Question

The vertical motion of mass Ais defined by the relation 10 sin 215cos 2100,xtt=++where xand tare expressed in mm and seconds, respectively. Determine (a) the position, velocity and acceleration of Awhen t=1 s, (b) the maximum velocity and acceleration of A.

Answer 1

x = 10sin 2t + 15cos 2t +100

v = dx/dy = 20cos 2t - 30sin 2t

a = dv/dt = 40sin 2t - 60cos 2t

For trigonometric function set calculator to radian:

(a) At t = 1 s x1 = 10sin 2 + 15cos 2 + 100 = 102.9

v1 = 20cos 2 - 30sin 2 = -35.6

a1 = -40sin 2 - 60cos 2 = -11.40

(b) Maximum velocity occure when a = 0.

-40sin 2t - 60cos 2t = 0

tan 2t = -60/40 = -1.5

2t = tan⁻¹ (-1.5) = -0.9828 and -0.9828 +
`\pi`

Reject the negative value. 2t = 2.1588

t = 1.0794
`s`

t = 1.0794
`s` for V
`max`

so
`Vmax` =20cos(2.1588) -30sin(2.1588)

= -36.056

Note that we could have also used

`Vmax` =
`\sqrt`20² +30²

= 36.056

by combining the sine and cosine terms.

For
`\alpha max` we can take the derivative and set equal to zero or just combine the sine and cosine terms.

`\alpha max` =
`\sqrt{40^(2) + 60^(2)`

= 72.1 mm/
`s^(2)`