A2 June 2024 Q5
5. A light elastic string has natural length \(2a\) and modulus of elasticity \(2mg\).
One end of the string is attached to a fixed point \(A\) on a horizontal ceiling.
The other end is attached to a particle \(P\) of mass \(m\).
The particle \(P\) hangs in equilibrium at the point \(E\), where \(AE = 3a\).
The particle \(P\) is then projected vertically downwards from \(E\) with speed \(\dfrac{3}{2}\sqrt{ag}\)
Air resistance is assumed to be negligible.
Find the elastic energy stored in the string, when \(P\) first comes to instantaneous rest.
Give your answer in the form \(kmga\), where \(k\) is a constant to be found. (7)
| Scheme | Marks | AO |
|---|---|---|
 | ||
| GPE from \(E\) to instantaneous rest eg \(mgx\), \(mg(d - a)\) | M1 | 1.1b |
| Use of conservation of energy principle | M1 | 3.1a |
For example,
| A1 A1 | 1.1b 1.1b |
| \(x = \dfrac{3a}{2}\) or \(d = \dfrac{5a}{2}\) | A1 | 1.1b |
| Use of EPE formula | M1 | 2.1 |
| \(\dfrac{25mga}{8}\) | A1 | 2.2a |
| (7) | ||
| (7 marks) |
Notes
M1: Use of GPE for unknown distance from \(E\) to instantaneous rest. May be implied by a difference of 2 GPE terms.
M1: Use of conservation of energy principle to form an equation with one KE term, one GPE term, two EPE terms. For the method mark only we will, condone only one EPE term. However, all terms are required for A marks. Dimensionally correct energy equation (energy terms must have the correct structure). Note there are common rearrangements.
Eg Initial Energy = Final Energy, Energy Loss = Energy Gain oe
Condone \(\pm\) sign errors. M0 for use of suvat. M0 if ‘E’ or similar is used to represent unknown EPE unless recovered.
A1: All 4 terms present in an energy equation with one unknown length.
At most one error. A0 if an energy term is missing.
A1: Fully correct equation in one unknown length.
A1: cao
M1: Use of EPE formula at least once. May be seen here or in earlier working. EPE must have the form \(\dfrac{\lambda x^2}{ka}\) where \(\lambda\) is modulus of elasticity, \(k\) is a constant and \(x\) is extension.
A1: Correct answer