| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Elastic string on rough inclined plane |
| Difficulty | Challenging +1.2 This is a multi-step M3 mechanics problem requiring energy methods with elastic strings, impulse-momentum, and friction on an inclined plane. While it involves several components (impulse giving initial velocity, work done against friction and gravity, elastic potential energy), the approach is methodical and follows standard M3 techniques without requiring novel insight. The 'comes to rest when slack' condition provides a clear endpoint that simplifies the problem. |
| Spec | 6.02g Hooke's law: T = k*x or T = lambda*x/l6.02i Conservation of energy: mechanical energy principle6.03e Impulse: by a force |
| Answer | Marks |
|---|---|
| B1 | Correct value for \(u\), seen explicitly or used. |
| M1A1 | Attempt the work done against friction. Weight must be resolved (sin/cos interchange accepted.) Distance moved to be 0.6 m. Mass can be 0.5 or \(m\) |
| B1 | Correct work done. Mass can be 0.5 or \(m\) |
| M1A1A1 | Allow both of the above marks if the work done against friction is embedded in some incorrect work e.g. including other forces to form a resultant force. Correct initial EPE. Need not be simplified. The work done and the EPE may not be shown explicitly. Check the equation if necessary. |
| A1 | Attempt a complete work-energy equation. Must have an EPE, a GPE, a KE and a (dimensionally correct) work against friction term. The final KE may be included provided it becomes 0 here or later. EPE term must be of the form \(k\frac{x^2}{2}\), \(\frac{1}{2}kx^2\) or equivalent. |
| A1ft | Deduct one per error. Follow through their EPE and work. |
| A1 | Correct value of \(\theta\), 2 or 3 sf only. |
**4(a)**
B1 | Correct value for $u$, seen explicitly or used.
M1A1 | Attempt the work done against friction. Weight must be resolved (sin/cos interchange accepted.) Distance moved to be 0.6 m. Mass can be 0.5 or $m$
B1 | Correct work done. Mass can be 0.5 or $m$
M1A1A1 | Allow both of the above marks if the work done against friction is embedded in some incorrect work e.g. including other forces to form a resultant force. Correct initial EPE. Need not be simplified. The work done and the EPE may not be shown explicitly. Check the equation if necessary.
A1 | Attempt a complete work-energy equation. Must have an EPE, a GPE, a KE and a (dimensionally correct) work against friction term. The final KE may be included provided it becomes 0 here or later. EPE term must be of the form $k\frac{x^2}{2}$, $\frac{1}{2}kx^2$ or equivalent.
A1ft | Deduct one per error. Follow through their EPE and work.
A1 | Correct value of $\theta$, 2 or 3 sf only.
[8]
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\caption{Figure 2}
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Figure 2 shows a light elastic string, of modulus of elasticity $\lambda$ newtons and natural length 0.6 m . One end of the string is attached to a fixed point $A$ on a rough plane which is inclined at $30 ^ { \circ }$ to the horizontal. The other end of the string is attached to a particle $P$ of mass 0.5 kg . The string lies along a line of greatest slope of the plane. The particle is held at rest on the plane at the point $B$, where $B$ is lower than $A$ and $A B = 1.2 \mathrm {~m}$. The particle then receives an impulse of magnitude 1.5 N s in the direction parallel to the string, causing $P$ to move up the plane towards $A$. The coefficient of friction between $P$ and the plane is 0.7 . Given that $P$ comes to rest at the instant when the string becomes slack, find the value of $\lambda$.
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\hfill \mbox{\textit{Edexcel M3 2018 Q4 [8]}}