QUESTION 1 The total mechanical energy of a harmonic oscillator that consists of

Question

QUESTION 1 The total mechanical energy of a harmonic oscillator that consists of a spring with spring constant k and mass m is E subscript t o t end subscript equals 1 half m v squared plus 1 half k x squared Here K equals 1 half m v squared is the kinetic energy of the mass and U subscript e l end subscript equals 1 half k x squared is the elastic potential energy of the spring. If there are no other dissipative forces acting (e.g. friction or airdrag are negligible), the total mechanical energy is conserved. This means the oscillator reaches maximum displacement xmax=A (amplitude) for v=0, and maximum speed vmax for x=0. Consider a harmonic oscillator with mass 0.47 kg and spring constant 190 N/m. If the speed of the mass at the equilibrium point is 1.86 m/s, what is the amplitude? (work this out using conservation of energy. Practise how to draw an energy bar chart for this.) Convert your answer to units of cm.

QUESTION 2 Consider a harmonic oscillator with mass 0.34 kg and spring constant 199 N/m. If the amplitude is 7.74 cm, what is the speed of the mass at the equilibrium point? (work this out using conservation of energy. Practise how to draw an energy bar chart for this.) Answer with units of m/s.

QUESTION 3 Consider a harmonic oscillator with frequency 5 Hz. If the amplitude of the oscillation is 5.73 cm, what is the speed of the mass at the equilibrium point? (work this out using conservation of energy. Practise how to draw an energy bar chart for this.) Answer with units of m/s.

QUESTION 4 Consider a harmonic oscillator with mass 0.26 kg and spring constant 175 N/m. If the amplitude is 8.5 cm, what is the speed of the mass at a point which is displaced by 72% of the amplitude off the equilibrium point? (work this out using conservation of energy. Practise how to draw an energy bar chart for this.) Answer with units of m/s. Answer with exactly 3 significant figures, (for example 1.23m/s or 0.345m/s or 0.0876m/s or 12.4m/s)

QUESTION 5 Consider a harmonic oscillator with period 0.07 s. If the amplitude is 8.14 cm, and at a certain time the mass is found to be moving at 1.67 m/s, what is the magnitude of the displacement from the equilibrium position (answer in units of cm). (work this out using conservation of energy. Practise how to draw an energy bar chart for this.) Answer with units of cm. Answer with exactly 3 significant figures (for example 1.23cm or 0.345cm or 0.0876cm or 12.4cm)

Explanation / Answer

1) At the maximum displacement, x = A, the total energy is P.E since velocity v = 0

At equilibrium total energy is K.E, since displacement x = 0.

Hence by conservation of energy total energy at equilibrium = total energy at maximum displacement,

i.e kinetic energy (1/2)mv2 gets converted into potential energy (1/2)kA2

=> (1/2)mv2 = (1/2)kA2 => A = sqrt(mv2/k) = sqrt(0.47*(1.86)2/190) = 0.092m = 9.2 cm

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