M3 June 2018 Q1
1. A rough disc is rotating in a horizontal plane with constant angular speed \(\omega\) about a vertical axis through the centre of the disc. A particle \(P\) is placed on the disc at a distance \(r\) from the axis. The coefficient of friction between \(P\) and the disc is \(\mu\).
Given that \(P\) does not slip on the disc, show that
\[\omega \leqslant \sqrt{\frac{\mu g}{r}}\](5)
NB: This is a “show that” question and candidates must make it clear that they are starting from the given information and deriving the given answer. It must be clear that the forces acting on the particle are being considered.
Consequently starting from \(mr\omega^2 \leqslant \mu mg\) (which can be obtained by working backwards from the answer) scores 0/5.
| Scheme | Marks |
|---|---|
| “\(F = ma\)” is sometimes seen. Do not penalise work that follows where \(F\) is used for friction. | |
| \(F \leqslant \mu mg\) or \(F = \mu mg\) or \(F \leqslant \mu R\) and \(R = mg\) or \(F = \mu R\) and \(R = mg\) | B1 |
| \(F = mr\omega^2\) or \(F \geqslant mr\omega^2\) | M1A1 |
| \(mr\omega^2 \leqslant \mu mg\) | dM1 |
| \(\omega \leqslant \sqrt{\dfrac{\mu g}{r}}\) * | A1cso |
| (5 marks) |
Notes
The following notes apply whatever method the candidate has attempted.
B1 \(F \leqslant \mu mg\) or \(F = \mu mg\) or \(F \leqslant \mu R\) and \(R = mg\) or \(F = \mu R\) and \(R = mg\) seen
Award for any of these four statements seen.
M1 Equation of motion horizontally. Acceleration in either form. Can be given in the form of an inequality. Must include \(F\)
A1 Correct equation or inequality, with acceleration \(r\omega^2\)
dM1 Eliminate \(F\) Must now have an inequality
A1cso Correct completion with no errors seen and clear notation. Candidates who work with = signs but have not specified the particle is on the point of slipping or seem to be using max friction but do not state this should not be awarded this mark.
Example 1
Here are 2 “perfect” examples. As written here they score 5/5. Same work but without reference to max friction or slipping would score 4/5
| \(F_{\max} = \mu mg\) or \(F_{\max} = \mu R\) and \(R = mg\) | B1 |
| \(F_{\max} \geqslant mr\omega^2\) | M1A1 |
| \(\mu mg \geqslant m\omega^2r\) | dM1 |
| \(\omega \leqslant \sqrt{\dfrac{\mu g}{r}}\) * | A1cso [5] |
Example 2
| On the point of slipping: \(F = \mu mg\) or \(F = \mu R\) and \(R = mg\) | B1 |
| \(F = mr\omega^2\) | M1A1 |
| \(\mu mg = m\omega^2r \quad \left(\Rightarrow \omega = \sqrt{\dfrac{\mu g}{r}}\right)\) | |
| Does not slip, \(\therefore \omega \leqslant \sqrt{\dfrac{\mu g}{r}}\) * | dM1A1cso [5] |
(Corrected from the printed mark scheme: in Example 1 a B1 is also printed against the introductory sentence, which would make 6 marks; the B1 belongs to the first line only.)