A-LevelMathematicsMechanics9709/52 · Feb/Mar 2017 · Q710 Marks

One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to a fixed point O. The other end of the string is attached to a particle P of mass 0.4 kg which hangs in equilibrium vertically below O.

(i)Calculate the extension of the string.[2]
(ii)P is projected vertically downwards from the equilibrium position with speed 5 m s⁻¹. Calculate the distance P travels before it is first at instantaneous rest.[4]
(iii)When P is first at instantaneous rest a stationary particle of mass 0.4 kg becomes attached to P. Find the greatest speed of the combined particle in the subsequent motion.[4]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2017). The mark or mark type is shown in brackets.

[M1] Uses T = λx/L (0.4g = 24e/0.6)
[A1] e = 0.1 m
[B1] Uses EE = λx²/2L (Initial EE = 24 x 0.1²/(2 x 0.6) (= 0.2 J))
[M1 A1] Set up a 4 term energy equation involving EE, PE and KE (0.4 x 5²/2 + 0.4gd=24(0.1 + d)²/(2 x 0.6) −24 x 0.1²/(2 x 0.6))
[A1] d = 0.5 m
[B1] e = 0.2
[M1 A1] Set up a 4 term energy equation in EE, PE and KE (0.8v²/2=24 x 0.6²/(2 x 0.6)– 24 x 0.2²/(2 x 0.6) – 0.8g x 0.4)
[A1] v = 2√2 = 2.83 ms⁻¹

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