an object of mass 1.5 kg on a spring of force constant 600N/m loses 3% of its energy in each cycle.THE SYSTEM IS DRIVEn by a sinosuidal force with maximim value of F0=0.5N.(a) What is Q for this system?.(b) What is the resonance (angluar)frequency? (c) If the driving frequendy is varied what is the width delta w of the resonance? (d) What is the amplitude at resonance?(c) What is the amplutude if the driving frequency is w=19 rad\s?
Picture the Problem: We can find q from 1=2 PI(delta E\E)cycle and then ise this result to find the width of the resonance deltaw=w\Q.The resonance frequency is the natural frequency.The amplitude can be found from Equation 14-49 both at resonance and off resonance with damping contant caculated from Q using Equation 14-41 in the from b=w0m\Q
To solve this problem, we will go step by step to find the answers to each question.
(a) To find Q for this system, we need to calculate the quality factor, which is a measure of the damping in the system. The relationship between Q and the percentage energy loss per cycle is given by the equation: 1 = 2π(ΔE/E)cycle. Rearranging this equation, we get: Q = 2π/(ΔE/E)cycle. Given that the energy loss per cycle is 3%, we can plug in this value to find Q.
Q = 2π/(0.03) = 207.9 (approx)
(b) The resonance frequency (angular frequency) can be found using the formula: ω0 = √(k/m), where k is the force constant of the spring and m is the mass of the object. Plugging in the values, we get:
ω0 = √(600/1.5) = 16.33 rad/s (approx)
(c) The width of the resonance, denoted by Δω, can be calculated using the formula: Δω = ω0/Q. We have already calculated ω0 and Q, so we can plug in these values to find Δω:
Δω = 16.33/207.9 = 0.0786 rad/s (approx)
(d) The amplitude at resonance can be calculated using the formula from Equation 14-49: A = F0/mω0Q, where F0 is the maximum value of the driving force, m is the mass of the object, ω0 is the resonance frequency, and Q is the quality factor. Plugging in the values given, we get:
A = 0.5/(1.5 * 16.33 * 207.9) = 0.00096 m (approx)
(e) To find the amplitude if the driving frequency is ω = 19 rad/s, we can use Equation 14-41: b = ω0m/Q. With the values we have calculated earlier, we can plug them into this formula:
b = 16.33 * 1.5 / 207.9 = 0.117 m (approx)
So, the amplitude if the driving frequency is ω = 19 rad/s is 0.117 m (approx).
Remember, these calculations are based on the given formulas and assumptions about the system. Make sure to double-check the equations and calculations to ensure accuracy.