The total energy of a system undergoing simple harmonic motion (such as an object at the end of a spring) is proportional to the square of its amplitude. Therefore, to double the total energy of the system, we need to increase the amplitude by a certain factor.
Let's denote the initial amplitude as A and the final amplitude as B. We want to find the factor by which we need to increase A to achieve a doubled total energy.
The total energy of the system is given by the formula:
Total Energy = (1/2) * k * A^2
where k is the spring constant.
If we double the total energy, we have:
2 * Total Energy = 2 * (1/2) * k * A^2
Doubling the energy means multiplying the right side by 2. Thus, we have:
2 * Total Energy = k * A^2
To find the final amplitude B, we need to determine the factor by which we need to increase A to achieve the doubled energy:
B^2 = A^2 * 2
Taking the square root of both sides, we get:
B = A * √2
Therefore, to double the total energy of the system, we need to increase the amplitude by a factor of √2 (approximately 1.414).