M3 June 2009 Q1
1. A light elastic string has natural length 8 m and modulus of elasticity 80 N. The ends of the string are attached to fixed points \(P\) and \(Q\) which are on the same horizontal level and 12 m apart. A particle is attached to the mid-point of the string and hangs in equilibrium at a point 4.5 m below \(PQ\).
(a) Calculate the weight of the particle. (6)
(b) Calculate the elastic energy in the string when the particle is in this position. (3)
| Scheme | Marks |
|---|---|
| Resolving vertically: \(2T\cos\theta = W\) | M1A2,1,0 |
| Hooke’s Law: \(T = \dfrac{80 \times 3.5}{4}\) | M1A1 |
| \(W = 84\)N | A1 |
| Scheme | Marks |
|---|---|
| EPE \(= 2 \times \dfrac{80 \times 3.5^2}{2 \times 4},\ = 245\) (or awrt 245) (alternative \(\dfrac{80 \times 7^2}{16} = 245\)) | M1A1ft,A1 |
| (9 marks) |