| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics A AS (Further Mechanics A AS) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Friction |
| Type | Particle on inclined plane motion |
| Difficulty | Standard +0.8 This is a multi-part mechanics question requiring resolution of forces in two directions with friction, including a proof and optimization. Parts (a)-(b) are standard, but parts (c)-(d) require careful algebraic manipulation and finding a limiting condition by analyzing when the inequality can never be satisfied, which demands deeper problem-solving insight than typical A-level mechanics questions. |
| Spec | 3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces3.03v Motion on rough surface: including inclined planes |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | Let the frictional force up the slope have magnitude F N and |
| Answer | Marks | Guidance |
|---|---|---|
| R m g c o s = and m g s i n F = | M1 | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| max | M1 | 3.4 |
| Answer | Marks | Guidance |
|---|---|---|
| m g c o s R | A1 | 1.1 |
| Answer | Marks |
|---|---|
| (b) | Let the block have acceleration a ms-2 down the slope. |
| Answer | Marks | Guidance |
|---|---|---|
| m a x | M1 | 3.5a |
| m g s i n 4 0 − 59 m g c o s 4 0 = m a a = 2 .1 2 8 6 3 | A1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| (c) | R m g c o s P s i n = + | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| P c o s m g s i n 59 ( m g c o s P s i n ) + + | M1 | 2.1 |
| Answer | Marks |
|---|---|
| argued later | M0 for sliding |
| Answer | Marks | Guidance |
|---|---|---|
| P ( c o s 59 s i n ) m g ( 59 c o s s i n ) − + | A1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | c o s 59 s i n 0 - = | |
| m in m in | M1 | 3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| m in | A1 | 1.1 |
Question 6:
6 | (a) | Let the frictional force up the slope have magnitude F N and
the normal contact force have magnitude R N.
R m g c o s = and m g s i n F = | M1 | 3.3 | Both soi
Condone θ used instead of α
Limiting equilibrium ⇒ 𝐹 = 𝐹 = 𝜇𝑅
max | M1 | 3.4 | soi
mgsinα = μmgcosα implies M1M1
Not inequality (unless recovered)
m g s i n 59 R
t a n 59 = =
m g c o s R | A1 | 1.1 | AG, requires proper explanation
throughout
(e.g. F = μR preceding eqn above)
[3]
(b) | Let the block have acceleration a ms-2 down the slope.
Block slides so F = F = 59 m g c o s 4 0
m a x | M1 | 3.5a | Accept F = μmgcos α
m g s i n 4 0 − 59 m g c o s 4 0 = m a a = 2 .1 2 8 6 3 | A1 | 1.1
[2]
(c) | R m g c o s P s i n = + | B1 | 3.3 | soi
So for equilibrium to be broken, we require
P c o s m g s i n 59 ( m g c o s P s i n ) + + | M1 | 2.1 | Any correct form Pcos α > … e.g.
P c o s m g s i n F + or
m a x
P c o s m g s i n R +
M0 for ‘Limiting when Pcos α = …’
unless inequality is correctly
argued later | M0 for sliding
down the slope
(F acting
upwards)
P ( c o s 59 s i n ) m g ( 59 c o s s i n ) − + | A1 | 2.2a | AG
[3]
(d) | c o s 59 s i n 0 - =
m in m in | M1 | 3.1a | Accept arguments made in terms of
inequalities.
6 0 .9 4 5 3 9 =
m in | A1 | 1.1 | Allow 1.0636978… radians.
[2]6 A block B of mass $m \mathrm {~kg}$ rests on a rough slope inclined at angle $\alpha$ to the horizontal. The coefficient of friction between $B$ and the slope is $\frac { 5 } { 9 }$.
\begin{enumerate}[label=(\alph*)]
\item When B is in limiting equilibrium, show that $\tan \alpha = \frac { 5 } { 9 }$.
\item If $\alpha = 40 ^ { \circ }$, determine the acceleration of B down the slope.
A horizontal force of magnitude $P \mathrm {~N}$ is now applied to B , as shown in the diagram below. At first B is at rest.\\
\includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-7_381_410_689_242}\\
$P$ is gradually increased.
\item Show that, for B to slide on the slope,
$$\mathrm { P } \left( \cos \alpha - \frac { 5 } { 9 } \sin \alpha \right) > \mathrm { mg } \left( \frac { 5 } { 9 } \cos \alpha + \sin \alpha \right) .$$
\item Determine, in degrees, the least value of $\alpha$ for which B will not slide no matter how large $P$ becomes.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2022 Q6 [10]}}