January Q1
7 marks Standard +0.3
  1. A particle \(P\) of mass 3 kg is moving in a straight line. At time \(t\) seconds, \(0 \leqslant t \leqslant 4\), the only force acting on \(P\) is a resistance to motion of magnitude \(\left( 9 + \frac { 15 } { ( t + 1 ) ^ { 2 } } \right) \mathrm { N }\). At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 4 , v = 0\).
Find the value of \(v\) when \(t = 0\).
Edexcel M3 2009 January Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-03_467_622_242_635} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a horizontal force of magnitude \(\frac { 4 } { 3 } m g\) applied to \(P\). This force acts in the vertical plane containing the string, as shown in Figure 1. Find (a) the tension in the string,
(b) the elastic energy stored in the string.
Edexcel M3 2009 January Q3
7 marks Moderate -0.3
3. A rough disc rotates about its centre in a horizontal plane with constant angular speed 80 revolutions per minute. A particle \(P\) lies on the disc at a distance 8 cm from the centre of the disc. The coefficient of friction between \(P\) and the disc is \(\mu\). Given that \(P\) remains at rest relative to the disc, find the least possible value of \(\mu\).
Edexcel M3 2009 January Q4
11 marks Standard +0.3
4. A small shellfish is attached to a wall in a harbour. The rise and fall of the water level is modelled as simple harmonic motion and the shellfish as a particle. On a particular day the minimum depth of water occurs at 1000 hours and the next time that this minimum depth occurs is at 2230 hours. The shellfish is fixed in a position 5 m above the level of the minimum depth of the water and 11 m below the level of the maximum depth of the water. Find
  1. the speed, in metres per hour, at which the water level is rising when it reaches the shellfish,
  2. the earliest time after 1000 hours on this day at which the water reaches the shellfish.
Edexcel M3 2009 January Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-07_311_716_249_612} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} One end \(A\) of a light elastic string, of natural length \(a\) and modulus of elasticity \(6 m g\), is fixed at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the string. Initially \(B\) is held at rest with the string lying along a line of greatest slope of the plane, with \(B\) below \(A\) and \(A B = a\). The ball is released and comes to instantaneous rest at a point \(C\) on the plane, as shown in Figure 2. Find
  1. the length \(A C\),
  2. the greatest speed attained by \(B\) as it moves from its initial position to \(C\).
Edexcel M3 2009 January Q6
14 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-09_433_376_242_781} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The region \(R\) is bounded by part of the curve with equation \(y = 4 - x ^ { 2 }\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 3. The unit of length on both axes is one metre. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Show that the centre of mass of \(S\) is \(\frac { 5 } { 8 } \mathrm {~m}\) from \(O\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-09_702_584_1138_676} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a cross section of a uniform solid \(P\) consisting of two components, a solid cylinder \(C\) and the solid \(S\). The cylinder \(C\) has radius 4 m and length \(l\) metres. One end of \(C\) coincides with the plane circular face of \(S\). The point \(A\) is on the circumference of the circular face common to \(C\) and \(S\). When the solid \(P\) is freely suspended from \(A\), the solid \(P\) hangs with its axis of symmetry horizontal.
  2. Find the value of \(l\).
Edexcel M3 2009 January Q7
15 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-11_671_1077_276_429} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle is projected from the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g a )\). The particle leaves the surface of the sphere at the point \(C\), where \(\angle A O C = \theta\), and strikes the plane at the point \(P\), as shown in Figure 5.
  1. Show that \(\cos \theta = \frac { 3 } { 4 }\).
  2. Find the angle that the velocity of the particle makes with the horizontal as it reaches \(P\).
Edexcel M3 2010 January Q1
7 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis. At time \(t\) seconds, \(P\) is moving under the action of a single force of magnitude \([ 4 + \cos ( \pi t ) ] \mathrm { N }\), directed away from the origin. When \(t = 1\), the particle \(P\) is moving away from the origin with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find the speed of \(P\) when \(t = 1.5\), giving your answer to 3 significant figures.
Edexcel M3 2010 January Q2
9 marks Standard +0.3
2. A particle \(P\) moves in a straight line with simple harmonic motion of period 2.4 s about a fixed origin \(O\). At time \(t\) seconds the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). When \(t = 0 , P\) is at \(O\). When \(t = 0.4 , v = 4\). Find
  1. the greatest speed of \(P\),
  2. the magnitude of the greatest acceleration of \(P\).
Edexcel M3 2010 January Q3
10 marks Challenging +1.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_556_576_224_687} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A bowl \(B\) consists of a uniform solid hemisphere, of radius \(r\) and centre \(O\), from which is removed a solid hemisphere, of radius \(\frac { 2 } { 3 } r\) and centre \(O\), as shown in Figure 1.
  1. Show that the distance of the centre of mass of \(B\) from \(O\) is \(\frac { 65 } { 152 } r\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_526_1014_1292_478} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The bowl \(B\) has mass \(M\). A particle of mass \(k M\) is attached to a point \(P\) on the outer rim of \(B\). The system is placed with a point \(C\) on its outer curved surface in contact with a horizontal plane. The system is in equilibrium with \(P , O\) and \(C\) in the same vertical plane. The line \(O P\) makes an angle \(\theta\) with the horizontal as shown in Figure 2. Given that \(\tan \theta = \frac { 4 } { 5 }\),
  2. find the exact value of \(k\). January 2010
Edexcel M3 2010 January Q4
10 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-08_388_521_279_710} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of weight 40 N is attached to one end of a light elastic string of natural length 0.5 m . The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 30 N is applied to \(P\), as shown in Figure 3. The particle \(P\) is in equilibrium and the elastic energy stored in the string is 10 J . Calculate the length \(O P\).
Edexcel M3 2010 January Q5
11 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-10_590_858_242_575} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} One end \(A\) of a light inextensible string of length \(3 a\) is attached to a fixed point. A particle of mass \(m\) is attached to the other end \(B\) of the string. The particle is held in equilibrium at a distance \(2 a\) below the horizontal through \(A\), with the string taut. The particle is then projected with speed \(\sqrt { } ( 2 a g )\), in the direction perpendicular to \(A B\), in the vertical plane containing \(A\) and \(B\), as shown in Figure 4. In the subsequent motion the string remains taut. When \(A B\) is at an angle \(\theta\) below the horizontal, the speed of the particle is \(v\) and the tension in the string is \(T\).
  1. Show that \(v ^ { 2 } = 2 \operatorname { ag } ( 3 \sin \theta - 1 )\).
  2. Find the range of values of \(T\).