Question

An object suspended from a spring vibrates with simple harmonic motion. Part A At an instant when the displacement of the object is equal to one-fourth the amplitude, what fraction of the total energy of the system is kinetic

Answer 1

Complete Question

An object suspended from a spring vibrates with simple harmonic motion.

a. At an instant when the displacement of the object is equal to one-half the amplitude, what fraction of the total energy of the system is kinetic?

b. At an instant when the displacement of the object is equal to one-half the amplitude, what fraction of the total energy of the system is potential?

Answer:

a

The fraction of the total energy of the system is kinetic energy
`(KE)/(T) = (3)/(4)`

b

The fraction of the total energy of the system is potential energy
`(PE)/(T) = (1)/(4)`

Step-by-step explanation:

From the question we are told that

The displacement of the system is
`e = (a)/(2)`

where a is the amplitude

Let denote the potential energy as PE which is mathematically represented as

`PE = (1)/(2) * k* x^2`

=>
`PE = (1)/(2) * k* [(a)/(2) ]^2`

`PE = k* [(a^2)/(8) ]`

Let denote the total energy as T which is mathematically represented as

`T = (1)/(2) * k * a^2`

Let denote the kinetic energy as KE which is mathematically represented as

`KE = T -PE`

=>
`KE =k [ (a^2)/(2) - (a^2)/(8) ]`

=>
`KE =k [ (3)/(8) a^2 ]`

Now the fraction of the total energy that is kinetic energy is

`(KE)/(T) = ( (3ka^2)/(8) )/((ka^2)/(2) )`

`(KE)/(T) = (3)/(4)`

Now the fraction of the total energy that is potential energy is

`(PE)/(T) = ((k a^2)/(8) )/((k a^2)/(2) )`

`(PE)/(T) = (1)/(4)`