For non-ideal oscillators (such as a real pendulum) energy is lost and the amplitude (which is the maximum displacement) is no longer constant but also decreases with time. How fast the energy lost is described by a TIME CONSTANT, τ = m/b. Typically after ONE time constant has elapsed, the system has ...(hint: this is when t = τ)
a)LOST half of its initial energy.
b)LOST 63% of its initial energy.
c)LOST 13% of its initial energy.
d)LOST 37% of its initial energy.
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B) LOST 63% of its initial energy.
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The correct answer is b) LOST 63% of its initial energy.
The time constant (τ) is defined as the time it takes for the system to decrease to a fraction of its initial value. For a non-ideal oscillator, such as a real pendulum, the time constant is given by τ = m/b, where m represents the mass of the oscillator and b represents the damping coefficient.
After one time constant has elapsed (when t = τ), the system has lost approximately 63% of its initial energy. This means that the amplitude (maximum displacement) of the oscillator has decreased to approximately 37% of its initial value. This is known as the 63% rule.
Therefore, the correct answer is b) LOST 63% of its initial energy.
To determine what happens after one time constant has elapsed in a non-ideal oscillator, we can refer to the given options and choose the one that matches the behavior.
The time constant, τ, is defined as the ratio of the mass (m) of the oscillator to its damping coefficient (b). It represents the characteristic time scale for the system's energy loss.
The answer options provide different percentages of energy loss. To determine which one is correct, we need to understand the relationship between energy loss and time in a non-ideal oscillator.
In most non-ideal oscillators, energy is lost exponentially with time. The equation governing the decay of energy over time is given by:
E(t) = E0 * e^(-t/τ)
Where E(t) is the energy at time t, E0 is the initial energy, e is the base of the natural logarithm, t is the time elapsed, and τ is the time constant.
Now, let's substitute t = τ into the equation and see which answer choice matches:
E(τ) = E0 * e^(-τ/τ)
E(τ) = E0 * e^(-1)
E(τ) = E0 * 0.36787944117
This means that after one time constant has elapsed, the system retains approximately 36.8% of its initial energy. Therefore, the answer that matches this behavior is option:
d) LOST 37% of its initial energy.
So, after one time constant has passed, a non-ideal oscillator typically loses approximately 37% of its initial energy.