if the kinetic energy and potential energy becomes equal for a particle executing SHM, given by y=10sin5πt, then at that instance the displacement of the particle from the mean position is

Asked by binayakbiswas | 16th Mar, 2020, 09:25: AM

Expert Answer:

Displacement y = 10 sin (5π t )  ................(1)
 
From eqn.(1), we get,  angular frequency ω = 5π rad/s
 
if kinetic energy and potential energy are equal, then we have
 
(1/2) m v2 = (1/2) k y2 .....................(2)
 
where m is mass of particle , v is speed at time t and k = mω2 is force constant.
 
hence we get from eqn.(2),  y = ± v/ω  ................(3)
 
speed of particle is obtained by differentiating eqn.(1), v = 10 (5π) cos( 5πt ) ...............(4)
 
hence from eqn.(3) and (4), we get,  y = ± 10 cos ( 5π t )  .....................(5)
 
Using eqn.(1), we write eqn.(5) as ,  10 sin ( 5πt ) = ± 10 cos ( 5π t )   or  tan (5πt) = ± 1  ............(6)
 
Hence we get , ( 5 π t ) = + π/4  or  ( 5 π t ) = +3π/4
 
Hence time to get equal kinetic energy and potential energy is obtained as
 
  t =  1/20 s  or  t = 3/20 s
 

Answered by Thiyagarajan K | 16th Mar, 2020, 03:24: PM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.