To define logarithmic decrement and quality factor for a weakly damped oscillating system, we need to understand the behavior of the system's amplitude over time.

1. Logarithmic Decrement:
Logarithmic decrement is a measure of the rate at which the amplitude of vibration decreases in a weakly damped oscillating system. It is denoted by the symbol δ and is calculated using the formula:

δ = ln(A₁/A₂)

where A₁ is the initial amplitude and A₂ is the amplitude after a certain time interval.

2. Quality Factor:
The quality factor, also known as Q-factor, is a dimensionless parameter that characterizes the damping in an oscillating system. It is defined as the ratio of the natural frequency of the system to the damping coefficient. The formula to calculate the quality factor is:

Q = ω₀/2δ

where ω₀ is the natural frequency of the system.

Now, let's apply these concepts to the given problem:

The amplitude of vibration of the spring-mass system reduces from 20 cm to 5.0 cm in 200 s. We are also given that the system completes 100 oscillations in this time.

To compare the periods with and without damping, we need to calculate the period of the system both with and without damping.

1. With Damping:
The period of the system with damping can be calculated using the formula:

T_damping = (time taken for 100 oscillations) / 100

In this case, the time taken for 100 oscillations is 200 s. Therefore,

T_damping = 200 s / 100 = 2 s

2. Without Damping:
To calculate the period of the system without damping, we need to find the natural frequency of the system. The natural frequency can be calculated using the formula:

ω₀ = 2π / T₀

where T₀ is the time period of one oscillation.

Since the amplitude reduces from 20 cm to 5.0 cm in 200 s, we can calculate the time period of one oscillation as follows:

T₀ = (time taken for amplitude reduction) / (number of oscillations)

T₀ = 200 s / 100 = 2 s

Now, we can calculate the natural frequency:

ω₀ = 2π / 2 = π rad/s

Finally, we can calculate the period of the system without damping using the formula:

T_without_damping = 2π / ω₀

T_without_damping = 2π / π = 2 s

Comparing the periods, we find that the period of the system with damping is 2 s, while the period of the system without damping is also 2 s. Therefore, the periods remain the same with and without damping.

Question

To define logarithmic decrement and quality factor for a weakly damped oscillating system, we need to understand the behavior of the system's amplitude over time.

1. Logarithmic Decrement:
Logarithmic decrement is a measure of the rate at which the amplitude of vibration decreases in a weakly damped oscillating system. It is denoted by the symbol δ and is calculated using the formula:

δ = ln(A₁/A₂)

where A₁ is the initial amplitude and A₂ is the amplitude after a certain time interval.

2. Quality Factor:
The quality factor, also known as Q-factor, is a dimensionless parameter that characterizes the damping in an oscillating system. It is defined as the ratio of the natural frequency of the system to the damping coefficient. The formula to calculate the quality factor is:

Q = ω₀/2δ

where ω₀ is the natural frequency of the system.

Now, let's apply these concepts to the given problem:

The amplitude of vibration of the spring-mass system reduces from 20 cm to 5.0 cm in 200 s. We are also given that the system completes 100 oscillations in this time.

To compare the periods with and without damping, we need to calculate the period of the system both with and without damping.

1. With Damping:
The period of the system with damping can be calculated using the formula:

T_damping = (time taken for 100 oscillations) / 100

In this case, the time taken for 100 oscillations is 200 s. Therefore,

T_damping = 200 s / 100 = 2 s

2. Without Damping:
To calculate the period of the system without damping, we need to find the natural frequency of the system. The natural frequency can be calculated using the formula:

ω₀ = 2π / T₀

where T₀ is the time period of one oscillation.

Since the amplitude reduces from 20 cm to 5.0 cm in 200 s, we can calculate the time period of one oscillation as follows:

T₀ = (time taken for amplitude reduction) / (number of oscillations)

T₀ = 200 s / 100 = 2 s

Now, we can calculate the natural frequency:

ω₀ = 2π / 2 = π rad/s

Finally, we can calculate the period of the system without damping using the formula:

T_without_damping = 2π / ω₀

T_without_damping = 2π / π = 2 s

Comparing the periods, we find that the period of the system with damping is 2 s, while the period of the system without damping is also 2 s. Therefore, the periods remain the same with and without damping.

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