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A rigid rod consists of two parts. The part \(B C\) is in the form of an arc of a circle of radius 2 m and centre \(O\), with angle \(B O C = \frac { 1 } { 4 } \pi\) radians. \(B C\) is uniform and has weight 3 N . The part \(A B\) is straight and of length 2 m ; it is uniform and has weight 4 N . The part \(A B\) of the rod is a tangent to the arc \(B C\) at \(B\). The end \(A\) of the rod is freely hinged to a fixed point of a vertical wall. The rod is held in equilibrium, with the straight part \(A B\) making an angle of \(\frac { 1 } { 4 } \pi\) radians with the wall, by means of a horizontal string attached to \(C\). The string is in the same vertical plane as the rod, and the tension in the string is \(T \mathrm {~N}\) (see diagram).
- Show that the centre of mass \(G\) of the part \(B C\) of the rod is at a distance of 2.083 m from the wall, correct to 4 significant figures.
- Find the value of \(T\).
- State the magnitude of the horizontal component and the magnitude of the vertical component of the force exerted on the rod by the hinge.
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A particle \(A\) is released from rest at time \(t = 0\), at a point \(P\) which is 7 m above horizontal ground. At the same instant as \(A\) is released, a particle \(B\) is projected from a point \(O\) on the ground. The horizontal distance of \(O\) from \(P\) is 24 m . Particle \(B\) moves in the vertical plane containing \(O\) and \(P\), with initial speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and initial direction making an angle of \(\theta\) above the horizontal (see diagram). Write down - an expression for the height of \(A\) above the ground at time \(t \mathrm {~s}\),
- an expression in terms of \(V , \theta\) and \(t\) for
(a) the horizontal distance of \(B\) from \(O\),
(b) the height of \(B\) above the ground.
At time \(t = T\) the particles \(A\) and \(B\) collide at a point above the ground. - Show that \(\tan \theta = \frac { 7 } { 24 }\) and that \(V T = 25\).
- Deduce that \(7 V ^ { 2 } > 3125\).