| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Coupled circular motions |
| Difficulty | Challenging +1.2 Part (a) is trivial substitution (1 mark). Part (b) requires equating centripetal force to gravitational force for circular orbits, yielding v ∝ 1/√r, which is standard M3 material (2 marks). Part (c) is more demanding: students must recognize that the wire constraint forces both satellites to have the same angular velocity (not their natural orbital speeds), then calculate the tension by considering the imbalance between required centripetal force and gravitational force for each satellite. This requires careful force analysis and algebraic manipulation over 8 marks, making it moderately challenging but still within standard M3 problem-solving scope. |
| Spec | 3.03k Connected particles: pulleys and equilibrium6.05a Angular velocity: definitions |
| Answer | Marks |
|---|---|
| (a) \(mg = k\frac{m\pi^2}{l}\) (given), so \(k = g\ell^2\) | B1 |
| (b) Equal periods, so \(\frac{2\pi(4R)}{v_a} = \frac{2\pi(3R)}{v_b}\) | M1 A1 |
| \(\frac{v_a}{v_b} = \frac{4}{3}\) | |
| (c) \(\frac{g\pi^2 m}{9} - T = \frac{m v_a^2}{3R}\); \(\frac{g\pi^2}{16} + T = \frac{16mv^2}{9(4R)}\) | M1 A1 A1 |
| \(\frac{mg}{9} - T = \frac{m v^2}{3R}\), \(\frac{mg}{16} + T = \frac{4mv^2}{9R}\) | |
| Solve: \(T = \frac{37}{1008}mg\) | M1 A1 M1 A1 |
**(a)** $mg = k\frac{m\pi^2}{l}$ (given), so $k = g\ell^2$ | B1 |
**(b)** Equal periods, so $\frac{2\pi(4R)}{v_a} = \frac{2\pi(3R)}{v_b}$ | M1 A1 |
$\frac{v_a}{v_b} = \frac{4}{3}$ | |
**(c)** $\frac{g\pi^2 m}{9} - T = \frac{m v_a^2}{3R}$; $\frac{g\pi^2}{16} + T = \frac{16mv^2}{9(4R)}$ | M1 A1 A1 |
$\frac{mg}{9} - T = \frac{m v^2}{3R}$, $\frac{mg}{16} + T = \frac{4mv^2}{9R}$ | |
Solve: $T = \frac{37}{1008}mg$ | M1 A1 M1 A1 |
**Total: 11 marks**The radius of the Earth is $R$ m. The force of attraction towards the centre of the Earth experienced by a body of mass $m$ kg at a distance $x$ m from the centre is of magnitude $\frac{km}{x^2}$ N, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = gR^2$. [1 mark]
\end{enumerate}
Two satellites $A$ and $B$, each of mass $m$ kg, are moving in circular orbits around the Earth at distances $3R$ m and $4R$ m respectively from the centre of the Earth. Given that the satellites move in the same plane and that they lie along the same radial line from the centre at any time,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item show that the ratio of the speed of $B$ to that of $A$ is $4:3$. [2 marks]
\end{enumerate}
If, in addition, the satellites are linked with a taut, straight wire in the same plane and along the same radial line,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find, in terms of $m$ and $g$, the magnitude of the force in the wire. [8 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q4 [11]}}