Work done by constant force - vector setup

3. \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. At a certain instant, a particle \(P\) of mass 1.8 kg is moving with velocity \(( 24 \mathrm { i } - 7 \mathrm { j } ) \mathrm { ms } ^ { - 1 }\).

1. Calculate the kinetic energy of \(P\) at this instant. \(P\) is now subjected to a constant retardation. After 10 seconds, the velocity of \(P\) is \(( - 12 \mathbf { i } + 3 \cdot 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
2. Calculate the work done by the retarding force over the 10 seconds.

1. A bead of mass 0.5 kg is threaded on a smooth straight wire. The only forces acting on the bead are a constant force ( \(4 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k }\) ) N and the normal reaction of the wire. The bead starts from rest at the point \(A\) with position vector \(( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and moves to the point \(B\) with position vector \(( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\).

Find the speed of the bead when it reaches \(B\).
(4)

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal, \(x - y\) plane.]

A bead \(P\) of mass 0.08 kg is threaded on a smooth straight horizontal wire which lies along the line with equation \(y = 2 x - 1\). The unit of length on both axes is the metre. Initially the bead is at rest at the point \(( a , b )\). A force \(( 6 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it along the wire so that \(P\) passes through the point \(( 5,9 )\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(a\) and the value of \(b\).

4 The diagram shows the path of a particle P of mass 2 kg as it moves from the origin O to C via A and B . The lengths of the sections \(\mathrm { OA } , \mathrm { AB }\) and BC are given in the diagram. The units of the axes are metres. \includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-4_670_1322_404_246} P , starting from O , moves along the path indicated in the diagram to C under the action of a constant force of magnitude \(T \mathrm {~N}\) acting in the positive \(x\)-direction. As P moves, it does \(R \mathrm {~J}\) of work for every metre travelled against resistances to motion. It is given that

• the speed of P at O is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• the speed of P at A is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• the speed of P at C is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

You should assume that both \(x\) - and \(y\)-axes lie in a horizontal plane.

1. By considering the entire path of P from O to C , show that $$20 \mathrm {~T} - 30 \mathrm { R } = 108 .$$
2. By formulating a second equation, determine the values of \(T\) and \(R\).
3. It is now given that the \(x\)-axis is horizontal, and the \(y\)-axis is directed vertically upwards. By considering the kinetic energy of P at B , show that the motion as described above is impossible.

3. Three forces \(( 4 \mathbf { i } - 7 \mathbf { j } + 9 \mathbf { k } ) \mathrm { N } , ( 5 \mathbf { i } + 3 \mathbf { j } - 8 \mathbf { k } ) \mathrm { N }\) and \(( - 2 \mathbf { i } + 6 \mathbf { j } - 11 \mathbf { k } ) \mathrm { N }\) act on a particle.

1. Find the resultant \(\mathbf { R }\) of the three forces.
2. The points \(A\) and \(B\) have position vectors \(( 3 \mathbf { i } + 4 \mathbf { j } - 12 \mathbf { k } ) \mathrm { m }\) and \(( a \mathbf { i } + 7 \mathbf { j } - 10 \mathbf { k } ) \mathrm { m }\) respectively, where \(a\) is a constant. The work done by \(\mathbf { R }\) in moving the particle from \(A\) to \(B\) is 21 J . Calculate the value of \(a\).
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\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane. A body of mass 1 kg moves under the action of a constant force \((4\mathbf{i} + 5\mathbf{j})\) N. The body moves from the point \(P\) with position vector \((-3\mathbf{i} - 15\mathbf{j})\) m to the point \(Q\) with position vector \(9\mathbf{i}\) m.

1. Find the work done by the force in moving the body from \(P\) to \(Q\). [5 marks]
2. Given that the body started from rest at \(P\), find its speed when it is at \(Q\). [5 marks]

A particle moves from the point \(A\) with position vector \((3i - j + 3k)\) m to the point \(B\) with position vector \((i - 2j - 4k)\) m under the action of the force \((2i - 3j - k)\) N. Find the work done by the force. [4]

A particle of mass 0.5 kg is at rest at the point with position vector \((2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})\) m. The particle is then acted upon by two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \((4\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})\) m with speed 12 m s\(^{-1}\). Given that \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j} - \mathbf{k})\) N, find \(\mathbf{F}_2\). [9]

A particle moves from the point \(A\) with position vector \((3\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) m to the point \(B\) with position vector \((\mathbf{i} - 2\mathbf{j} - 4\mathbf{k})\) m under the action of the force \((2\mathbf{i} - 3\mathbf{j} - \mathbf{k})\) N. Find the work done by the force. [4]

A bead of mass 0.125 kg is threaded on a smooth straight horizontal wire. The bead moves from rest at the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j} - \mathbf{k})\) m relative to a fixed origin \(O\) to a point with position vector \((3\mathbf{i} - 4\mathbf{j} - \mathbf{k})\) m relative to \(O\) under the action of a force \(\mathbf{F} = (14\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) N. Find

1. the work done by \(\mathbf{F}\) as the bead moves from \(A\) to \(B\), [3]
2. the speed of the bead at \(B\). [2]