Edexcel FM2 AS (Further Mechanics 2 AS) 2018 June

1. Figure 1 A thin uniform rod, of total length \(30 a\) and mass \(M\), is bent to form a frame. The frame is in the shape of a triangle \(A B C\), where \(A B = 12 a , B C = 5 a\) and \(C A = 13 a\), as shown in Figure 1.

1. Show that the centre of mass of the frame is \(\frac { 3 } { 2 } a\) from \(A B\). The frame is freely suspended from \(A\). A horizontal force of magnitude \(k M g\), where \(k\) is a constant, is applied to the frame at \(B\). The line of action of the force lies in the vertical plane containing the frame. The frame hangs in equilibrium with \(A B\) vertical.
2. Find the value of \(k\).

1. A car moves round a bend which is banked at a constant angle of \(\theta ^ { \circ }\) to the horizontal.

When the car is travelling at a constant speed of \(80 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius 500 m .

1. Find the value of \(\theta\).
2. Identify one limitation of this model. The speed of the car is increased so that it is now travelling at a constant speed of \(90 \mathrm { kmh } ^ { - 1 }\) The car is still modelled as a particle moving in a horizontal circle of radius 500 m .
3. Describe the extra force that will now be acting on the car, stating the direction of this force.

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3. \begin{figure}[h] \end{figure} The lamina \(L\), shown in Figure 2, consists of a uniform square lamina \(A B D F\) and two uniform triangular laminas \(B D C\) and \(F D E\). The square has sides of length \(2 a\). The two triangles are identical. The straight lines \(B D E\) and \(F D C\) are perpendicular with \(B D = D F = 2 a\) and \(D C = D E = a\).
The mass per unit of area of the square is \(M\).
The mass per unit area of each triangle is \(3 M\).
The centre of mass of \(L\) is at the point \(G\).

1. Without doing any calculations, explain why \(G\) lies on \(A D\).
2. Show that the distance of \(G\) from \(D\) is \(\frac { \sqrt { 2 } } { 2 } a\) The lamina \(L\) is freely suspended from \(B\) and hangs in equilibrium.
3. Find the size of the angle between \(B E\) and the downward vertical.

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1. A particle, \(P\), moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the direction of \(x\) increasing.

When \(t = 0\) the particle is at rest at the origin \(O\).
Given that \(a = \frac { 5 } { 2 } ( 5 - v )\)

1. show that \(v = 5 \left( 1 - \mathrm { e } ^ { - 2.5 t } \right)\)
2. state the limiting value of \(v\) as \(t\) increases. At the instant when \(v = 2.5\), the particle is \(d\) metres from \(O\).
3. Show that \(d = 2 \ln 2 - 1\)