To double the total energy of an oscillating mass-spring system, the amplitude must increase by a factor of √2 (approximately 1.414).
The total energy of an oscillating mass-spring system is directly proportional to the square of the amplitude. Therefore, if the total energy is doubled, the amplitude must be increased by a factor of √2. This means that the displacement from the equilibrium position will need to be increased by approximately 1.414 times its original value.
Regarding the effect on the frequency, the frequency of an oscillating mass-spring system remains unchanged when the amplitude is altered. The frequency of the system depends on the mass of the object and the stiffness of the spring, but not on the amplitude. Therefore, doubling the total energy and increasing the amplitude by a factor of √2 will not affect the frequency of the oscillation.