Question

When the displacement of a mass on a spring is 1/2a the half of the amplitude, what fraction of the mechanical energy is kinetic energy?

Answer 1

Total energy is a spring:

`E = (1)/(2) kx^2 + (1)/(2) mv^2 = (1)/(2) ka^2`

At x = 0.5a:

`(1)/(2) k (a)/(2) ^2 + (1)/(2) mv^2 = (1)/(2) ka^2 \\ (1)/(2) mv^2 = (1)/(2) ka^2 - (1)/(8) ka^2 = (3)/(8) ka^2`

The ration:

`( (3)/(8)ka^2 )/( (1)/(2) ka^2) = (3)/(4)`

Answer 2

`KE : TE = 3 : 4`

Step-by-step explanation:

As we know that the total mechanical energy of the object which is executing SHM is given by

`E_(total) = (1)/(2)KA^2`

here we know that

A = amplitude of SHM

K = spring constant

now we know that total mechanical energy of the spring is always constant so here we can say

kinetic energy + Potential energy = total mechanical energy

we know that potential energy of the spring at any given distance is

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

at given position x = A/2

`U = (1)/(2)K((A)/(2))^2 = (1)/(8)KA^2`

now we have

`KE + (1)/(8)KA^2 = (1)/(2)KA^2`

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

now ratio of kinetic energy and total mechanical energy will be given as

`KE : TE = (3)/(8)KA^2 : (1)/(2)KA^2`

`KE : TE = 3 : 4`