| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Advanced work-energy problems |
| Type | Particle on sphere or circular surface |
| Difficulty | Challenging +1.2 This is a substantial M3 circular motion problem requiring energy conservation, normal reaction analysis, and projectile motion. While it has many parts (6 sub-questions), each step follows standard mechanics techniques: (a) uses energy conservation, (b-d) apply the condition for leaving the surface (N=0), and (e-f) are routine projectile calculations. The problem is longer than average and requires careful bookkeeping, but doesn't demand novel insight—it's a thorough application of well-practiced M3 methods. |
| Spec | 3.02h Motion under gravity: vector form6.02i Conservation of energy: mechanical energy principle6.05e Radial/tangential acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Energy: \(\frac{1}{2}mv^2 = \frac{1}{2}mu^2 + mga(1-\cos\theta)\) | M1 A1 | |
| \(v^2 = \frac{1}{2}ga + 2ga(1-\cos\theta)\) | ||
| \(= \frac{1}{2}ga(5 - 4\cos\theta)\) | A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(R(\nearrow)\): \(mg\cos\theta - R = \frac{mv^2}{a}\) | M1 A1 | |
| so \(R = mg(3\cos\theta - \frac{5}{2})\) | A1 | |
| \(\cos\theta = 0.9 \Rightarrow R = mg(2.7 - 2.5)\) | M1 | |
| \(= 0.2mg > 0 \Rightarrow P\) still on hemisphere | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P\) leaves hemisphere when \(R = 0 \Rightarrow 3\cos\theta - \frac{5}{2} = 0 \Rightarrow \cos\theta = \frac{5}{6}\) | M1 A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\cos\theta = \frac{5}{6} \Rightarrow v^2 = \frac{1}{2}ga\left(5 - 4 \times \frac{5}{6}\right)\) | M1 | |
| \(= \frac{5ga}{6}\), \(v = \sqrt{\frac{5ga}{6}}\) | A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| At \(B\), speed \(v\) given by \(v^2 = u^2 + 2ga = \frac{5}{2}ga\), \(v = \sqrt{\frac{5ga}{2}}\) | M1 A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| After leaving hemisphere, horizontal component of velocity remains constant \(= \sqrt{\frac{5ga}{6}} \cdot \frac{5}{6}\) | B1 | |
| \(\cos\phi = \dfrac{\frac{5}{6}\sqrt{\frac{5ga}{6}}}{\sqrt{\frac{5ga}{2}}} = \frac{5}{6\sqrt{3}}\) | M1 | |
| \(\Rightarrow \phi = 61.2°\) or \(61°\) to horizontal | A1 | (3) |
## Question 7:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Energy: $\frac{1}{2}mv^2 = \frac{1}{2}mu^2 + mga(1-\cos\theta)$ | M1 A1 | |
| $v^2 = \frac{1}{2}ga + 2ga(1-\cos\theta)$ | | |
| $= \frac{1}{2}ga(5 - 4\cos\theta)$ | A1 | (3) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R(\nearrow)$: $mg\cos\theta - R = \frac{mv^2}{a}$ | M1 A1 | |
| so $R = mg(3\cos\theta - \frac{5}{2})$ | A1 | |
| $\cos\theta = 0.9 \Rightarrow R = mg(2.7 - 2.5)$ | M1 | |
| $= 0.2mg > 0 \Rightarrow P$ still on hemisphere | A1 | (5) |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P$ leaves hemisphere when $R = 0 \Rightarrow 3\cos\theta - \frac{5}{2} = 0 \Rightarrow \cos\theta = \frac{5}{6}$ | M1 A1 | (2) |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\cos\theta = \frac{5}{6} \Rightarrow v^2 = \frac{1}{2}ga\left(5 - 4 \times \frac{5}{6}\right)$ | M1 | |
| $= \frac{5ga}{6}$, $v = \sqrt{\frac{5ga}{6}}$ | A1 | (2) |
### Part (e):
| Answer/Working | Marks | Guidance |
|---|---|---|
| At $B$, speed $v$ given by $v^2 = u^2 + 2ga = \frac{5}{2}ga$, $v = \sqrt{\frac{5ga}{2}}$ | M1 A1 | (2) |
### Part (f):
| Answer/Working | Marks | Guidance |
|---|---|---|
| After leaving hemisphere, horizontal component of velocity remains constant $= \sqrt{\frac{5ga}{6}} \cdot \frac{5}{6}$ | B1 | |
| $\cos\phi = \dfrac{\frac{5}{6}\sqrt{\frac{5ga}{6}}}{\sqrt{\frac{5ga}{2}}} = \frac{5}{6\sqrt{3}}$ | M1 | |
| $\Rightarrow \phi = 61.2°$ or $61°$ to horizontal | A1 | (3) |
**(17 marks)**7. A smooth solid hemisphere is fixed with its plane face on a horizontal table and its curved surface uppermost. The plane face of the hemisphere has centre $O$ and radius $a$. The point $A$ is the highest point on the hemisphere. A particle $P$ is placed on the hemisphere at $A$. It is then given an initial horizontal speed $u$, where $u ^ { 2 } = \frac { 1 } { 2 } ( a g )$. When $O P$ makes an angle $\theta$ with $O A$, and while $P$ remains on the hemisphere, the speed of $P$ is $v$.\\
(a) Find an expression for $v ^ { 2 }$.\\
(b) Show that, when $\theta = \arccos 0.9 , P$ is still on the hemisphere.\\
(c) Find the value of $\cos \theta$ when $P$ leaves the hemisphere.\\
(d) Find the value of $v$ when $P$ leaves the hemisphere.
After leaving the hemisphere $P$ strikes the table at $B$.\\
(e) Find the speed of $P$ at $B$.\\
(f) Find the angle at which $P$ strikes the table.
\section*{Alternative Question 2:}
\begin{enumerate}
\item Two light elastic strings $A B$ and $B C$ are joined at $B$. The string $A B$ has natural length 1 m and modulus of elasticity 15 N . The string $B C$ has natural length 1.2 m and modulus of elasticity 30 N . The ends $A$ and $C$ are attached to fixed points 3 m apart and the strings rest in equilibrium with $A B C$ in a straight line.
\end{enumerate}
Find the tension in the combined string $A C$.
\hfill \mbox{\textit{Edexcel M3 Q7 [17]}}