A2 June 2020 Paper 1 Q13
13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3c\) kg, which rests on a smooth horizontal table.
The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart.
\(ABC\) is a horizontal line.
String \(AB\) has a natural length of 4 metres and a stiffness of \(5c\) newtons per metre.
String \(BC\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\dfrac{1}{3}\) metre from its equilibrium position towards \(A\), and released from rest.
(a) Show that the particle moves with simple harmonic motion. [8 marks]
(b) Find the speed of the particle when it is at a point \(P\), a distance \(\dfrac{1}{4}\) metre from the equilibrium position. Give your answer to two significant figures. [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Deduces the total extension of both strings is 3. PI | B1 | 2.2a |
| Finds an expression for the tension in one string in terms of \(c\) and an extension. | B1 | 3.4 |
| Forms a two-term force equation at equilibrium. | M1 | 3.4 |
| Obtains the two correct equilibrium extensions. | A1 | 1.1b |
| Forms an equation of motion in terms of a general displacement with at least one correct extension FT their equilibrium extensions. | M1 | 3.4 |
| Obtains correct equation of motion FT their equilibrium extensions. | A1F | 1.1b |
| Simplifies their equation of motion correctly to the form \(\ddot{x} = -kx\) (May use \(a\), \(\frac{dv}{dt}\) or any other correct symbol for acceleration) | M1 | 1.1a |
| Correctly concludes that the particle moves with SHM with a clear reason from their equation of the correct form e.g. comparison with the standard form \(\ddot{x} = -\omega^2x\) | R1F | 2.1 |
Typical solution
\[3 = y_{AB} + y_{BC}\]\[5cy_{AB} = cy_{BC}\]\[6y_{AB} = 3\]\[y_{AB} = \frac{1}{2},\ y_{BC} = \frac{5}{2}\]\[5c\left(\frac{1}{2} - x\right) - c\left(\frac{5}{2} + x\right) = 3c\ddot{x}\]\[\ddot{x} = -2x\]Of form \(\ddot{x} = -\omega^2x\), therefore SHM
| Scheme | Marks | AO |
|---|---|---|
| Obtains the correct value for \(\omega\) FT their final equation in (a) | B1F | 1.1b |
| States or uses the correct value for the amplitude. | B1 | 3.1b |
| Uses a correct complete method to find the speed. | M1 | 3.1b |
| Obtains the correct speed with correct units FT their \(\omega\) and accurate to 2 or more sf | A1F | 3.2a |
| (12 marks) |