Question

When the displacement of a mass on a spring is 12A the half of the amplitude, what fraction of the mechanical energy is kinetic energy?At what displacement, as a fraction of A, is the mechanical energy half kinetic and half potential?

Answer 1

When the displacement of a mass on a spring is half the amplitude, the fraction of the mechanical energy that is kinetic energy is 1. At half the amplitude, the mechanical energy is divided equally between kinetic and potential energy.

Step-by-step explanation:

In a mass-spring system, the mechanical energy is the sum of the kinetic energy and the potential energy.

Given that the displacement of the mass on the spring is 12A, which is half the amplitude, we can determine the fraction of the mechanical energy that is kinetic energy.

The potential energy at this displacement is zero since the spring is not stretched or compressed. Therefore, the entire mechanical energy is in the form of kinetic energy.

So, the fraction of the mechanical energy that is kinetic energy is 1.

To find the displacement at which the mechanical energy is half kinetic and half potential, we can consider that the total energy of the system is constant and proportional to the amplitude squared.

When the displacement is at half the amplitude, the potential energy is equal to the kinetic energy. So, at this point, the mechanical energy is divided equally between kinetic and potential.

Answer 2

As the object displacement is half of the amplitude then the potential energy stored in the spring is given by

`U = (1)/(2) kx^2`

here we know that

`x = (A)/(2)`

`U = (1)/(2)k(A/2)^2`

total mechanical energy is given as

`ME = (1)/(2)KA^2`

now KE is given by

`KE = ME - U`

`KE = (1)/(2)KA^2 - (1)/(2)K(A/2)^2`

`KE = (3)/(8)KA^2`

now fraction of KE with respect to ME is given as

`f = (KE)/(ME) = (3)/(4)`

now if the mechanical energy is divided equally in KE and PE

so now we will have

`KE = (1)/(2)KA^2 - (1)/(2)Kx^2`

`PE = (1)/(2)kx^2`

now we have

`KE = PE`

`(1)/(2)KA^2 - (1)/(2)Kx^2 = (1)/(2)Kx^2`

`A^2 - x^2 = x^2`

`x = (A)/(\sqrt2)`