| Exam Board | Edexcel |
|---|---|
| Module | FM1 (Further Mechanics 1) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Motion on a slope |
| Type | Motion up rough slope |
| Difficulty | Standard +0.3 This is a standard Further Mechanics 1 question requiring resolution of forces on a slope (using tan θ = 3/4 to find sin θ and cos θ), calculation of friction force, and application of work-energy principle. Part (a) is routine force resolution with a 'show that' answer provided, and part (b) is straightforward application of work-energy. While it requires multiple techniques, all are standard FM1 procedures with no novel insight needed, making it slightly easier than average. |
| Spec | 3.03v Motion on rough surface: including inclined planes6.02c Work by variable force: using integration6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Resolve perpendicular to the plane and use \(F = \mu R\) | M1 | Condone sin/cos confusion on weight component. All required terms present and no extras. Dimensionally correct. |
| \(\frac{1}{7}mg\cos\theta\) | A1 | Correct unsimplified expression for friction. Allow with \(\cos\theta\) or \(\frac{4}{5}\) |
| \(\frac{1}{7}mg \times \frac{4}{5} = \frac{4mg}{35}\) or \(\frac{4}{35}mg\) | A1* | Given answer. Working must include both \(\frac{1}{7}\) and \(\frac{4}{5}\) in the same line before reaching the given answer. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use of work-energy principle | M1 | Correct number of terms: 1 work, 1 KE, 1 GPE. Condone \(\pm\) sign errors. Must use given answer from (a) in work term. M0 if friction not multiplied by distance. M0 for incorrect trig e.g. \(d\tan\theta\), \(\frac{d}{\sin\theta}\), \(\frac{d}{\cos\theta}\) |
| \(\frac{4mgd}{35} = \frac{1}{2}m \times 2ag - mgd\sin\theta\) | A1 | Correct equation with at most one error |
| (above equation fully correct) | A1 | Correct equation |
| \(d = \frac{7a}{5}\) | A1 | Correct answer for \(d\). Any equivalent fraction or decimal multiple of \(a\) |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Resolve perpendicular to the plane **and** use $F = \mu R$ | M1 | Condone sin/cos confusion on weight component. All required terms present and no extras. Dimensionally correct. |
| $\frac{1}{7}mg\cos\theta$ | A1 | Correct unsimplified expression for friction. Allow with $\cos\theta$ or $\frac{4}{5}$ |
| $\frac{1}{7}mg \times \frac{4}{5} = \frac{4mg}{35}$ or $\frac{4}{35}mg$ | A1* | **Given** answer. Working must include both $\frac{1}{7}$ and $\frac{4}{5}$ in the same line before reaching the given answer. |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of work-energy principle | M1 | Correct number of terms: 1 work, 1 KE, 1 GPE. Condone $\pm$ sign errors. Must use given answer from (a) in work term. M0 if friction not multiplied by distance. M0 for incorrect trig e.g. $d\tan\theta$, $\frac{d}{\sin\theta}$, $\frac{d}{\cos\theta}$ |
| $\frac{4mgd}{35} = \frac{1}{2}m \times 2ag - mgd\sin\theta$ | A1 | Correct equation with at most one error |
| (above equation fully correct) | A1 | Correct equation |
| $d = \frac{7a}{5}$ | A1 | Correct answer for $d$. Any equivalent fraction or decimal multiple of $a$ |
---\begin{enumerate}
\item A rough plane is inclined to the horizontal at an angle $\theta$, where $\tan \theta = \frac { 3 } { 4 }$
\end{enumerate}
A particle $P$ of mass $m$ is at rest at a point on the plane.
The particle is projected up the plane with speed $\sqrt { 2 a g }$\\
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 $\frac { 1 } { 7 }$\\
(a) Show that the magnitude of the frictional force acting on $P$ as it moves up the plane is $\frac { 4 m g } { 35 }$
Air resistance is assumed to be negligible.\\
Using the work-energy principle,\\
(b) find $d$ in terms of $a$.
\hfill \mbox{\textit{Edexcel FM1 2024 Q2 [7]}}