| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Work done and energy |
| Type | Particle on smooth curved surface |
| Difficulty | Challenging +1.2 This M3 question involves standard circular motion energy concepts with calculus optimization. Parts (a) and (b) are routine formula applications (KE = ½mv², PE = mgh), while part (c) requires differentiating total energy with respect to time and finding the maximum—a multi-step problem requiring chain rule and understanding that power relates to the rate of energy change. The conceptual leap to recognize that dE/dt = Power = T·v·sin(angle) and optimizing this is moderately challenging but follows standard M3 techniques. |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.05a Angular velocity: definitions |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(K.E. = \frac{1}{2}mv^2 = \frac{1}{2}m\ell^2\omega^2\) J | M1 A1 | |
| (b) \(P.E. = mg\ell(1 - \cos\theta)\) J | M1 A1 | |
| (c) Total energy \(E = \frac{1}{2}m\ell^2\omega^2 + mg\ell(1 - \cos\theta)\) J | M1 A1 | |
| \(\frac{dE}{d\theta} = mg\ell\sin\theta\frac{d\theta}{dt} = mg\ell\omega\sin\theta\) | A1 | Maximum when \(\theta = \frac{\pi}{2}\) |
| Total: 7 marks |
(a) $K.E. = \frac{1}{2}mv^2 = \frac{1}{2}m\ell^2\omega^2$ J | M1 A1 |
(b) $P.E. = mg\ell(1 - \cos\theta)$ J | M1 A1 |
(c) Total energy $E = \frac{1}{2}m\ell^2\omega^2 + mg\ell(1 - \cos\theta)$ J | M1 A1 |
$\frac{dE}{d\theta} = mg\ell\sin\theta\frac{d\theta}{dt} = mg\ell\omega\sin\theta$ | A1 | Maximum when $\theta = \frac{\pi}{2}$ |
| **Total: 7 marks** |A particle of mass $m$ kg is attached to one end of a light inextensible string of length $l$ m whose other end is fixed to a point $O$. The particle is made to move in a vertical circle with centre $O$, with constant angular velocity $\omega$ rad s$^{-1}$. At a certain instant it is in the position shown, where the string makes an angle $\theta$ radians with the downward vertical through $O$.
\includegraphics{figure_1}
\begin{enumerate}[label=(\alph*)]
\item Find an expression, in terms of $m$, $l$ and $\omega$, for the kinetic energy of the particle at this instant. [2 marks]
\item Find an expression, in terms of $m$, $g$, $l$ and $\theta$, for the potential energy of the particle relative to the horizontal plane through the lowest point $A$. [2 marks]
\item Determine the position of the particle when the rate of increase of its total energy, with respect to time, is a maximum. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q1 [7]}}