M5 June 2017 Q1
1. [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 \(y = 2x - 1\). The unit of length on both axes is the metre. Initially the bead is at rest at the point \((a, b)\). A force \((6\mathbf{i} - 2\mathbf{j})\) N acts on \(P\) and moves it along the wire so that \(P\) passes through the point \((5, 9)\) with speed 10 m s\(^{-1}\).
Find the value of \(a\) and the value of \(b\). (7)
| Scheme | Marks |
|---|---|
| \(b = 2a - 1\) | B1 |
| \((6\mathbf{i} - 2\mathbf{j})\cdot((5 - a)\mathbf{i} + (9 - b)\mathbf{j}) = \tfrac{1}{2} \times 0.08 \times 10^2\) | M1 A2 |
| \(-3a + b = -4\) | |
| \(a = 3,\ b = 5\) | DM1 A1, A1 |
| (7) |
Notes
Using Work-energy
B1 for \(b = 2a - 1\)
First M1 for use of work-energy principle with usual rules
A2 for a correct equn in any form with dot product evaluated
Second DM1 for solving two simultaneous equation, for either \(a\) or \(b\)
Third A1 for \(a = 3\)
Fourth A1 for \(b = 5\)
See Alternative
ALTERNATIVE
| \((6\mathbf{i} - 2\mathbf{j})\cdot\dfrac{1}{\sqrt{5}}(\mathbf{i} + 2\mathbf{j}) = 0.08a\) | M1 |
| \(a = 5\sqrt{5}\) | A1 |
| \(10^2 = 2 \times 5\sqrt{5} \times s\) | DM1 |
| \(s = 2\sqrt{5}\) | A1 |
| \(\mathbf{s} = 2(\mathbf{i} + 2\mathbf{j})\) | B1 |
| \(a = 3,\ b = 5\) | A1 A1 |
ALTERNATIVE using force - accln
First M1 for use of \(\mathbf{F} = m\mathbf{a}\) along the wire with usual rules
First A1 for correct \(a\)
Second DM1 for using \(v^2 = u^2 + 2as\) along wire
Second A1 for \(s = 2\sqrt{5}\)
B1 for use of \(y = 2x - 1\)
Third A1 for \(a = 3\)
Fourth A1 for \(b = 5\)
ALTERNATIVE
| \(\lambda(\mathbf{i} + 2\mathbf{j})\) | B1 |
| \((6\mathbf{i} - 2\mathbf{j})\cdot\lambda(\mathbf{i} + 2\mathbf{j}) = \dfrac{1}{2}0.08 \times 10^2\) | M1 A2 |
| \(6\lambda - 4\lambda = 4 \Rightarrow \lambda = 2\) | DM1 |
| \((2\mathbf{i} + 4\mathbf{j}) = (5 - a)\mathbf{i} + (9 - b)\mathbf{j}\) | |
| \(a = 3,\ b = 5\) | A1 A1 |