10 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A particle \(R\) of mass 2 kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf { F }\) N. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf { v } = \left( p t ^ { 2 } - 3 t \right) \mathbf { i } + ( 8 t + q ) \mathbf { j }\), where \(p\) and \(q\) are constants and \(p < 0\).
- Given that when \(t = 0.5\) the magnitude of \(\mathbf { F }\) is 20 , find the value of \(p\).
When \(t = 0 , R\) is at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m }\).
- Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\).
When \(t = 1 , R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2 \mathbf { i } - 8 \mathbf { j }\).
- Find the value of \(q\).
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The diagram shows a ladder \(A B\), of length \(2 a\) and mass \(m\), resting in equilibrium on a vertical wall of height \(h\). The ladder is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The end \(A\) is in contact with horizontal ground. An object of mass \(2 m\) is placed on the ladder at a point \(C\) where \(A C = d\).
The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth. - Show that the normal contact force between the ladder and the wall is \(\frac { m g ( a + 2 d ) \sqrt { 3 } } { 4 h }\).
It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 8 } \sqrt { 3 }\).
- Show that \(h = k ( a + 2 d )\), where \(k\) is a constant to be determined.
- Hence find, in terms of \(a\), the greatest possible value of \(d\).
- State one improvement that could be made to the model.