Edexcel M3 (Mechanics 3) 2011 June

1. A particle \(P\) of mass 0.5 kg moves on the positive \(x\)-axis under the action of a single force directed towards the origin \(O\). At time \(t\) seconds the distance of \(P\) from \(O\) is \(x\) metres, the magnitude of the force is \(0.375 x ^ { 2 } \mathrm {~N}\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

When \(t = 0 , O P = 8 \mathrm {~m}\) and \(P\) is moving towards \(O\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 260 - \frac { 1 } { 2 } \chi ^ { 3 }\).
2. Find the distance of \(P\) from \(O\) at the instant when \(v = 5\).

2. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = 9 - x ^ { 2 }\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 1. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of \(S\).

3. \begin{figure}[h] \end{figure} A solid consists of a uniform solid right cylinder of height \(5 l\) and radius \(3 l\) joined to a uniform solid hemisphere of radius \(3 l\). The plane face of the hemisphere coincides with a circular end of the cylinder and has centre \(O\), as shown in Figure 2. The density of the hemisphere is twice the density of the cylinder.

1. Find the distance of the centre of mass of the solid from \(O\).
   (5) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-04_618_807_1327_571} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The solid is now placed with its circular face on a plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal, as shown in Figure 3. The plane is sufficiently rough to prevent the solid slipping. The solid is on the point of toppling.
2. Find the value of \(\theta\).

4. \begin{figure}[h] \end{figure} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves in a horizontal circle of radius \(3 a\) with angular speed \(\omega\). The centre of the circle is \(C\) where \(C\) lies on \(A B\) and \(A C = 4 a\), as shown in Figure 4. Both parts of the string are taut.

1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 3 a \omega ^ { 2 } + g \right)\).
2. Find the tension in \(B P\).
3. Deduce that \(\omega \geqslant \frac { 1 } { 2 } \sqrt { } \left( \frac { g } { a } \right)\).

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity \(3 m g\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. The particle lies at rest at the point \(A\) on the table, where \(O A = \frac { 7 } { 6 } l\). The coefficient of friction between \(P\) and the table is \(\mu\).
   1. Show that \(\mu \geqslant \frac { 1 } { 2 }\).
   The particle is now moved along the table to the point \(B\), where \(O B = \frac { 3 } { 2 } l\), and released from rest. Given that \(\mu = \frac { 1 } { 2 }\), find
2. the speed of \(P\) at the instant when the string becomes slack,
3. the total distance moved by \(P\) before it comes to rest again.

6. \begin{figure}[h] \end{figure} A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal. The point \(B\) is vertically above \(O\) and the point \(C\) is vertically below \(O\), with \(O B = O C = a\), as shown in Figure 5. The particle is projected vertically upwards with speed \(3 \sqrt { } ( a g )\).

1. Show that \(P\) will pass through \(B\).
2. Find the speed of \(P\) as it reaches \(C\). As \(P\) passes through \(C\) it receives an impulse. Immediately after this, the speed of \(P\) is \(\frac { 5 } { 12 } \sqrt { } ( 11 a g )\) and the direction of motion of \(P\) is unchanged.
3. Find the angle between the string and the downward vertical when \(P\) comes to instantaneous rest.

1. A particle \(P\) of mass 0.5 kg is attached to the mid-point of a light elastic string of natural length 1.4 m and modulus of elasticity 2 N . The ends of the string are attached to the points \(A\) and \(B\) on a smooth horizontal table, where \(A B = 2 \mathrm {~m}\). The mid-point of \(A B\) is \(O\) and the point \(C\) is on the table between \(O\) and \(B\) where \(O C = 0.2 \mathrm {~m}\). At time \(t = 0\) the particle is released from rest at \(C\). At time \(t\) seconds the length of the string \(A P\) is \(( 1 + x ) \mathrm { m }\).
   1. Show that the tension in \(B P\) is \(\frac { 2 } { 7 } ( 3 - 10 x ) \mathrm { N }\).
   2. Find, in terms of \(x\), the tension in \(A P\).
   3. Show that \(P\) performs simple harmonic motion with period \(2 \pi \sqrt { } \left( \frac { 7 } { 80 } \right)\) s.
   4. Find the greatest speed of \(P\) during the motion.
   The point \(D\) lies between \(O\) and \(A\), where \(O D = 0.1 \mathrm {~m}\).
2. Find the time taken by \(P\) to move directly from \(C\) to \(D\).