1 The fixed point A is at a height \(4 b\) above a smooth horizontal surface, and C is the point on the surface which is vertically below A. A light elastic string, of natural length \(3 b\) and modulus of elasticity \(\lambda\), has one end attached to A and the other end attached to a block of mass \(m\). The block is released from rest at a point B on the surface where \(\mathrm { BC } = 3 b\), as shown in Fig. 1. You are given that the block remains on the surface and moves along the line BC .
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\caption{Fig. 1}
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- Show that immediately after release the acceleration of the block is \(\frac { 2 \lambda } { 5 m }\).
- Show that, when the block reaches C , its speed \(v\) is given by \(v ^ { 2 } = \frac { \lambda b } { m }\).
- Show that the equation \(v ^ { 2 } = \frac { \lambda b } { m }\) is dimensionally consistent.
The time taken for the block to move from B to C is given by \(k m ^ { \alpha } b ^ { \beta } \lambda ^ { \gamma }\), where \(k\) is a dimensionless constant.
- Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
When the string has natural length 1.2 m and modulus of elasticity 125 N , and the block has mass 8 kg , the time taken for the block to move from B to C is 0.718 s .
- Find the time taken for the block to move from B to C when the string has natural length 9 m and modulus of elasticity 20 N , and the block has mass 75 kg .