Question

The vertical motion of mass A is defined by the relation x 5 10 sin 2t 1 15 cos 2t 1 100, where x and t are expressed in millimeters and seconds, respectively. Determine (a) the position, velocity, and acceleration of A when t 5 1 s, (b) the maximum velocity and acceleration of A.

Answer 1

Step-by-step explanation:

Given that,

The vertical motion of mass A is defined by the relation as :

`x=10\ sin2t+15\ cos2t+100`

At t = 1 s

`x=10\ sin2+15\ cos2+100`

x = 115.33 mm

(a) We know that,

Velocity,
`v=(dx)/(dt)`

`v=(d(10\ sin2t+15\ cos2t+100))/(dt)`

`v=20\ cos2t-30\ sin2t`

At t = 1 s

`v=20\ cos2-30\ sin2`

v = 18.94 mm/s

We know that,

Acceleration,
`a=(dv)/(dt)`

`a=(d(20\ cos2-30\ sin2))/(dt)`

`a=-40\ cos2t-60\ cos2t`

At t = 1 s

`v=-40\ cos2-60\ cos2`

`a=-99.93\ mm/s^2`

(b) For maximum velocity,
`(dv)/(dt)=a=0`

`-40\ cos2t-60\ cos2t=0`

t = 45 seconds

For maximum acceleration,
`(da)/(dt)=0`

`80\ sin2t+120\ cos2t=0`

t = 61.8 seconds

Hence, this is the required solution.