Ratio of tensions/forces

6 A particle \(P\), of mass \(m\), 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 \(P\) moves in complete vertical circles about \(O\) with the string taut. The points \(A\) and \(B\) are on the path of \(P\) with \(A B\) a diameter of the circle. \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(\sqrt { 5 a g }\). The ratio of the tension in the string when \(P\) is at \(A\) to the tension in the string when \(P\) is at \(B\) is \(9 : 5\).

1. Find the value of \(\cos \theta\).
2. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) during its motion. \includegraphics[max width=\textwidth, alt={}, center]{b10c65ef-dafd-4746-be5b-789130b7d030-12_613_718_251_676} The smooth vertical walls \(A B\) and \(C B\) are at right angles to each other. A particle \(P\) is moving with speed \(u\) on a smooth horizontal floor and strikes the wall \(C B\) at an angle \(\alpha\). It rebounds at an angle \(\beta\) to the wall \(C B\). The particle then strikes the wall \(A B\) and rebounds at an angle \(\gamma\) to that wall (see diagram). The coefficient of restitution between each wall and \(P\) is \(e\).
   1. Show that \(\tan \beta = e \tan \alpha\).
   2. Express \(\gamma\) in terms of \(\alpha\) and explain what this result means about the final direction of motion of \(P\).
      As a result of the two impacts the particle loses \(\frac { 8 } { 9 }\) of its initial kinetic energy.
   3. Given that \(\alpha + \beta = 90 ^ { \circ }\), find the value of \(e\) and the value of \(\tan \alpha\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A particle \(P\) of mass \(m\) 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 \(O\). When \(P\) is hanging vertically below \(O\), it is given a horizontal speed \(u\). In the subsequent motion, \(P\) moves in a complete circle. When \(O P\) makes an angle \(\theta\) with the downward vertical, the tension in the string is \(T\). Show that $$T = \frac { m u ^ { 2 } } { a } + m g ( 3 \cos \theta - 2 )$$ Given that the ratio of the maximum value of \(T\) to the minimum value of \(T\) is \(3 : 1\), find \(u\) in terms of \(a\) and \(g\). Assuming this value of \(u\), find the value of \(\cos \theta\) when the tension is half of its maximum value.

2 A particle \(P\) of mass \(m\) 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 moving in a complete vertical circle about \(O\). The points \(A\) and \(B\) are on the circle, at opposite ends of a diameter, and such that \(O A\) makes an acute angle \(\alpha\) with the upward vertical through \(O\). The speed of \(P\) as it passes through \(A\) is \(\frac { 3 } { 2 } \sqrt { } ( a g )\). The tension in the string when \(P\) is at \(B\) is four times the tension in the string when \(P\) is at \(A\).

1. Show that \(\cos \alpha = \frac { 3 } { 4 }\).
2. Find the tension in the string when \(P\) is at \(B\).

3 A particle \(P\) of mass \(m\) 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 path of the particle is a complete vertical circle with centre \(O\). When \(P\) is at its lowest point, its speed is \(u\). When \(P\) is at the point \(A\), the tension in the string is \(T\) and the string makes an angle \(\theta\) with the downward vertical, where \(\cos \theta = \frac { 3 } { 5 }\). When \(P\) is at the point \(B\), above the level of \(O\), the tension in the string is \(\frac { 1 } { 8 } T\) and angle \(B O A = 90 ^ { \circ }\). Find \(u\) in terms of \(a\) and \(g\).

\includegraphics{figure_5} A particle of mass \(m\) 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 is moving in complete vertical circles with the string taut. When the particle is at the point \(P\), where \(OP\) makes an angle \(\alpha\) with the upward vertical through \(O\), its speed is \(u\). When the particle is at the point \(Q\), where angle \(QOP = 90°\), its speed is \(v\) (see diagram). It is given that \(\cos \alpha = \frac{4}{5}\).

1. Show that \(v^2 = u^2 + \frac{14}{5}ag\). [2]

The tension in the string when the particle is at \(Q\) is twice the tension in the string when the particle is at \(P\).

1. Obtain another equation relating \(u^2\), \(v^2\), \(a\) and \(g\), and hence find \(u\) in terms of \(a\) and \(g\). [5]
2. Find the least tension in the string during the motion. [3]

A particle \(Q\) of mass \(m\) is attached to a fixed point \(O\) by a light inextensible string of length \(a\). The particle moves in complete vertical circles about \(O\). The points \(A\) and \(B\) are on the path of \(Q\) with \(AB\) a diameter of the circle. \(OA\) makes an angle of \(60°\) with the downward vertical through \(O\) and \(OB\) makes an angle of \(60°\) with the upward vertical through \(O\). The speed of \(Q\) when it is at \(A\) is \(2\sqrt{ag}\). Given that \(T_A\) and \(T_B\) are the tensions in the string at \(A\) and \(B\) respectively, find the ratio \(T_A : T_B\). [6]

A particle \(P\) of mass \(m\) 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 completes vertical circles with centre \(O\). The points A and B are on the path of \(P\), both on the same side of the vertical through \(O\). \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at A is \(u\) and the speed of \(P\) when it is at B is \(\sqrt{ag}\). The tensions in the string at A and B are \(T_A\) and \(T_B\) respectively. It is given that \(T_A = 7T_B\). Find the value of \(\theta\) and find an expression for \(u\) in terms of \(a\) and \(g\). [8]

A particle \(P\), of mass \(m\), 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 \(P\) moves in complete vertical circles about \(O\) with the string taut. The points \(A\) and \(B\) are on the path of \(P\) with \(AB\) a diameter of the circle. \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(\sqrt{5ag}\). The ratio of the tension in the string when \(P\) is at \(A\) to the tension in the string when \(P\) is at \(B\) is \(9 : 5\).

1. Find the value of \(\cos \theta\). [6]
2. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) during its motion. [2]