If the displacement of a body is given by ` x = 3 cos [ 2 pi t + pi/4]` m, then the acceleration of the body at t=2s is

Options

A. 0
B. `-6 sqrt2 pi ^(2) m//s ^(2)`
C. `-10 pi^(2) m//s ^(2)`
D. `-12 sqrt2 pi^(2) m//s ^(2)`

Answer

B

Explanation

Given,
Displacement of a body, `x =3 cos [ 2pi t + (pi)/(4)]m`
Acceleration of body at `t = 2s,`
`|a = (d ^(2) x)/(dt ^(2))|_(2s)`
`a = [ (d ^(2))/(dt ^(2) ) 3 cos (2pit + (pi)/(4)) ] _(2),`
`= [(d)/(dt) ((d)/(dt ) 3 cos (2pi t + (pi)/(4)))] _(2s)`
`= [(d)/(dt) (-3 sin (2 pi t + (pi)/(4) ) 2pi )]_(2s)`
`=[-6pi (d)/(dt) sin (2pit + (pi)/(4) ) ] _(2s)`
`[- 6pi cos (2pit + (pi)/(4) ) (2pi) ] _( 2s )`
`=- 12 pi ^(2) cos (2pi t + (pi)/(4)) _(2s)`
`=-12 pi ^(2)cos (2 pi (2) + (pi)/(4))`
`=- 12 pi ^(2) cos (4pi + (pi)/(4))`
`=- 12 pi ^(2) cos ((pi)/(4))`
`=- 12 pi^(2) xx (1)/(sqrt2)`
`therefore a =-6 sqrt2 pi ^(2) m//s`