Edexcel FM2 (Further Mechanics 2) 2022 June

1. Three particles of masses \(2 m , 3 m\) and \(k m\) are placed at the points with coordinates (3a, 2a), (a, -4a) and (-3a, 4a) respectively.

The centre of mass of the three particles lies at the point with coordinates \(( \bar { x } , \bar { y } )\).

1. 1. Find \(\bar { x }\) in terms of \(a\) and \(k\)
   2. Find \(\bar { y }\) in terms of \(a\) and \(k\) Given that the distance of the centre of mass of the three particles from the point ( 0,0 ) is \(\frac { 1 } { 3 } a\)
2. find the possible values of \(k\)

1. A cyclist and her cycle have a combined mass of 60 kg . The cyclist is moving along a straight horizontal road and is working at a constant rate of 200 W .

When she has travelled a distance \(x\) metres, her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the magnitude of the resistance to motion is \(3 v ^ { 2 } \mathrm {~N}\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 200 - 3 v ^ { 3 } } { 60 v ^ { 2 } }\) The distance travelled by the cyclist as her speed increases from \(2 \mathrm {~ms} ^ { - 1 }\) to \(4 \mathrm {~ms} ^ { - 1 }\) is \(D\) metres.
2. Find the exact value of \(D\)

3. \begin{figure}[h] \end{figure} Nine uniform rods are joined together to form the rigid framework \(A B C D E F A\), with \(A B = B C = D F = 3 a , B F = C D = D E = 4 a\) and \(A F = F E = C F = 5 a\), as shown in Figure 1. All nine rods lie in the same plane. The mass per unit length of each of the rods \(B F , C F\) and \(D F\) is twice the mass per unit length of each of the other six rods.

1. Find the distance of the centre of mass of the framework from \(A C\) The mass of the framework is \(M\). A particle of mass \(k M\) is attached to the framework at \(E\) to form a loaded framework. When the loaded framework is freely suspended from \(F\), it hangs in equilibrium with \(C E\) horizontal.
2. Find the exact value of \(k\)

4. \begin{figure}[h] \end{figure} A small smooth ring \(R\) of mass \(m\) is threaded onto a light inextensible string. One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to the fixed point \(B\) such that \(B\) is vertically above \(A\) and \(A B = 6 a\) The ring moves with constant angular speed \(\omega\) in a horizontal circle with centre \(A\). The string is taut and \(B R\) makes a constant angle \(\theta\) with the downward vertical, as shown in Figure 2. The ring is modelled as a particle.
Given that \(\tan \theta = \frac { 8 } { 15 }\)

1. find, in terms of \(m\) and \(g\), the magnitude of the tension in the string,
2. find \(\omega\) in terms of \(a\) and \(g\)

5. \begin{figure}[h] \end{figure} The uniform plane lamina shown in Figure 3 is formed from two squares, \(A B C O\) and \(O D E F\), and a sector \(O D C\) of a circle with centre \(O\). Both squares have sides of length \(3 a\) and \(A O\) is perpendicular to \(O F\). The radius of the sector is \(3 a\)
[0pt] [In part (a) you may use, without proof, any of the centre of mass formulae given in the formulae booklet.]

1. Show that the distance of the centre of mass of the sector \(O D C\) from \(O C\) is \(\frac { 4 a } { \pi }\)
2. Find the distance of the centre of mass of the lamina from \(F C\) The lamina is freely suspended from \(F\) and hangs in equilibrium with \(F C\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
3. Find the value of \(\theta\)

6. \begin{figure}[h] \end{figure} The shaded region shown in Figure 4 is bounded by the \(x\)-axis, the line with equation \(x = 9\) and the line with equation \(y = \frac { 1 } { 3 } x\). This shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm . The cone is non-uniform and the mass per unit volume of the cone at the point ( \(x , y , z\) ) is \(\lambda x \mathrm {~kg} \mathrm {~cm} ^ { - 3 }\), where \(0 \leqslant x \leqslant 9\) and \(\lambda\) is constant.

1. Find the distance of the centre of mass of the cone from its vertex. A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm , so that the centres of the two plane surfaces coincide. The weight of the cone is \(W\) newtons and the weight of the hemisphere is \(k W\) newtons.
   When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
2. Find the value of \(k\)

7. \begin{figure}[h] \end{figure} A package \(P\) of mass \(m\) is attached to one end of a string of length \(\frac { 2 a } { 5 }\). The other end of the string is attached to a fixed point \(O\). The package hangs at rest vertically below \(O\) with the string taut and is then projected horizontally with speed \(u\), as shown in Figure 5. When \(O P\) has turned through an angle \(\theta\) and the string is still taut, the tension in the string is \(T\) The package is modelled as a particle and the string as being light and inextensible.

1. Show that \(T = 3 m g \cos \theta - 2 m g + \frac { 5 m u ^ { 2 } } { 2 a }\) Given that \(P\) moves in a complete vertical circle with centre \(O\)
2. find, in terms of \(a\) and \(g\), the minimum possible value of \(u\) Given that \(u = 2 \sqrt { a g }\)
3. find, in terms of \(g\), the magnitude of the acceleration of \(P\) at the instant when \(O P\) is horizontal.
4. Apart from including air resistance, suggest one way in which the model could be refined to make it more realistic.

1. Throughout this question, use \(\boldsymbol { g } = \mathbf { 1 0 m ~ s } ^ { \mathbf { - 2 } }\)

A light elastic string has natural length 1.25 m and modulus of elasticity 25 N .
A particle \(P\) of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). Particle \(P\) hangs freely in equilibrium with \(P\) vertically below \(A\) The particle is then pulled vertically down to a point \(B\) and released from rest.

1. Show that, while the string is taut, \(P\) moves with simple harmonic motion with period \(\frac { \pi } { \sqrt { 10 } }\) seconds. The maximum kinetic energy of \(P\) during the subsequent motion is 2.5 J .
2. Show that \(A B = 2 \mathrm {~m}\) The particle returns to \(B\) for the first time \(T\) seconds after it was released from rest at \(B\)
3. Find the value of \(T\)