M3 January 2007 Q3
3. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3.6mg\). The other end of the string is fixed at a point \(O\) on a rough horizontal table. The particle is projected along the surface of the table from \(O\) with speed \(\sqrt{(2ag)}\). At its furthest point from \(O\), the particle is at the point \(A\), where \(OA = \tfrac{4}{3}a\).
(a) Find, in terms of \(m\), \(g\) and \(a\), the elastic energy stored in the string when \(P\) is at \(A\). (3)
(b) Using the work-energy principle, or otherwise, find the coefficient of friction between \(P\) and the table. (6)
| Scheme | Marks |
|---|---|
| E.P.E. \(= \dfrac{1}{2}\dfrac{3.6mg}{a}x^2 = \dfrac{1}{2}\dfrac{3.6mg}{a}\left(\dfrac{a}{3}\right)^2\) | M1 A1 |
| \(= 0.2mga\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Friction \(= \mu mg \Rightarrow\) work done by friction \(= \mu mg\left(\dfrac{4a}{3}\right)\) | M1 A1 |
| Work-energy: \(\tfrac{1}{2}m.2ga = \mu mgd + 0.2mga\) (3 relevant terms) | M1 A1ft |
| Solving to find \(\mu\): \(\mu = 0.6\) | ↓ M1 A1 |
| (6) | |
| (9 marks) |
Notes
↓ marks a mark that depends on the M mark above it (an arrow in the scheme).
1st M1: allow for attempt to find work done by frictional force (i.e. not just finding friction).
2nd M1: “relevant” terms, i.e. energy or work terms!
A1 f.t. on their work done by friction