The percentage of mechanical energy of the oscillator lost in each cycle is 5.91 %.
The formula for mechanical energy in an oscillator is given by
E = 1/2 kA²
It is the sum of elastic potential energy and kinetic energy.
Amplitude is nothing but the maximum displacement moved by a point on vibrating body.
It is given that the amplitude A decreases by 3%, then
(E₂ - E₁) / E₁ = [1/2 k (A₂²- A₁²)] /(1/2 k A₁²) = (A₂²- A₁²)/ A₁² = (97² - 100²)/ 100²
⇒ 5.91% of the mechanical energy is lost in each cycle.
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It makes calculations with very large and small numbers easier.
Scientific notation is a system used in order to It makes calculations with very large and small numbers easier. It is useful as it allows very large number that would take a lot of space to write otherwise, and it allows them to be calculated easier.
for example is a incredible large number, but written in this form is immediately understandable and useful for calculation.
Answer:
0.088 seconds
0.0880000273785 second
0.08800054757 seconds
Explanation:
In the question it is given each century adds 1 ms to a day due to the slowing rotation of the Earth
In 88 centuries the length of the first day of the year will be
88 × 1 = 88 ms = 0.088 seconds
1 ms = 1 century
1 century = 100 years × 365.25 days
⇒1 ms = 36525 days

Sum of the gain on the first day would be
0.088 + 2.7378×10⁻⁸ = 0.0880000273785 second
Sum of the gain on the second day would be
0.088 + 2.7378×10⁻⁸+2.7378×10⁻⁸ = 0.08800054757 seconds