Edexcel FM2 AS (Further Mechanics 2 AS) 2019 June

1. \begin{figure}[h] \end{figure} Five identical uniform rods are joined together to form the rigid framework \(A B C D\) shown in Figure 1. Each rod has weight \(W\) and length 4a. The points \(A , B , C\) and \(D\) all lie in the same plane. The centre of mass of the framework is at the point \(G\).

1. Explain why \(G\) is the midpoint of \(A C\). The framework is suspended from the ceiling by two vertical light inextensible strings. One string is attached to the framework at \(A\) and the other string is attached to the framework at \(B\). The framework hangs freely in equilibrium with \(A B\) horizontal.
2. Find
   1. the tension in the string attached at \(A\),
   2. the tension in the string attached at \(B\). A particle of weight \(k W\) is now attached to the framework at \(D\) and a particle of weight \(2 k W\) is now attached to the framework at \(C\). The framework remains in equilibrium with \(A B\) horizontal and the strings vertical. Either string will break if the tension in it exceeds \(6 W\).
3. Find the greatest possible value of \(k\).

1. A car moves in a straight line along a horizontal road. The car is modelled as a particle. At time \(t\) seconds, where \(t \geqslant 0\), the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\)

At the instant when \(t = 0\), the car passes through the point \(A\) with speed \(2 \mathrm {~ms} ^ { - 1 }\)
The acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the car is modelled by $$a = \frac { 4 } { 2 + v }$$ in the direction of motion of the car.

1. Use algebraic integration to show that \(v = \sqrt { 8 t + 16 } - 2\) At the instant when the car passes through the point \(B\), the speed of the car is \(4 \mathrm {~ms} ^ { - 1 }\)
2. Use algebraic integration to find the distance \(A B\).

1. A light inextensible string has length \(8 a\). One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to a fixed point \(B\), with \(A\) vertically above \(B\) and \(A B = 4 a\). A small ball of mass \(m\) is attached to a point \(P\) on the string, where \(A P = 5 a\).

The ball moves in a horizontal circle with constant speed \(v\), with both \(A P\) and \(B P\) taut.
The string will break if the tension in it exceeds \(\frac { 3 m g } { 2 }\)
By modelling the ball as a particle and assuming the string does not break,

1. show that \(\frac { 9 a g } { 4 } < v ^ { 2 } \leqslant \frac { 27 a g } { 4 }\)
2. find the least possible time needed for the ball to make one complete revolution.

4. \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C D E\) is such that angle \(C E A = 90 ^ { \circ } , C E = 9 a\) and \(E A = 6 a\). The point \(D\) lies on \(C E\), with \(D E = 3 a\). The point \(B\) on \(C A\) is such that \(D B\) is parallel to \(E A\) and \(D B = 4 a\). The triangular lamina is folded along the line \(D B\) to form the folded lamina \(A B D E C F\), as shown in Figure 2. The distance of the centre of mass of the triangular lamina from \(D C\) is \(d _ { 1 }\)
The distance of the centre of mass of the folded lamina from \(D C\) is \(d _ { 2 }\)

1. Explain why \(d _ { 1 } = d _ { 2 }\) The folded lamina is freely suspended from \(B\) and hangs in equilibrium with \(B A\) inclined at an angle \(\alpha\) to the downward vertical through \(B\).
2. Find, to the nearest degree, the size of angle \(\alpha\).