A bee flies in a straight line from \(A\) to \(B\), where \(\overrightarrow { A B } = \left( 3 \frac { 1 } { 2 } \mathbf { i } - 12 \mathbf { j } \right) \mathrm { m }\), in 5 seconds at a constant speed. Find
the straight-line distance \(A B\),
the speed of the bee,
the velocity vector of the bee.
A small ball \(B\), of mass 0.8 kg , is suspended from a horizontal ceiling by two light inextensible strings. \(B\) is in equilibrium under gravity with both strings inclined at \(30 ^ { \circ }\) to the horizontal, as shown.
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Find the tension, in N , in either string.
Calculate the magnitude of the least horizontal force that must be applied to \(B\) in this position to cause one string to become slack.
A particle \(P\) moves in a straight line through a fixed point \(O\) with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\). 3 seconds after passing through \(O , P\) is 6 m from \(O\).
After a further 6 seconds, \(P\) has travelled a further 33 m in the same direction. Calculate
the value of \(a\),
the speed with which \(P\) passed through \(O\).
A force of magnitude \(F \mathrm {~N}\) is applied to a block of mass \(M \mathrm {~kg}\) which is initially at rest on a horizontal plane. The block starts to move with acceleration \(3 \mathrm {~ms} ^ { - 2 }\). Modelling the block as a particle,
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if the plane is smooth, find an expression for \(F\) in terms of \(M\).
If the plane is rough, and the coefficient of friction between the block and the plane is \(\mu\),
express \(F\) in terms of \(M , \mu\) and \(g\).
Calculate the value of \(\mu\) if \(F = \frac { 1 } { 2 } M g\).