Question

If 27 J of work are needed to stretch a spring from 15 cm to 21 cm and 45 J are needed to stretch it from 21 cm to 27 cm, what is the natural length of the spring?

Answer 1

9 cm.

Step-by-step explanation:

The energy used for stretch the spring from
`15 cm` to
`21 cm` will be ,
`E_(1)=27J`

The energy used for stretch the spring from
`21 cm` to
`27 cm` will be ,
`E_(2)=45J`

using the energy of spring formula ,we find that

`27 = (1)/(2)K((21-L^(2))-(15-L^(2)))`

`45 = (1)/(2)K((27-L^(2))-(21-L^(2)))`

Dividing both the equation will get,

`(3)/(5)=((21-L)^(2)-(15-L)^(2))/((27-L)^(2)-(21-L)^(2))\\5((21-L)^(2)-(15-L)^(2))=3((27-L)^(2)-(21-L)^(2))\\3(729 - 54L + L^(2)- 441 + 42L - L^(2) ) = 5(441 - 42L + L^(2) - 225 + 30L - L^(2) )\\3(288 - 12L) = 5(216 - 12L)\\24L = 216\\L = 9 cm`

Therefore, the natural length of the spring is, 9 cm.