3 Fixed points A and B are 4.8 m apart on the same horizontal level. The midpoint of AB is M . A light elastic string, with natural length 3.9 m and modulus of elasticity 573.3 N , has one end attached to A and the other end attached to \(\mathbf { B }\).
- Find the elastic energy stored in the string.
A particle P is attached to the midpoint of the string, and is released from rest at M . It comes instantaneously to rest when P is 1.8 m vertically below M .
- Show that the mass of P is 15 kg .
- Verify that P can rest in equilibrium when it is 1.0 m vertically below M .
In general, a light elastic string, with natural length \(a\) and modulus of elasticity \(\lambda\), has its ends attached to fixed points which are a distance \(d\) apart on the same horizontal level. A particle of mass \(m\) is attached to the midpoint of the string, and in the equilibrium position each half of the string has length \(h\), as shown in Fig. 3.
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\caption{Fig. 3}
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When the particle makes small oscillations in a vertical line, the period of oscillation is given by the formula
$$\sqrt { \frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } } } m ^ { \alpha } a ^ { \beta } \lambda ^ { \gamma }$$ - Show that \(\frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } }\) is dimensionless.
- Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
- Hence find the period when the particle P makes small oscillations in a vertical line centred on the position of equilibrium given in part (iii).