A particle executing SHM possesses two types of energy. Potential energy is on account of displacement of particle from the mean positoin, anad kinetic energy is ono account of velocity of the particle. At any time t
`P.E. =U=(1)/(2)momega^(2)y^(2)=(1)/(2)momega^(2)a^(2)sin^(2)omegat, ` and
`K.E..=K=(1)/(2)momega^(2)(a^(2)-y^(2))=(1)/(2)momega^(2)a^(2)cos^(2)omegat`
`T.E. =E=U+K=(1)/(2)momega^(2)a^(2)=` constant
Read the above passage and answer the following questions:
(i) At what distance from the mean position will K.E. of a particle be twice its P.E.?
(ii) What are the implications of this study in day life?

Explanation

(i) As K.E. `=2P.E`.
`:. (1)/(2)momega^(2)(a^(2)-y^(2))=2xx(1)/(2)momega^(2)y^(2) or a^(2)-y^(2)=2y^(2) or 3y^(2)=a^(2), y=(a)/(sqrt(3))`
(ii) This study shows that sum total of P.E. and K.E. of a particle in SHM stays constant at all position and at all times. However, P.E. and K.E. both keep on changing with position or time. The same is true in daya to day life. We can acquire one form of energy by spending some other form of energy, and vice-versa. For example, when we work too har, we forget about rest and relaxation. And when we relax, we store energy to work again. This is what goes on in life.