One end of a light inextensible string of length \(2 r \mathrm {~m}\) is attached to a fixed point \(O\). A particle of mass \(m \mathrm {~kg}\) is attached to the other end \(Q\) of the string, so that it can move in a vertical plane. The string is held taut and horizontal and the particle is projected vertically downwards with a speed \(\sqrt { } ( g r ) \mathrm { ms } ^ { - 1 }\). When the string is vertical it begins to wrap round a small, smooth peg \(X\) at a distance \(r \mathrm {~m}\) vertically below \(O\). The particle continues to move.
Find the speed of the particle when it reaches \(O\), in terms of \(g\) and \(r\).
Show that, when \(Q X\) is horizontal, the tension in the string is 3 mgN .
A particle moving along the \(x\)-axis describes simple harmonic motion about the origin \(O\). The period of its motion is \(\frac { \pi } { 2 }\) seconds. When it is at a distance 1 m from \(O\), its speed is \(3 \mathrm {~ms} ^ { - 1 }\). Calculate
the amplitude of its motion,
the maximum acceleration of the particle,
the least time that it takes to move from \(O\) to a point 0.25 m from \(O\).
A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length \(8 l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The two ends of the string are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(A B = 81 \mathrm {~m} . P\) is released from rest at the mid-point of \(A B\).
If \(P\) comes to instantaneous rest at a depth \(3 / \mathrm { m }\) below \(A B\), find an expression for \(\lambda\) in terms of \(m\) and \(g\).
Using this value of \(\lambda\), show that the speed \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) when it passes through the point \(2 l \mathrm {~m}\) below \(A B\) is given by \(v ^ { 2 } = 4 ( 24 \sqrt { 5 } - 53 ) g l\).
A particle \(P\) of mass 0.8 kg moves along a straight line \(O L\) and is acted on by a resistive force of magnitude \(R \mathrm {~N}\) directed towards the fixed point \(O\). When the displacement of \(P\) from \(O\) is \(x \mathrm {~m} , R = \frac { 0 \cdot 8 x v ^ { 2 } } { 1 + x ^ { 2 } }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) at that instant.
Write down a differential equation for the motion of \(P\).
Given that \(v = 2\) when \(x = 0\),
find the speed with which \(P\) passes through the point \(A\), where \(O A = 1 \mathrm {~m}\).
\section*{MECHANICS 3 (A) TEST PAPER 3 Page 2}
The diagram shows a uniform solid right circular cone of mass \(m \mathrm {~kg}\), height \(h \mathrm {~m}\) and base radius \(r \mathrm {~m}\) suspended by two vertical strings attached to the points \(P\) and \(Q\) on the circumference of the base. The vertex \(O\) of the cone is vertically below \(P\).
Show that the tension in the string attached at \(Q\) is \(\frac { 3 m g } { 8 } \mathrm {~N}\).
\includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_296_277_269_1668}
Find, in terms of \(m\) and \(g\), the tension in the other string.
Two identical particles \(P\) and \(Q\) are connected by a light inextensible string passing through a small smooth-edged hole in a smooth table, as shown.
\(P\) moves on the table in a horizontal circle of radius 0.2 m and \(Q\) hangs at rest.
\includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_309_430_859_1476}
Calculate the number of revolutions made per minute by \(P\).
(5 marks)
\(Q\) is now also made to move in a horizontal circle of radius 0.2 m below the table. The part of the string between \(Q\) and the table makes an angle of \(45 ^ { \circ }\) with the vertical.
Show that the numbers of revolutions per minute made by \(P\) and \(Q\) respectively are in the ratio \(2 ^ { 1 / 4 } : 1\).
\includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_293_428_1213_1499}
A particle \(P\) of mass \(m \mathrm {~kg}\) is fixed to one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(k m g \mathrm {~N}\). The other end of the string is fixed to a point \(X\) on a horizontal plane. \(P\) rests at \(O\), where \(O X = l \mathrm {~m}\), with the string just taut. It is then pulled away from \(X\) through a distance \(\frac { 3 l } { 4 } \mathrm {~m}\) and released from rest. On this side of \(O\), the plane is smooth.
Show that, as long as the string is taut, \(P\) performs simple harmonic motion.
Given that \(P\) first returns to \(O\) with speed \(\sqrt { } ( g l ) \mathrm { ms } ^ { - 1 }\), find the value of \(k\).
On the other side of \(O\) the plane is rough, the coefficient of friction between \(P\) and the plane being \(\mu\). If \(P\) does not reach \(X\) in the subsequent motion, show that \(\mu > \frac { 1 } { 2 }\). ( 4 marks)
If, further, \(\mu = \frac { 3 } { 4 }\), show that the time which elapses after \(P\) is released and before it comes to rest is \(\frac { 1 } { 24 } ( 9 \pi + 32 ) \sqrt { \frac { l } { g } }\) s.
(6 marks)