Edexcel M3 (Mechanics 3)

1. A small bead is threaded onto a smooth circular hoop, of radius \(r \mathrm {~m}\), fixed in a vertical plane. It is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) from the lowest point of the hoop. Find \(u\) in terms of \(g\) and \(r\) if
   1. the bead just reaches the highest point of the hoop,
   2. the reaction on the bead is zero when it is at the highest point of the hoop.
   3. An ornamental tower is made from a solid right circular cylinder of mass \(M\) and height \(h\) by removing three identical cylindrical sections, each of height \(\frac { h } { 8 }\), equally spaced above a base of height \(\frac { h } { 4 }\), as shown. The tower is held in position by light, thin vertical strips \(A B\) and \(C D\).
      \includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-1_312_328_703_1622}
   Find the distance of the centre of mass of the tower from its horizontal base.

3. Two particles \(A\) and \(B\), of masses \(M \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively, are connected by a light inextensible string passing over a smooth fixed pulley. \(B\) is placed on a smooth horizontal table and \(A\) hangs freely, as shown. \(B\) is attached to a spring of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\), whose other end is fixed to a vertical wall.
\includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-1_333_405_1160_1572}
The system starts to move from rest when the string is taut and the spring neither extended nor compressed. \(A\) does not reach the ground, nor does \(B\) reach the pulley, during the motion.

1. Show that the maximum extension of the spring is \(\frac { 2 M g l } { \lambda } \mathrm {~m}\).
2. If \(M = 3 , m = 1.5\) and \(\lambda = 35 l\), find the speed of \(A\) when the extension in the spring is 0.5 m .

4. A particle \(P\) of mass \(m \mathrm {~kg}\) moves along a straight line under the action of a force of magnitude \(\frac { k m } { x ^ { 2 } } \mathrm {~N}\), where \(k\) is a constant, directed towards a fixed point \(O\) on the line, where \(O P = x \mathrm {~m} P\) starts from rest at \(A\), at a distance \(a \mathrm {~m}\) from \(O\). When \(O P = x \mathrm {~m}\), the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\).

1. Show that \(v = \sqrt { \frac { 2 k ( a - x ) } { a x } }\).
   \(B\) is the point half-way between \(O\) and \(A\). When \(k = \frac { 1 } { 2 }\) and \(a = 1\), the time taken by \(P\) to travel from \(A\) to \(B\) is \(T\) seconds
   Assuming the result that, for \(0 \leq x \leq 1 , \int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x = \arcsin ( \sqrt { } x ) - \sqrt { } \left( x - x ^ { 2 } \right) +\) constant,
2. find the value of \(T\). \section*{MECHANICS 3 (A) TEST PAPER 9 Page 2}

1. A car moves round a circular racing track of radius 100 m , which is banked at an angle of \(4 ^ { \circ }\) to the horizontal.
   1. Show that when its speed is \(8.28 \mathrm {~ms} ^ { - 1 }\), there is no sideways force acting on the car.
      (4 marks)
   2. When the speed of the car is \(12.5 \mathrm {~ms} ^ { - 1 }\), find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip.
   3. The diagram shows a particle \(P\) of mass \(m \mathrm {~kg}\) moving on the inner surface of a smooth fixed hemispherical bowl of radius \(r \mathrm {~m}\) which is fixed with its axis vertical. \(P\) moves at a constant speed in a horizontal circle, at a depth \(h \mathrm {~m}\) below the top of the bowl.
      \includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-2_254_431_786_1528}
   4. Show that the force \(R\) exerted on \(P\) by the bowl has magnitude \(\frac { m g r } { h } \mathrm {~N}\).
   5. Find, in terms of \(g , h\) and \(r\), the constant speed of \(P\).
   The bowl is now inverted and \(P\) moves on the smooth outer surface at a height \(h\) above the plane face under the action of a force of magnitude \(m g\) applied tangentially as shown. The reaction of the
   \includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-2_209_355_1224_1636}
   surface of the sphere on \(P\) now has magnitude \(S \mathrm {~N}\).
2. Given that \(r = 2 h\), prove that \(S < \frac { 1 } { 6 } R\).

7. A particle \(P\) of mass \(m \mathrm {~kg}\) is fixed to one end of a light elastic string of modulus \(m g \mathrm {~N}\) and natural length \(l \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. Initially \(P\) is at rest in limiting equilibrium on the table at the point \(X\) where \(O X = \frac { 5 l } { 4 } \mathrm {~m}\).

1. Find the coefficient of friction between \(P\) and the table.
   \(P\) is now given a small displacement \(x \mathrm {~m}\) horizontally along \(O X\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.
2. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. If \(P\) is held at the point where the extension in the string is \(l m\) and then released,
3. show that the string becomes slack after a time \(\left( \frac { \pi } { 2 } + \arcsin \left( \frac { 1 } { 3 } \right) \right) \sqrt { \frac { l } { g } } \mathrm {~s}\).
4. Determine the speed of \(P\) when it reaches \(O\).