Edexcel M3 (Mechanics 3) 2016 January

1. \begin{figure}[h] \end{figure} A hemispherical bowl of internal radius \(2 r\) is fixed with its circular rim horizontal. A particle \(P\) is moving in a horizontal circle of radius \(r\) on the smooth inner surface of the bowl, as shown in Figure 1. Particle \(P\) is moving with constant angular speed \(\omega\). Show that \(\omega = \sqrt { \frac { g \sqrt { 3 } } { 3 r } }\)

2. A particle \(P\) is moving in a straight line. At time \(t\) seconds, the distance of \(P\) from a fixed point \(O\) on the line is \(x\) metres and the acceleration of \(P\) is \(( 6 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the direction of \(x\) increasing. When \(t = 0 , P\) is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the velocity of \(P\) in terms of \(t\).
2. Find the total distance travelled by \(P\) in the first 4 seconds.

3. A car of mass 800 kg is driven at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) round a bend in a race track. Around the bend, the track is banked at \(20 ^ { \circ }\) to the horizontal and the path followed by the car can be modelled as a horizontal circle of radius 20 m . The car is modelled as a particle. The coefficient of friction between the car tyres and the track is 0.5 Given that the tyres do not slip sideways on the track, find the maximum value of \(v\).

4. Fixed points \(A\) and \(B\) are on a horizontal ceiling, where \(A B = 4 a\). A light elastic string has natural length \(3 a\) and modulus of elasticity \(\lambda\). One end of the string is attached to \(A\) and the other end is attached to \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at the point \(C\), where \(C\) is at a distance \(\frac { 3 } { 2 } a\) vertically below the ceiling.

1. Show that \(\lambda = \frac { 5 m g } { 4 }\)
   (5) The point \(D\) is the midpoint of \(A B\). The particle is now raised vertically upwards to \(D\), and released from rest.
2. Find the speed of \(P\) as it passes through \(C\).
   

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5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The particle rests in equilibrium on the plane at the point \(B\) with the string lying along a line of greatest slope of the plane, as shown in Figure 2. Given that \(A B = \frac { 6 } { 5 } l\)

1. show that \(\lambda = 3 \mathrm { mg }\) The particle is pulled down the line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 2 } l\), and released from rest.
2. Show that, while the string remains taut, \(P\) moves with simple harmonic motion about centre \(B\).
3. Find the greatest magnitude of the acceleration of \(P\) while the string remains taut. The point \(D\) is the midpoint of \(B C\). The time taken by \(P\) to move directly from \(D\) to the point where the string becomes slack for the first time is \(k \sqrt { \frac { l } { g } }\), where \(k\) is a constant.
4. Find, to 2 significant figures, the value of \(k\).

6. (a) Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 } { 8 } r\) from the centre of its plane face.
[0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]
(5) \begin{figure}[h] \end{figure} A uniform solid hemisphere of mass \(m\) and radius \(r\) is joined to a uniform solid right circular cone to form a solid \(S\). The cone has mass \(M\), base radius \(r\) and height \(4 r\). The vertex of the cone is \(O\). The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3.
(b) Find the distance of the centre of mass of \(S\) from \(O\). The point \(A\) lies on the circumference of the base of the cone. The solid is placed on a horizontal table with \(O A\) in contact with the table. The solid remains in equilibrium in this position.
(c) Show that \(M \geqslant \frac { 1 } { 10 } m\)

7. \begin{figure}[h] \end{figure} A particle of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about a horizontal axis through \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), as shown in Figure 4. The particle moves in complete vertical circles. Given that \(\cos \alpha = \frac { 4 } { 5 }\)

1. show that \(u > \sqrt { \frac { 2 g l } { 5 } }\) As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(4 T\).
2. Show that \(u = \sqrt { \frac { 17 } { 5 } g l }\)
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