Edexcel M3 2016 January — Question 3 9 marks

Exam BoardEdexcel
ModuleM3 (Mechanics 3)
Year2016
SessionJanuary
Marks9
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TopicCircular Motion 1
TypeBanked track – with friction (find maximum/minimum speed or friction coefficient)
DifficultyStandard +0.8 This is a challenging M3 banked track problem requiring resolution of forces in two directions (parallel and perpendicular to the plane), incorporation of limiting friction, and solving simultaneous equations involving trigonometric terms. While the setup is standard for M3, the algebra and coordination of multiple force components makes it significantly harder than average A-level questions.
Spec3.03t Coefficient of friction: F <= mu*R model6.05c Horizontal circles: conical pendulum, banked tracks
3. A car of mass 800 kg is driven at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) round a bend in a race track. Around the bend, the track is banked at \(20 ^ { \circ }\) to the horizontal and the path followed by the car can be modelled as a horizontal circle of radius 20 m . The car is modelled as a particle. The coefficient of friction between the car tyres and the track is 0.5 Given that the tyres do not slip sideways on the track, find the maximum value of \(v\).
Question 3:
AnswerMarks Guidance
WorkingMarks Guidance
\(R(\uparrow): \ R\cos 20° = F\cos 70° + 800g\)M1A1A1 Resolve vertically; A1A1 completely correct; A1A0 one error; A0A0 two or more errors
\(NL2(\rightarrow): \ R\cos 70° + F\cos 20° = 800\frac{v^2}{20}\)M1A1A1 Equation of motion along radius; LHS correct; RHS correct inc acceleration in form \(\frac{v^2}{r}\)
\(F = \mu R = 0.5R\)
\(R\cos 20° = 0.5R\cos 70° + 800g\)
\(R = \frac{800g}{(\cos 20° - 0.5\cos 70°)}\)
\(v^2 = \frac{1}{40}\frac{800g(\cos 70° + 0.5\cos 20°)}{(\cos 20° - 0.5\cos 70°)}\)ddM1 Use \(F=\mu R\) and eliminate \(R\) to obtain \(v^2\); depends on both previous M marks
\(v = 14.38... = 14.4\) or \(14 \ \text{m s}^{-1}\)dddM1A1cao [9] Complete to numerical value; correct value of \(v\) to 2 or 3 sf 
## Question 3:

| Working | Marks | Guidance |
|---------|-------|----------|
| $R(\uparrow): \ R\cos 20° = F\cos 70° + 800g$ | M1A1A1 | Resolve vertically; A1A1 completely correct; A1A0 one error; A0A0 two or more errors |
| $NL2(\rightarrow): \ R\cos 70° + F\cos 20° = 800\frac{v^2}{20}$ | M1A1A1 | Equation of motion along radius; LHS correct; RHS correct inc acceleration in form $\frac{v^2}{r}$ |
| $F = \mu R = 0.5R$ | | |
| $R\cos 20° = 0.5R\cos 70° + 800g$ | | |
| $R = \frac{800g}{(\cos 20° - 0.5\cos 70°)}$ | | |
| $v^2 = \frac{1}{40}\frac{800g(\cos 70° + 0.5\cos 20°)}{(\cos 20° - 0.5\cos 70°)}$ | ddM1 | Use $F=\mu R$ and eliminate $R$ to obtain $v^2$; depends on both previous M marks |
| $v = 14.38... = 14.4$ or $14 \ \text{m s}^{-1}$ | dddM1A1cao [9] | Complete to numerical value; correct value of $v$ to 2 or 3 sf |

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3. A car of mass 800 kg is driven at constant speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ round a bend in a race track. Around the bend, the track is banked at $20 ^ { \circ }$ to the horizontal and the path followed by the car can be modelled as a horizontal circle of radius 20 m . The car is modelled as a particle. The coefficient of friction between the car tyres and the track is 0.5

Given that the tyres do not slip sideways on the track, find the maximum value of $v$.\\

\hfill \mbox{\textit{Edexcel M3 2016 Q3 [9]}}
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