Question Solved1 Answer A particle of mass \( \frac{1}{2} \mathrm{~kg} \) is attached to one end of a model spring which is hanging vertically from a fixed point \( A \). The spring has stiffness \( 4 \mathrm{Nm}^{-1} \) and a natural length of 1 metre. The system is oscillating in a vertical line with the particle below \( A \). Use the approximation that the magnitude of the acceleration due to gravity is \( 10 \mathrm{~ms}^{-2} \). (a) Find an expression for the total mechanical energy function for the system, carefully defining your coordinate system and the datum for gravitational potential energy. (b) When the particle is 2 metres below \( A \) it has speed \( 2 \mathrm{~ms}^{-1} \). Use conservation of mechanical energy to establish whether the spring is ever in compression during the motion. (Hint: try to determine the speed of the particle when the spring has its natural length.)

O1KLWF The Asker · Advanced Mathematics

Transcribed Image Text: A particle of mass \( \frac{1}{2} \mathrm{~kg} \) is attached to one end of a model spring which is hanging vertically from a fixed point \( A \). The spring has stiffness \( 4 \mathrm{Nm}^{-1} \) and a natural length of 1 metre. The system is oscillating in a vertical line with the particle below \( A \). Use the approximation that the magnitude of the acceleration due to gravity is \( 10 \mathrm{~ms}^{-2} \). (a) Find an expression for the total mechanical energy function for the system, carefully defining your coordinate system and the datum for gravitational potential energy. (b) When the particle is 2 metres below \( A \) it has speed \( 2 \mathrm{~ms}^{-1} \). Use conservation of mechanical energy to establish whether the spring is ever in compression during the motion. (Hint: try to determine the speed of the particle when the spring has its natural length.)
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Transcribed Image Text: A particle of mass \( \frac{1}{2} \mathrm{~kg} \) is attached to one end of a model spring which is hanging vertically from a fixed point \( A \). The spring has stiffness \( 4 \mathrm{Nm}^{-1} \) and a natural length of 1 metre. The system is oscillating in a vertical line with the particle below \( A \). Use the approximation that the magnitude of the acceleration due to gravity is \( 10 \mathrm{~ms}^{-2} \). (a) Find an expression for the total mechanical energy function for the system, carefully defining your coordinate system and the datum for gravitational potential energy. (b) When the particle is 2 metres below \( A \) it has speed \( 2 \mathrm{~ms}^{-1} \). Use conservation of mechanical energy to establish whether the spring is ever in compression during the motion. (Hint: try to determine the speed of the particle when the spring has its natural length.)
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content: ".";}mjx-c.mjx-c45::before{padding: 0.68em 0.681em 0 0; content: "E";}Solution (a) m=12kg,k=4NmLength (l) = 1 m g=10msWhen mass is hung then equilibrium position will be K(x−1)=mg4(x−1)=12×104x−4=5⇒x=94Now mechanical energy of the system is =12K(94−x)2+12mv2=12×4(94&#87 ... 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