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A point-like mass m is constrained to move along the z-axis only. It is attached to two springs, each of spring constant k, and each of rest length 0. One spring has one of its ends fixed at x = 0 and y = h; the other spring is fixed at one end to z = 0 and y = -h as shown in the figure below. Gravity does not act on the system. The springs do not bend but contract and extend along straight lines between their end points. y m (d) Determine the equilibrium position of the system and the frequencies of small oscillations about this equilibrium position. Hint: use approximations (i.e. Taylor expansion) to simplify the equation of motion near the equilibrium position to that of a harmonic oscillator. (e) Repeat part (d) for the case where the two springs have different spring constants k₁ and k₂ and are placed at different distances on the y-axis h₁ and h₂.

A point-like mass m is constrained to move along the z-axis only. It is attached to two springs, each of spring constant k, and each of rest length 0. One spring has one of its ends fixed at x = 0 and y = h; the other spring is fixed at one end to z = 0 and y = -h as shown in the figure below. Gravity does not act on the system. The springs do not bend but contract and extend along straight lines between their end points. y m (d) Determine the equilibrium position of the system and the frequencies of small oscillations about this equilibrium position. Hint: use approximations (i.e. Taylor expansion) to simplify the equation of motion near the equilibrium position to that of a harmonic oscillator. (e) Repeat part (d) for the case where the two springs have different spring constants k₁ and k₂ and are placed at different distances on the y-axis h₁ and h₂.

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A point-like mass m is constrained to move along the z-axis only. It is attached to two springs, each of spring constant k, and each of rest length 0. One spring has one of its ends fixed at x = 0 and y = h; the other spring is fixed at one end to z = 0 and y=-h as shown in the figure below. Gravity does not act on the system. The springs do not bend but contract and extend along straight lines between their end points. y m (d) Determine the equilibrium position of the system and the frequencies of small oscillations about this equilibrium position. Hint: use approximations (i.e. Taylor expansion) to simplify the equation of motion near the equilibrium position to that of a harmonic oscillator. (e) Repeat part (d) for the case where the two springs have different spring constants k₁ and k₂ and are placed at different distances on the y-axis h₁ and h₂.
Transcribed Image Text:A point-like mass m is constrained to move along the z-axis only. It is attached to two springs, each of spring constant k, and each of rest length 0. One spring has one of its ends fixed at x = 0 and y = h; the other spring is fixed at one end to z = 0 and y=-h as shown in the figure below. Gravity does not act on the system. The springs do not bend but contract and extend along straight lines between their end points. y m (d) Determine the equilibrium position of the system and the frequencies of small oscillations about this equilibrium position. Hint: use approximations (i.e. Taylor expansion) to simplify the equation of motion near the equilibrium position to that of a harmonic oscillator. (e) Repeat part (d) for the case where the two springs have different spring constants k₁ and k₂ and are placed at different distances on the y-axis h₁ and h₂.
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