Edexcel M3 (Mechanics 3)

1. A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal circle at one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(m g \mathrm {~N}\). The other end of the string is attached to a fixed point \(O\). Given that the string makes an angle of \(60 ^ { \circ }\) with the vertical,
   1. show that \(O P = 31 \mathrm {~m}\).
   2. Find, in terms of \(l\) and \(g\), the angular speed of \(P\).
   3. A particle \(P\) of mass \(m \mathrm {~kg}\) moves vertically upwards under gravity, starting from ground level. It is acted on by a resistive force of magnitude \(m \mathrm { f } ( x ) \mathrm { N }\), where \(\mathrm { f } ( x )\) is a function of the height \(x \mathrm {~m}\) of \(P\) above the ground. When \(P\) is at this height, its upward speed \(v \mathrm {~ms} ^ { - 1 }\) is given by \(v ^ { 2 } = 2 \mathrm { e } ^ { - 2 g x } - 1\).
   4. Write down a differential equation for the motion of \(P\) and hence determine \(\mathrm { f } ( x )\) in terms of \(g\) and \(x\).
   5. Show that the greatest height reached by \(P\) above the ground is \(\frac { 1 } { 2 g } \ln 2 \mathrm {~m}\).
   Given that the work, in J , done by \(P\) against the resisting force as it moves from ground level to a point \(H \mathrm {~m}\) above the ground is equal to \(\int _ { 0 } ^ { H } m \mathrm { f } ( x ) \mathrm { d } x\),
2. show that the total work done by \(P\) against the resistance during its upward motion is \(\frac { 1 } { 2 } m ( 1 - \ln 2 ) \mathrm { J }\).

3. A car of mass \(m \mathrm {~kg}\) moves round a curve of radius \(r \mathrm {~m}\) on a road which is banked at an angle \(\theta\) to the horizontal. When the speed of the car is \(u \mathrm {~ms} ^ { - 1 }\), the car experiences no sideways frictional force. Given that \(\tan \theta = \frac { u ^ { 2 } } { g r }\), show that the sideways frictional force on the car when its speed is \(\frac { u } { 2 } \mathrm {~ms} ^ { - 1 }\) has magnitude \(\frac { 3 } { 4 } m g \sin \theta \mathrm {~N}\).

4. Two light elastic strings, each of length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\),are attached to a particle \(P\) of mass \(m \mathrm {~kg}\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on the same horizontal level,
\includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-1_202_362_2068_1633}
where \(A B = 2 l \mathrm {~m} . P\) is held vertically below the mid-point of \(A B\), with each string taut and inclined at \(30 ^ { \circ }\) to the horizontal, and released from rest. Given that \(P\) comes to instantaneous rest when each string makes an angle of \(60 ^ { \circ }\) with the horizontal, show that \(\lambda = \frac { 3 m g } { 6 - 2 \sqrt { } 3 }\). \section*{MECHANICS 3 (A) TEST PAPER 8 Page 2}

1. A particle \(P\) is projected horizontally with speed \(u \mathrm {~ms} ^ { - 1 }\) from the highest point of a smooth sphere of radius \(r \mathrm {~m}\) and centre \(O\). It moves on the surface in a vertical plane, and at a particular instant the radius \(O P\) makes an angle \(\theta\) with the upward vertical, as shown. At this instant \(P\) has speed \(v \mathrm {~ms} ^ { - 1 }\) and
   \includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-2_264_261_294_1693}
   the magnitude of the reaction between \(P\) and the sphere is \(X \mathrm {~N}\).
   1. Assuming that \(u ^ { 2 } < g r\), show that
      1. \(v ^ { 2 } = u ^ { 2 } + 2 g r ( 1 - \cos \theta )\),
      2. \(X = m g \left( 3 \cos \theta - 2 - \frac { y ^ { 2 } } { g r } \right)\).
         (2 marks)
         (4 marks)
   2. Show that \(P\) leaves the surface of the sphere when \(\cos \theta = \frac { u ^ { 2 } + 2 g r } { 3 g r }\).
   3. Discuss what happens if \(u ^ { 2 } \geq g r\).
   4. A particle \(P\) of mass \(m \mathrm {~kg}\) hangs in equilibrium at one end of a light spring, of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\), whose other end is fixed at a point vertically above \(P\). In this position the length of the spring is \(( l + e ) \mathrm { m }\). When \(P\) is displaced vertically through a small distance and released, it performs simple harmonic motion with 5 oscillations per second.
   5. Show that \(\frac { \lambda } { l } = 100 \pi ^ { 2 } \mathrm {~m}\).
   6. Express \(e\) in terms of \(g\).
   7. Determine, in terms of \(m\) and \(l\), the magnitude of the tension in the spring when it is stretched to twice its natural length.
   8. (a) Prove that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac { 3 h } { 4 }\) from the vertex.
   An item of confectionery consists of a thin wafer in the form of a hollow right circular cone of height \(h\) and mass \(m\), filled with solid chocolate, also of mass \(m\), to a depth of \(k h\) as shown. The centre of mass of the item is at \(O\), the centre of the horizontal plane face
   \includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-2_314_332_1896_1636}
   of the chocolate.
2. Show that \(k = \frac { 8 h } { 15 }\). (3 marks) In the packaging process, the cone has to move on a conveyor belt inclined at an angle \(\alpha\) to the horizontal as shown. If the belt is rough enough to prevent sliding, and the maximum value of \(\alpha\) for which
   \includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-2_284_445_2271_1560}
   the cone does not topple is \(45 ^ { \circ }\),
3. find the radius of the base of the cone in terms of \(h\).