CAIE Further Paper 3 (Further Paper 3) 2020 November

3 An object consists of a uniform solid circular cone, of vertical height \(4 r\) and radius \(3 r\), and a uniform solid cylinder, of height \(4 r\) and radius \(3 r\). The circular base of the cone and one of the circular faces of the cylinder are joined together so that they coincide. The cone and the cylinder are made of the same material.

1. Find the distance of the centre of mass of the object from the end of the cylinder that is not attached to the cone.
2. Show that the object can rest in equilibrium with the curved surface of the cone in contact with a horizontal surface.

4 A particle \(P\) of mass \(m\) is moving in a horizontal circle with angular speed \(\omega\) on the smooth inner surface of a hemispherical shell of radius \(r\). The angle between the vertical and the normal reaction of the surface on \(P\) is \(\theta\).

1. Show that \(\cos \theta = \frac { \mathrm { g } } { \omega ^ { 2 } \mathrm { r } }\).
   The plane of the circular motion is at a height \(x\) above the lowest point of the shell. When the angular speed is doubled, the plane of the motion is at a height \(4 x\) above the lowest point of the shell.
2. Find \(x\) in terms of \(r\).

5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t \mathrm {~s}\) are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Starting from the equation of the trajectory given in the List of formulae (MF19), show that $$\mathrm { y } = \mathrm { x } \tan \theta - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \left( 1 + \tan ^ { 2 } \theta \right)$$ When \(\theta = \tan ^ { - 1 } 2 , P\) passes through the point with coordinates \(( 10,16 )\).
2. Show that there is no value of \(\theta\) for which \(P\) can pass through the point with coordinates \(( 18,30 )\).

6 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k\), is attached to a particle \(P\) of mass \(m\). The other end of the string is attached to a fixed point \(Q\). The particle \(P\) is projected vertically upwards from \(Q\). When \(P\) is moving upwards and at a distance \(\frac { 4 } { 3 } a\) directly above \(Q\), it has a speed \(\sqrt { 2 g a }\). At this point, its acceleration is \(\frac { 7 } { 3 } g\) downwards. Show that \(\mathrm { k } = 4 \mathrm { mg }\) and find in terms of \(a\) the greatest height above \(Q\) reached by \(P\).

7 A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal straight line against a resistive force of magnitude \(\mathrm { mkv } ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) after it has moved a distance \(x \mathrm {~m}\) and \(k\) is a positive constant. The initial speed of \(P\) is \(u \mathrm {~ms} ^ { - 1 }\).

1. Show that \(\mathrm { x } = \frac { 1 } { \mathrm { k } } \ln 2\) when \(\mathrm { v } = \frac { 1 } { 2 } \mathrm { u }\).
   Beginning at the instant when the speed of \(P\) is \(\frac { 1 } { 2 } u\), an additional force acts on \(P\). This force has magnitude \(\frac { 5 \mathrm {~m} } { \mathrm { v } } \mathrm { N }\) and acts in the direction of increasing \(x\).
2. Show that when the speed of \(P\) has increased again to \(u \mathrm {~ms} ^ { - 1 }\), the total distance travelled by \(P\) is given by an expression of the form $$\frac { 1 } { 3 k } \ln \left( \frac { A - k u ^ { 3 } } { B - k u ^ { 3 } } \right) ,$$ stating the values of the constants \(A\) and \(B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.