Edexcel M5 (Mechanics 5) 2015 June

1. A particle \(P\) moves from the point \(A\), with position vector ( \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\) ) m , where \(a\) is a positive constant, to the point \(B\), with position vector ( \(- \mathbf { i } + a \mathbf { j } - \mathbf { k }\) ) m , under the action of a constant force \(\mathbf { F } = ( 2 \mathbf { i } + a \mathbf { j } - 3 \mathbf { k } )\) N. The work done by \(\mathbf { F }\), as it moves the particle \(P\) from \(A\) to \(B\), is 3 J . Find the value of \(a\).
   (6)

2. A particle \(P\) moves so that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, where \(0 \leqslant t < \frac { \pi } { 2 }\), satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - ( \tan t ) \mathbf { r } = ( \sin t ) \mathbf { i }$$ When \(t = 0 , \mathbf { r } = - \frac { 1 } { 2 } \mathbf { i }\).
Find \(\mathbf { r }\) in terms of \(t\).

1. A rigid body is in equilibrium under the action of three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 } \mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act at the points with position vectors \(\mathbf { r } _ { 1 }\) and \(\mathbf { r } _ { 2 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N } \quad \mathbf { r } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { m } \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { N } \quad \mathbf { r } _ { 2 } = ( - \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\)
   1. Find the magnitude of \(\mathbf { F } _ { 3 }\)
   2. Find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(t\) is a scalar parameter.
      
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4. A particle \(P\), whose initial mass is \(m _ { 0 }\), is projected vertically upwards from the ground at time \(t = 0\) with speed \(\frac { g } { k }\), where \(k\) is a constant. As the particle moves upwards it gains mass by picking up small droplets of moisture from the atmosphere. The droplets are at rest before they are picked up. At time \(t\) the speed of \(P\) is \(v\) and its mass has increased to \(m _ { 0 } \mathrm { e } ^ { k t }\). Assuming that, during the motion, the acceleration due to gravity is constant,

1. show that, while \(P\) is moving upwards, $$k v + \frac { \mathrm { d } v } { \mathrm {~d} t } = - g$$
2. find, in terms of \(m _ { 0 }\), the mass of \(P\) when it reaches its greatest height above the ground.
   (6)

1. A uniform circular disc, of mass \(m\) and radius \(a\), is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) is a tangent to the disc at the point \(A\). The centre \(O\) of the disc moves in a vertical plane that is perpendicular to \(L\).

The disc is held at rest with its plane horizontal and released.

1. Find the angular acceleration of the disc when it has turned through an angle of \(\frac { \pi } { 3 }\)
2. Find the magnitude of the component, in a direction perpendicular to the disc, of the force of the axis \(L\) acting on the disc at \(A\), when the disc has turned through an angle of \(\frac { \pi } { 3 }\)

1. A pendulum is modelled as a uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), which has a particle of mass \(2 m\) attached at \(B\). The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\). The vertical plane is perpendicular to the axis \(L\).
   1. Find the period of small oscillations of the pendulum about its position of stable equilibrium.
   The pendulum is hanging at rest in a vertical position, with \(B\) below \(A\), when it is given a horizontal impulse of magnitude \(J\). The impulse acts at \(B\) in a vertical plane which is perpendicular to the axis \(L\). Given that the pendulum turns through an angle of \(60 ^ { \circ }\) before first coming to instantaneous rest,
2. find \(J\).

7. (a) Find, using integration, the moment of inertia of a uniform solid hemisphere, of mass \(m\) and radius \(a\), about a diameter of its plane face.
[0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about a diameter is \(\frac { 1 } { 4 } m r ^ { 2 }\).]
(b) Hence find the moment of inertia of a uniform solid sphere, of mass \(M\) and radius \(a\), about a diameter.