4 Fig. 4 shows a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Two fixed points A and B on the plane are 4.55 m apart with B higher than A on a line of greatest slope. A particle P of mass 0.25 kg is in contact with the plane and is connected to A and to B by two light elastic strings. The string AP has natural length 1.5 m and modulus of elasticity 7.35 N ; the string BP has natural length 2.5 m and modulus of elasticity 7.35 N . The particle P moves along part of the line AB , with both strings taut throughout the motion.
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\caption{Fig. 4}
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- Show that, when \(\mathrm { AP } = 1.55 \mathrm {~m}\), the acceleration of P is zero.
- Taking \(\mathrm { AP } = ( 1.55 + x ) \mathrm { m }\), write down the tension in the string AP , in terms of \(x\), and show that the tension in the string BP is \(( 1.47 - 2.94 x ) \mathrm { N }\).
- Show that the motion of P is simple harmonic, and find its period.
The particle P is released from rest with \(\mathrm { AP } = 1.5 \mathrm {~m}\).
- Find the time after release when P is first moving down the plane with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).