M3 June 2015 Q1
1. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(A\) on a ceiling. The particle is hanging freely in equilibrium at a distance 1.5 m vertically below \(A\).
The particle is now raised to the point \(B\), where \(B\) is vertically below \(A\) and \(AB = 0.8\) m. The spring remains straight. The particle is released from rest and first comes to instantaneous rest at the point \(C\).
| Scheme | Marks |
|---|---|
| \(0.5g = T = \dfrac{\lambda \times 0.3}{1.2}\) | M1A1 |
| \(\lambda = 2g = 19.6\) | A1 |
| (3) |
Notes
M1 Use Hooke's law to obtain the tension and equate to the weight
A1 Correct equation
A1 Solve to get \(\lambda = 19.6\) Accept 20 or \(2g\)
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{2} \times \dfrac{19.6 \times x^2}{1.2} - \dfrac{1}{2} \times \dfrac{19.6 \times 0.4^2}{1.2} = 0.5 \times g \times (x + 0.4)\) | M1A1ftA1 |
| \(5x^2 - 3x - 2 = 0\) | |
| \((5x + 2)(x - 1) = 0\) or use of diff of 2 squares to obtain and then solve a linear equation | |
| \(x = 1\) \((x = -0.4\) need not be seen\()\) | |
| \(AC = 2.2\) m | A1 |
| (4) | |
| (7 marks) |
Notes
M1 Attempt an energy equation with the difference of 2 EPE terms and a loss of GPE
EPE formula must be of the form \(k\dfrac{\lambda x^2}{l}\)
A1ft EPE terms correct follow through their \(\lambda\)
A1 GPE term correct, including all signs in the equation correct If \(x\) used for EPE and GPE A0 here
A1 Correct length \(AC\) If \(\lambda = 20\) is used, this is p.a. and so scores A0
ALT
| Find \(BC\) first: \(\dfrac{1}{2} \times \dfrac{19.6 \times (h - 0.4)^2}{1.2} - \dfrac{1}{2} \times \dfrac{19.6 \times 0.4^2}{1.2} = 0.5gh\) | M1A1A1 |
| \(BC = 1.4\) \(AC = 2.2\) | A1 |
Methods depending on SHM must prove SHM first, but if correct answer only is given award B1 (M1 on e-PEN)
By integration: Integrating and substituting yields an equation equivalent to the one shown - mark from here M1A1A1ft -1 each error ft on \(\lambda\)