Edexcel M3 (Mechanics 3) 2018 June

1. A light elastic string of modulus of elasticity 29.4 N has one end attached to a fixed point \(A\). A particle \(P\) of mass 1.5 kg is attached to the other end of the string and \(P\) hangs freely in equilibrium 0.5 m vertically below \(A\). Find the natural length of the string.
   

2. A particle \(P\) is moving in a straight line with simple harmonic motion about the fixed point \(O\) as centre. When \(P\) is a distance 0.02 m from \(O\), the speed of \(P\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the acceleration of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Find the period of the motion. The amplitude of the motion is \(a\) metres. Find
2. the value of \(a\),
3. the total length of time during each complete oscillation for which \(P\) is within \(\frac { 1 } { 2 } a\)
   metres of \(O\). metres of \(O\).

3. \begin{figure}[h] \end{figure} A light inextensible string of length \(7 l\) has one end attached to a fixed point \(A\) and the other end attached to a fixed point \(B\), where \(A\) is vertically above \(B\) and \(A B = 5\) l. A particle of mass \(m\) is attached to the string at the point \(C\) where \(A C = 4 l\), as shown in Figure 1. The particle moves in a horizontal circle with constant angular speed \(\omega\). Both parts of the string are taut.

1. Find, in terms of \(m , g , l\) and \(\omega\),
   1. the tension in \(A C\),
   2. the tension in \(B C\). The time taken by the particle to complete one revolution is \(R\).
      Given that \(R \leqslant k \pi \sqrt { \frac { l } { 5 g } }\)
2. find the least possible value of \(k\).

4. \begin{figure}[h] \end{figure} Figure 2 shows a light elastic string, of modulus of elasticity \(\lambda\) newtons and natural length 0.6 m . One end of the string is attached to a fixed point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass 0.5 kg . The string lies along a line of greatest slope of the plane. The particle is held at rest on the plane at the point \(B\), where \(B\) is lower than \(A\) and \(A B = 1.2 \mathrm {~m}\). The particle then receives an impulse of magnitude 1.5 N s in the direction parallel to the string, causing \(P\) to move up the plane towards \(A\). The coefficient of friction between \(P\) and the plane is 0.7 . Given that \(P\) comes to rest at the instant when the string becomes slack, find the value of \(\lambda\).

1. A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis in the positive \(x\) direction under the action of a resultant force. This force acts in the direction of \(x\) increasing. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O , P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the force has magnitude \(\frac { 4 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}\).

When \(t = 0 , P\) is at rest at \(O\).

1. Show that \(v ^ { 2 } = 5 \left( \frac { ( x + 1 ) ^ { 2 } - 1 } { ( x + 1 ) ^ { 2 } } \right)\) When \(t = 2 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
2. Using algebraic integration, find the distance \(A B\).

6. A uniform solid right circular cone has base radius \(r\) and height \(h\).

1. Use algebraic integration to show that the distance of the centre of mass of the cone from its vertex is \(\frac { 3 } { 4 } h\).
   [0pt] [You may assume that the volume of a cone of base radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ] \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-20_394_716_632_621} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A solid \(S\) is formed by joining a uniform right circular solid cone of mass \(5 m\) to a uniform solid hemisphere, of radius \(r\) and mass \(k m\) where \(k < 20\). The cone has base radius \(r\) and height \(6 r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the cone is \(O\) and the point \(A\) is on the circular edge of this plane face, as shown in Figure 3.
2. Find the distance from \(O\) to the centre of mass of \(S\). The solid is suspended from \(A\) and hangs freely in equilibrium. The angle between the axis of the cone and the horizontal is \(30 ^ { \circ }\).
3. Find, to the nearest whole number, the value of \(k\).

7. \begin{figure}[h] \end{figure} A smooth solid sphere, with centre \(O\) and radius \(r\), is fixed with its lowest point on a horizontal plane. A particle is placed on the surface of the sphere at the highest point of the sphere. The particle is then projected horizontally with speed \(u\) and starts to move on the surface of the sphere. The particle leaves the surface of the sphere at the point \(A\) where \(O A\) makes an angle \(\alpha , \alpha > 0\), with the upward vertical, as shown in Figure 4.

1. Show that \(\cos \alpha = \frac { 1 } { 3 g r } \left( u ^ { 2 } + 2 g r \right)\)
2. Show that \(u < \sqrt { g r }\) After leaving the surface of the sphere, the particle strikes the plane with speed \(3 \sqrt { \frac { g r } { 2 } }\)
3. Find the value of \(\cos \alpha\).