The equation of motion of a particle started at t=0 is given by `x=5sin(20t+pi/3)`, where x is in centimetre and t in second. When does the particle
a. first come rest
b. first have zero acceleration
c. first have maximum speed?

Answer

So, at `pi/120 ` sec it first comes to rest

`t=pi/30sec`
`rarr t=pi/30sec`

Explanation

`x=5(20t+pi/3)`
a. Max displacement from the mean position = Amplitude of the particle.
At the extreme position the velocity becomes 0
`:. x=5=Amplitude`
`:. 5=5sin(20t+pi/3)`
`sin(20t+pi/3)=1=sinpi/2`
`rarr 20t+pi/3=pi/2`
`rarr t=pi/120sec`e
`So, at pi/120 ` sec it first comes to rest
`b. a=omega^2x`
`=omega^2[5sin(20t+pi/3)]`
Fro `a=0,`
`5sin(20t+pi/3)=0`
`rarr sin(20t+pi/3)=sinx`
`rarr 20t=pi-pi/3=(2pi)/3`
`t=pi/30sec`
c. `v=Aomegacos(omegat+pi/3)` lt. when omega is maximum i.e. `cos(20t+pi/3)=-1=cospi`
`20t=pi-pi/3=(2pi)/3`
`rarr t=pi/30sec`