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, let's break it down into steps:
Step 1: Find Q for the system.
Q is the quality factor of the system, which represents the ratio of energy stored to energy dissipated per cycle. In this case, we are given that the object loses 3% of its energy in each cycle.
Q is given by the formula: Q = 1 / (2π * (ΔE / E)cycle)
ΔE / E represents the fraction of energy lost per cycle. In this case, ΔE / E = 0.03 (3%).
Plug in the values and solve for Q.
Step 2: Find the resonance (angular) frequency.
The resonance frequency is the natural frequency of the system. In this case, the system is driven by a sinusoidal force with a maximum value of F0 = 0.5 N.
The resonance frequency is given by the formula: ω0 = √(k / m)
Where k is the force constant of the spring (600 N/m) and m is the mass of the object (1.5 kg).
Plug in the values and solve for ω0.
Step 3: Find the width (Δω) of the resonance.
The width of the resonance is given by the formula: Δω = ω0 / Q
Use the values of ω0 and Q calculated in steps 2 and 1, respectively, and solve for Δω.
Step 4: Find the amplitude at resonance.
The amplitude at resonance can be found using Equation 14-49, which accounts for damping. The damping constant can be calculated from Q using Equation 14-41 in the form b = ω0 * m / Q.
Use the values of ω0, m, and Q calculated in steps 2 and 1, respectively, and solve for b. Then, plug in the values of ω0, m, and b into Equation 14-49 to find the amplitude at resonance.
Step 5: Find the amplitude if the driving frequency is ω = 19 rad/s.
The amplitude off resonance can also be found using Equation 14-49. In this case, we are given a specific driving frequency (ω = 19 rad/s).
Use the values of ω, m, and b calculated in step 5, and plug them into Equation 14-49 to find the amplitude off resonance.
By following these steps, you should be able to find the answers to all the questions posed in the problem.