| Exam Board | OCR |
|---|---|
| Module | Further Mechanics (Further Mechanics) |
| Year | 2018 |
| Session | March |
| Marks | 6 |
| Topic | Advanced work-energy problems |
| Type | Variable force along axis work-energy |
| Difficulty | Standard +0.3 This is a straightforward application of work-energy principles with integration. Part (i) requires integrating a polynomial force function over a given interval (standard technique), and part (ii) applies the work-energy theorem directly. While it's a Further Maths topic, the mathematical steps are routine with no conceptual surprises, making it slightly easier than average. |
| Spec | 6.02b Calculate work: constant force, resolved component6.02c Work by variable force: using integration6.02d Mechanical energy: KE and PE concepts |
| Answer | Marks | Guidance |
|---|---|---|
| \(WD = \int_0^4 (12x - 3x^2) dx\) | Work done by the force = 32 J | M1, A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Initial KE = \(\frac{1}{2} \times 4.2 \times 2^2\) (=8.4) | B1, M1 | Need not be evaluated here; Work-Energy principle used |
| Work done = change in KE | M1 | |
| \(\frac{1}{2} \times 4.2v^2 = 32 + 8.4\) | A1FT | |
| Speed is 4.39 m s\(^{-1}\) | A1 [4] |
## (i)
$WD = \int_0^4 (12x - 3x^2) dx$ | Work done by the force = 32 J | M1, A1 [2] | BC
## (ii)
Initial KE = $\frac{1}{2} \times 4.2 \times 2^2$ (=8.4) | B1, M1 | Need not be evaluated here; Work-Energy principle used
Work done = change in KE | M1 |
$\frac{1}{2} \times 4.2v^2 = 32 + 8.4$ | A1FT |
Speed is 4.39 m s$^{-1}$ | A1 [4] |
---1 A particle $P$ of mass 4.2 kg is free to move along the $x$-axis which is horizontal. $P$ is projected from the origin, $O$, in the positive $x$ direction with a speed of $2 \mathrm {~ms} ^ { - 1 }$. As $P$ moves between $O$ and the point $A$ where $x = 4$, it is acted upon by a variable force of magnitude $\left( 12 x - 3 x ^ { 2 } \right) \mathrm { N }$ acting in the direction $O A$.\\
(i) Calculate the work done by the force as $P$ moves from $O$ to $A$.\\
(ii) Hence, assuming that no other force acts on $P$, calculate the speed of $P$ at $A$.
\hfill \mbox{\textit{OCR Further Mechanics 2018 Q1 [6]}}