A particle \(P\), of mass 2.5 kg , initially at rest at the point \(O\), moves on a smooth horizontal surface with constant acceleration \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively. Find
- the velocity vector of \(P\) at time \(t\) seconds after it leaves \(O\),
- the magnitude and direction of the velocity of \(P\) when \(t = 7\),
- the magnitude, in N , of the force acting on \(P\).
- An iron bar \(A B\), of length 4 m , is kept in a horizontal position by a support at \(A\) and a wire attached to the point \(P\) on the bar, where \(P B = 0.85 \mathrm {~m}\). The bar is modelled as a non-uniform rod whose centre of mass is at \(G\), where \(A G = 1.4 \mathrm {~m}\), and the wire is modelled as a light inextensible string. Given that the tension in the wire is 12 N , calculate
- the weight of the bar,
- the magnitude of the reaction on the bar at \(A\).
- State briefly how you have used the given modelling assumption about the bar.
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A small packet, of mass 1.2 kg , is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between the packet and the plane is \(\frac { 1 } { 8 }\).
When a force of magnitude 8.4 N , acting parallel to the plane, is applied to the packet as shown, the packet is just on the point of moving up the plane. Modelling the packet as a particle,