Obtain an expression for potential of a particle performing S.H.M. What is the value of potential energy at (i) Mean position and (ii) Extreme position.
A stretched sonometer wiere is in unison with a tunning fork. When the length of the wire is increases by 5%, the number of bats heard paer second is 10. Final the frequency of the tuning fork.

Explanation

Potential Energy :

Consider a particle of mass m performing S.H.N. along AB about man position 'O' at a distance x-from mean position, `OB=OA=a`.
Let, F =-kx be restoring force.
Where, k is force constant.
Let dx, be the particle displacmenet,
Work done
`:. dW=-Fdx.`
`=-(-kx).dx`
`=kx.dx`
Total workdone is
`int dW = int _(x=0)^(x) x.dx`
`W=(1)/(2)kx^(2)=(1)/(2) m omega^(2)x^(2) ( :. omega^(2)=(k)/(m))`
This workdone is stroed in the from of P.E.
`P.E.=(1)/(2)kx^(2)=(1)/(2)m omega^(2)x^(2)`
Cases (i) At mean position : when x=0
`P.E.=0`
(ii) At extrem Position : When `x=+pma`
`:. P.E.=(1)/(2)m omega^(2)a^(2)`
Numerical :
Given `L_(1)=1m=100 cm`
`L_(2)=105 cm ( :. ` It's increased by 5%)
`L_(2)=1.05L_(1)`
Since `L_(2)gtL_(1), :. n_(2)lt n_(1)`
`:. n_(1)-n_(2)=10`
`n_(1)L_(1)= n_(2)L_(2)`
`n_(1)L_(1)=(n_(1)-10)L_(2)`
`:. n_(1)L_(1)=(n_(1)-10)(1.05L_(1))`
`n_(1)=210 Hz`