A2 June 2024 Q2
2. A rough plane is inclined to the horizontal at an angle \(\theta\), where \(\tan\theta = \dfrac{3}{4}\)
A particle \(P\) of mass \(m\) is at rest at a point on the plane.
The particle is projected up the plane with speed \(\sqrt{2ag}\)
The particle moves up a line of greatest slope of the plane and comes to instantaneous rest after moving a distance \(d\).
The coefficient of friction between \(P\) and the plane is \(\dfrac{1}{7}\)
Air resistance is assumed to be negligible.
Using the work-energy principle,
| Scheme | Marks | AO |
|---|---|---|
| Resolve perpendicular to the plane and use \(F = \mu R\) | M1 | 3.1a |
| \(\tfrac{1}{7}mg\cos\theta\) | A1 | 1.1b |
| \(\dfrac{1}{7}mg \times \dfrac{4}{5} = \dfrac{4mg}{35}\) * or \(\dfrac{4}{35}mg\) * | A1* | 1.1b |
| (3) |
Notes
M1: Resolve perpendicular to the plane to find an expression for \(R\) and use \(\mu R\). Condone sin/cos confusion on weight component. All required terms present and no extras. Dimensionally correct.
A1: Correct unsimplified expression for friction. Allow with \(\cos\theta\) or \(\tfrac{4}{5}\)
A1*: Given answer correctly obtained. Working out must include both \(\tfrac{1}{7}\) and \(\tfrac{4}{5}\) in the same line before reaching the given answer.
| Scheme | Marks | AO |
|---|---|---|
| Use of work-energy principle | M1 | 3.3 |
| \(\dfrac{4mgd}{35} = \dfrac{1}{2}m \times 2ag - mgd\sin\theta\) | A1 A1 | 1.1b 1.1b |
| \(d = \dfrac{7a}{5}\) oe | A1 | 1.1b |
| (4) | ||
| (7 marks) |
Notes
M1: Use of work-energy principle with correct number of terms: 1 work, 1 KE, 1 GPE. Condone \(\pm\) sign errors on terms. All required terms present and no extras.
Resolve only when required and condone cos/sin confusion for the method mark.
Must use given answer from (a) in work term.
M0 if Friction is not multiplied by distance (dimensionally incorrect)
M0 if a term is missing.
M0 for incorrect use of trig eg \(d\tan\theta,\ \dfrac{d}{\sin\theta},\ \dfrac{d}{\cos\theta}\ \ldots\)
A1: Correct equation with at most one error
A1: Correct equation
A1: Correct answer for \(d\). Any equivalent fraction or decimal multiple of \(a\)