The motion of a particle of mass m is given by x = 0 for `t lt 0` s, x(t) = `A sin 4pit` for `0 lt t lt (1/4) s (A gt 0)` and x =0 for `t gt(1/4)` s which of the following statements is true ?

Options

A. The force at `t=(1/8)` on the particle is `-16 pi^(2)A-m`
B. The particle is acted upon by on impulse of magnitude `4pi^(2)` A - m at t = 0 s and `t=(1/4)s`
C. The particle is not acted upon by any force
D. The particle is not acted upon by a constant force

Answer

A, B, D

Explanation

(A,B,D )
x= 0 for `t le 0 ` sec
for 0 `lt t lt (1)/(4)s`
`v= (dx )/(dt ) = 4 pi A cos ( 4pi t ) `
At t = `(1)/( 8)` sec
`=-16 pi^(2) A sin ((pi)/( 2))`
Now force F = ma (t )
`=m[-16 pi^(2) A]`
`=-16 pi^(2) Am`
Now impulse of force change in mometum
`=(-16pi^(2) mA sin pi ) ^(1/4) = 0`
The impulse change in linear momentum a
`t= 0` is same as at `t= (1)/(4) ` sec