Show work and units Consider a harmonic oscillator with mass 0.48 kg and spring

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Show work and units

Consider a harmonic oscillator with mass 0.48 kg and spring constant 187 Wm. If the amplitude is 5.26 cm, what is the speed of the mass at the equilibrium point? (work this out using conservation of energy. Practice how to draw an energy bar chart for this.) Answer with units of m/s. Consider a harmonic oscillator with frequency 6.41 Hz. If the amplitude of the oscillation is 8.53 cm, what is the speed of the mass at the equilibrium point? (work this out using conservation of energy. Practice how to draw an energy bar chart for this.) Answer with units of m/s. Consider a harmonic oscillator with mass 0.3 kg and spring constant 191 N/m. If the amplitude is 8.34 cm, what is the speed of the mass at a point which is displaced by 29% of the amplitude off the equilibrium point? (work this out using conservation of energy. Practice how to draw an energy bar chart for this.) Answer with units of m/s. Consider a harmonic oscillator with period 0.07 s. If the amplitude is 5.71 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). Answer with units of cm.

Explanation / Answer

1. total energy of system = kA^2 /2 = 187 x 0.0526^2 / 2 = 0.259 J

at equilibirum point, PE = 0

so total energy = 0 + mv^2 /2

0.259 = 0.48 v^2 / 2

v = 1.04 m/s

2. angular frequency = 2pif =2pi x 6.41 = 40.27 rad/s

speed at equilibrium = Aw = 0.0853 x 40.27 = 3.43 m/s

3. total energy of system = kA^2 /2 = 191 x 0.0834^2 / 2 = 0.664 J

at point, x = 0.29A = 0.0247 m

PE = kx^2 /2 = 191 x 0.0247^2 / 2 = 0.0584 m

so total energy = 0.0584 + mv^2 /2

v = 2 m/s

4. angular speed w = 2pi / T = 89.76 rad/s

x = A cos(wt)

and v = - Aw sin(wt)

1.67 = 89.76 x 0.0571 sin(wt)

wt = 19.05 deg

x = 5.71 cos(19.05) =5.40 cm

v = 1.04 m/s

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