Edexcel M3 (Mechanics 3)

1. A particle \(P\) of mass 1.5 kg moves from rest at the origin such that at time \(t\) seconds it is subject to a single force of magnitude \(( 4 t + 3 ) \mathrm { N }\) in the direction of the positive \(x\)-axis.
   1. Find the magnitude of the impulse exerted by the force during the interval \(1 \leq t \leq 4\).
   Given that at time \(T\) seconds, \(P\) has a speed of \(22 \mathrm {~ms} ^ { - 1 }\),
2. find the value of \(T\) correct to 3 significant figures.

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest at the highest point \(A\) of a smooth sphere, centre \(O\), of radius 1.25 m which is fixed to a horizontal surface. When \(P\) is slightly disturbed it slides along the surface of the sphere. Whilst \(P\) is in contact with the sphere it has speed \(v \mathrm {~ms} ^ { - 1 }\) when \(\angle A O P = \theta\) as shown in Figure 1.

1. Show that \(v ^ { 2 } = 24.5 ( 1 - \cos \theta )\).
2. Find the value of \(\cos \theta\) when \(P\) leaves the surface of the sphere.

3. A car starts from rest at the point \(O\) and moves along a straight line. The car accelerates to a maximum velocity, \(V \mathrm {~ms} ^ { - 1 }\), before decelerating and coming to rest again at the point \(A\). The acceleration of the car during this journey, \(a \mathrm {~ms} ^ { - 2 }\), is modelled by the formula $$a = \frac { 500 - k x } { 150 }$$ where \(x\) is the distance in metres of the car from \(O\).
Using this model and given that the car is travelling at \(16 \mathrm {~ms} ^ { - 1 }\) when it is 40 m from \(O\),

1. find \(k\),
2. show that \(V = 41\), correct to 2 significant figures,
3. find the distance \(O A\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\).
\(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),

1. show that the work done against friction is 10.78 J ,
2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
3. find \(\lambda\).

5. \begin{figure}[h] \end{figure} A flask is modelled as a uniform solid formed by removing a cylinder of radius \(r\) and height \(h\) from a cylinder of radius \(\frac { 4 } { 3 } r\) and height \(\frac { 3 } { 2 } h\) with the same axis of symmetry and a common plane as shown in Figure 3.

1. Show that the centre of mass of the flask is a distance of \(\frac { 9 } { 10 } h\) from the open end of the flask. The flask is made from a material of density \(\rho\) and is filled to the level of the open plane face with a liquid of density \(k \rho\). Given that the centre of mass of the flask and liquid together is a distance of \(\frac { 15 } { 22 } h\) from the open end of the flask,
2. find the value of \(k\).
3. Explain why it may be advantageous to make the base of the flask from a more dense material.
   (2 marks)

6. A particle \(P\) of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points \(A\) and \(B\) on a smooth horizontal table. When \(P\) is 3 m from \(O\), the centre of the oscillations, its speed is \(6 \mathrm {~ms} ^ { - 1 }\). When \(P\) is 2.25 m from \(O\), its speed is \(8 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(A B = 7.5 \mathrm {~m}\).
2. Find the period of the motion.
3. Find the kinetic energy of \(P\) when it is 2.7 m from \(A\).
4. Show that the time taken by \(P\) to travel directly from \(A\) to the midpoint of \(O B\) is \(\frac { \pi } { 4 }\) seconds.