A2 June 2022 Q7
7. A spring of natural length \(a\) has one end attached to a fixed point \(A\). The other end of the spring is attached to a package \(P\) of mass \(m\).
The package \(P\) is held at rest at the point \(B\), which is vertically below \(A\) such that \(AB = 3a\).
After being released from rest at \(B\), the package \(P\) first comes to instantaneous rest at \(A\).
Air resistance is modelled as being negligible.
By modelling the spring as being light and modelling \(P\) as a particle,
In reality, the spring is not light.
| Scheme | Marks | AO |
|---|---|---|
| EPE at \(A = \dfrac{\lambda a^2}{2a}\) or EPE at \(B = \dfrac{\lambda(2a)^2}{2a}\) | M1 | 2.1 |
| Form work-energy equation: | M1 | 3.3 |
| \(\dfrac{\lambda a^2}{2a} + mg \times 3a = \dfrac{\lambda(2a)^2}{2a}\) \(\left(\dfrac{\lambda a}{2} + 3mga = 2\lambda a\right)\) | A1 A1 | 1.1b 1.1b |
| \(3mg = \dfrac{3\lambda}{2} \Rightarrow \lambda = 2mg\) * | A1* | 2.2a |
| (5) |
Notes
M1: Correct method for EPE seen or implied
Need something of the form \(\dfrac{1}{2}kx^2\) where \(k = \dfrac{\lambda}{a}\)
Must be using the formula for EPE correctly at least once
M1: Require all terms. Dimensionally correct. Condone their EPE. Condone sign errors
A1 A1: Unsimplified equation with at most one error. A repeated error in EPE formula is one error
Correct unsimplified equation.
A1*: Obtain given answer from correct working
| Scheme | Marks | AO |
|---|---|---|
| Extension at equilibrium: | M1 | 2.1 |
| \(\dfrac{2mgx}{a} = mg \Rightarrow x = \dfrac{a}{2}\) * | A1* | 1.1b |
| Use work-energy equation to find max speed: | M1 | 3.4 |
| \(\dfrac{2mgx^2}{2a} + mg \times (2a - x) + \dfrac{1}{2}mV^2 = \dfrac{2mg(2a)^2}{2a}\) \(\left(\dfrac{ag}{4} + \dfrac{3ag}{2} + \dfrac{1}{2}V^2 = 4ag\right)\) | A1 A1 | 1.1b 1.1b |
| \(V = 3\sqrt{\dfrac{ag}{2}}\) | A1 | 2.2a |
| (6) |
Notes
M1: Use correct method for tension to find the extension at equilibrium. Need to see the formula for tension used.
Allow verification with an appropriate conclusion
If they use SHM they must use \(F = ma\) to prove that \(P\) is moving with SHM, otherwise 0/2.
A1*: Correct answer from correct work
Allow verification with an appropriate conclusion
M1: Use given \(x\) to form work-energy equation. Need all terms, and dimensionally correct. Condone sign errors.
Accept with values of \(\lambda\) and \(x\) not substituted
A1 A1: Unsimplified equation with at most one error. Need given \(\lambda\) and given \(x\) substituted at some point. A repeated error in the formula for EPE is one error.
Correct unsimplified equation with given \(\lambda\) and given \(x\) substituted at some point
A1: Use correct method for tension to find the extension at equilibrium. Any equivalent form. \(2.1\sqrt{ag}\) or better
Alternative for the first M1A1
| Scheme | Marks | AO |
|---|---|---|
| Use the work-energy equation to obtain \(\dfrac{\mathrm{d}V^2}{\mathrm{d}x}\) and set the derivative equal to zero | M1 | |
| \(\dfrac{1}{a} \times 2x - 1 = 0 \Rightarrow x = \dfrac{a}{2}\) | A1 |
Alt: M1: Or an equivalent method for finding the turning point of a quadratic
Alt: A1*: Correct answer from correct work
| Scheme | Marks | AO |
|---|---|---|
| e.g. for B1 Need to include the GPE of the spring The extension of the spring at equilibrium will be different The spring will have KE You would need to include the KE of the spring in the energy equation You would need to include the GPE of the spring in the energy equation The GPE of the system changes It would take work to raise the spring so the package would have less KE If the spring has mass then GPE of the spring would need to be included | B1 | 3.5b |
| (1) | ||
| (12 marks) |
Notes
B1: Any valid response.
B0 if answer includes an additional incorrect factor. Must be specific e.g. not just “the GPE changes”, but the GPE of the system changes is OK.
Must relate to an effect on the energy equation
E.g. for B0
The extension changes
\(AB\) will increase
The tension/energy/GPE/work done etc would increase
The KE/GPE/EPE/acceleration/extension/velocity changes
The mass of the spring would drag down and the EPE would change
The EPE/KE/GPE etc would be variable
There would be tension in the spring as well
It has weight
The velocity would decrease as energy is converted