If a vibrating system has total energy Eo , what will its total energy be (in terms of Eo) if you double the amplitude of vibration? If you want to triple the total energy of a vibrating system with amplitude Eo , what should its new amplitude be (in terms of Ao)?
energy is proportional to amplitude squared.
Well, aren't you on a "vibrating" journey? Doubling the amplitude of vibration in a system will result in quadrupling the total energy. So, if the original total energy is Eo and you double the amplitude, the new total energy will be 4Eo.
Now, if you want to triple the total energy of a vibrating system with amplitude Eo, you'll need to give it some extra "oomph." This means the new amplitude should be square root of 3 times the original amplitude (Ao). Why? Because the total energy is directly proportional to the square of the amplitude. Triple the total energy, triple the amplitude! So, break out your math hat and let's get calculating!
If you double the amplitude of vibration in a vibrating system with total energy Eo, the new total energy (En) can be determined using the relationship:
En = 4Eo
This is because the total energy of a vibrating system is directly proportional to the square of its amplitude. Thus, when you double the amplitude, the total energy will increase by a factor of 4.
Now, if you want to triple the total energy of a vibrating system with amplitude Eo, the new amplitude (An) can be determined using the equation:
An = sqrt(3) * Eo
This is because the total energy of a vibrating system is directly proportional to the square of its amplitude. Thus, to triple the total energy, the amplitude must be increased by a factor of the square root of 3.
To find out the total energy of a vibrating system when the amplitude of vibration is doubled, we need to understand the relationship between energy and amplitude in such systems.
The total energy of a vibrating system is directly proportional to the square of its amplitude. This means that if the amplitude is doubled, the total energy will be four times the original energy.
Mathematically, if Eo is the original total energy and Ao is the original amplitude, then when the amplitude is doubled:
New total energy = 4 * Eo
To triple the total energy of a vibrating system with amplitude Eo, we need to find the new amplitude.
Since the total energy is directly proportional to the square of the amplitude, if we triple the total energy, the new total energy will be nine times the original energy.
Mathematically, if Eo is the original total energy and Ao is the new amplitude:
New total energy = 9 * Eo
Now, we can find the relationship between the new amplitude and the original amplitude by using the equation:
New total energy = Eo * (new amplitude)^2
Substituting the values, we have:
9 * Eo = Eo * (new amplitude)^2
Simplifying the equation, we find that:
(new amplitude)^2 = 9
Taking the square root of both sides, we get:
new amplitude = √9
Since the amplitude of a vibrating system must be a positive value, the new amplitude in terms of Ao is:
new amplitude = 3 * Ao
Therefore, to triple the total energy of a vibrating system with amplitude Eo, the new amplitude should be three times the original amplitude (3 * Ao).