Briefly define the following terms used in modelling in Mechanics:
lamina,
uniform rod,
smooth surface,
particle.
(4 marks)
Two forces \(\mathbf { F }\) and \(\mathbf { G }\) are given by \(\mathbf { F } = ( 6 \mathbf { i } - 5 \mathbf { j } ) \mathbf { N } , \mathbf { G } = ( 3 \mathbf { i } + 17 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the \(x\) and \(y\) directions respectively and the unit of length on each axis is 1 cm .
(a) Find the magnitude of \(\mathbf { R }\), the resultant of \(\mathbf { F }\) and \(\mathbf { G }\).
(b) Find the angle between the direction of \(\mathbf { R }\) and the positive \(x\)-axis.
\(\mathbf { R }\) acts through the point \(P ( - 4,3 )\). \(O\) is the origin \(( 0,0 )\).
(c) Use the fact that \(O P\) is perpendicular to the line of action of \(\mathbf { R }\) to calculate the moment of \(\mathbf { R }\) about an axis through the origin and perpendicular to the \(x - y\) plane.
A string is attached to a packing case of mass 12 kg , which is at rest on a rough horizontal plane. When a force of magnitude 50 N is applied at the other end of the string, and
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the string makes an angle of \(35 ^ { \circ }\) with the vertical as shown, the case is on the point of moving.
(a) Find the coefficient of friction between the case and the plane.
The force is now increased, with the string at the same angle, and the case starts to move along the plane with constant acceleration, reaching a speed of \(2 \mathrm {~ms} ^ { - 1 }\) after 4 seconds.
(b) Find the magnitude of the new force.
(c) State any modelling assumptions you have made about the case and the string.