CAIE Further Paper 3 (Further Paper 3) 2020 November

1 A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3 m g\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{0581e302-2fc5-46f0-b597-e5cae1f664a2-04_515_707_267_685} 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 \(P\) is held with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac { 4 } { 5 } \sqrt { 5 a g }\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos \theta\).

3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt { \frac { \mathrm { g } } { \mathrm { a } } }\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \(( \mathrm { k } + 1 ) \mathrm { a }\).

1. Find the value of \(k\).
2. Find the value of \(\cos \theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{0581e302-2fc5-46f0-b597-e5cae1f664a2-06_581_695_267_667} The diagram shows the cross-section \(A B C D\) of a uniform solid object which is formed by removing a cone with cross-section \(D C E\) from the top of a larger cone with cross-section \(A B E\). The perpendicular distance between \(A B\) and \(D C\) is \(h\), the diameter \(A B\) is \(6 r\) and the diameter \(D C\) is \(2 r\).
3. Find an expression, in terms of \(h\), for the distance of the centre of mass of the solid object from \(A B\).
   The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(A B\) and the downward vertical through \(B\) is \(\theta\).
4. Given that \(h = \frac { 13 } { 4 } r\), find the value of \(\tan \theta\).

5 A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) 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 \(t\) are denoted by \(x\) and \(y\) respectively.

1. Derive the equation of the trajectory of \(P\) in the form $$\mathrm { y } = \mathrm { x } \tan \alpha - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \sec ^ { 2 } \alpha$$ The point \(Q\) is the highest point on the trajectory of \(P\) in the case where \(\alpha = 45 ^ { \circ }\).
2. Show that the \(x\)-coordinate of \(Q\) is \(\frac { \mathrm { u } ^ { 2 } } { 2 \mathrm {~g} }\).
3. Find the other value of \(\alpha\) for which \(P\) would pass through the point \(Q\).

6 Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2 m\) respectively. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is moving on the floor with velocity \(u\) and collides directly with \(B\). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) after the collision.
   Subsequently, \(B\) collides with a fixed vertical wall which makes an angle \(\theta\) with the direction of motion of \(B\), where \(\tan \theta = \frac { 3 } { 4 }\). The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 3 }\). Immediately after \(B\) collides with the wall, the kinetic energy of \(A\) is \(\frac { 5 } { 32 }\) of the kinetic energy of \(B\).
2. Find the possible values of \(e\).

7 A particle \(P\) moving in a straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line at time \(t \mathrm {~s}\). The acceleration of \(P\), in \(\mathrm { ms } ^ { - 2 }\), is given by \(\frac { 200 } { x ^ { 2 } } - \frac { 100 } { x ^ { 3 } }\) for \(x > 0\). When \(t = 0 , x = 1\) and \(P\) has velocity \(10 \mathrm {~ms} ^ { - 1 }\) directed towards \(O\).

1. Show that the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) is given by \(\mathrm { v } = \frac { 10 ( 1 - 2 \mathrm { x } ) } { \mathrm { x } }\).
2. Show that \(x\) and \(t\) are related by the equation \(\mathrm { e } ^ { - 40 \mathrm { t } } = ( 2 \mathrm { x } - 1 ) \mathrm { e } ^ { 2 \mathrm { x } - 2 }\) and deduce what happens to \(x\) as \(t\) becomes large.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.