Edexcel M3 (Mechanics 3) 2003 June

1. A particle \(P\) of mass \(m\) is held at a point \(A\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac { 2 } { 3 }\). The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\). The other end of the string is attached to a fixed point \(O\) on the plane, where \(O A = \frac { 3 } { 2 } a\). The particle \(P\) is released from rest and comes to rest at a point \(B\), where \(O B < a\).

Using the work-energy principle, or otherwise, calculate the distance \(A B\).
(6)

2. A car moves round a bend which is banked at a constant angle of \(10 ^ { \circ }\) to the horizontal. When the car is travelling at a constant speed of \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius \(r\) metres. Calculate the value of \(r\).
(6)

3. A toy car of mass 0.2 kg is travelling in a straight line on a horizontal floor. The car is modelled as a particle. At time \(t = 0\) the car passes through a fixed point \(O\). After \(t\) seconds the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car is at a point \(P\) with \(O P = x\) metres. The resultant force on the car is modelled as \(\frac { 1 } { 10 } x ( 4 - 3 x ) \mathrm { N }\) in the direction \(O P\). The car comes to instantaneous rest when \(x = 6\). Find

1. an expression for \(v ^ { 2 }\) in terms of \(x\),
2. the initial speed of the car. \section*{4.} \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{c0a3336d-b0ca-4588-80d1-445e2a5e493c-3_1022_633_268_760}
   A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings \(A P\) and \(B P\) each of length \(l\). The ends \(A\) and \(B\) are attached to fixed points, with \(A\) vertically above \(B\) and \(A B = \frac { 3 } { 2 } l\), as shown in Fig. 1. The particle \(P\) moves in a horizontal circle with constant angular speed \(\omega\). The centre of the circle is the mid-point of \(A B\) and both strings remain taut.
3. Show that the tension AP is \(\frac { 1 } { 6 } m \left( 3 l \omega ^ { 2 } + 4 g \right)\).
4. Find, in terms of \(m , l , \omega\) and \(g\), an expression for the tension in \(B P\).
5. Deduce that \(\omega ^ { 2 } \geq \frac { 4 g } { 3 l }\).

5. A particle \(P\) of mass 0.8 kg is attached to one end \(A\) of a light elastic spring \(O A\), of natural length 60 cm and modulus of elasticity 12 N . The spring is placed on a smooth horizontal table and the end \(O\) is fixed. The particle \(P\) is pulled away from \(O\) to a point \(B\), where \(O B = 85 \mathrm {~cm}\), and is released from rest.

1. Prove that the motion of \(P\) is simple harmonic with period \(\frac { 2 \pi } { 5 }\) s.
2. Find the greatest magnitude of the acceleration of \(P\) during the motion. Two seconds after being released from rest, \(P\) passes through the point \(C\).
3. Find, to 2 significant figures, the speed of \(P\) as it passes through \(C\).
4. State the direction in which \(P\) is moving 2 s after being released.

6. \begin{figure}[h] \end{figure} A particle is at the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(u\), where \(u < \sqrt { } ( a g )\). The particle leaves the sphere at the point \(C\), where \(O C\) makes an angle \(\theta\) with the upward vertical, as shown in Fig. 2.

1. Find an expression for \(\cos \theta\) in terms of \(u , g\) and \(a\). The particle strikes the plane with speed \(\sqrt { \left( \frac { 9 a g } { 2 } \right) }\).
2. Find, to the nearest degree, the value of \(\theta\).

7. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = \frac { 1 } { 2 } ( x - 2 ) ^ { 2 }\), the \(x\)-axis and the \(y\)-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm . A uniform solid \(S\) is made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis. Using integration,

1. calculate the volume of the solid \(S\), leaving your answer in terms of \(\pi\),
2. show that the centre of mass of \(S\) is \(\frac { 1 } { 3 } \mathrm {~cm}\) from its plane face. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{c0a3336d-b0ca-4588-80d1-445e2a5e493c-5_411_772_1357_568}
   \end{figure} A tool is modelled as having two components, a solid uniform cylinder \(C\) and the solid \(S\). The diameter of \(C\) is 4 cm and the length of \(C\) is 8 cm . One end of \(C\) coincides with the plane face of \(S\). The components are made of different materials. The weight of \(C\) is \(10 W\) newtons and the weight of \(S\) is \(2 W\) newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports \(A\) and \(B\), which are at the ends of the cylinder, as shown in Fig. 4.
3. Find the magnitude of the force of the support \(A\) on the tool.