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

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, $v=\dfrac{dx}{dt}$

$v=\dfrac{d(10\ sin2t+15\ cos2t+100)}{dt}$

$v=20\ cos2t-30\ sin2t$

At t = 1 s

$v=20\ cos2-30\ sin2$

v = 18.94 mm/s

We know that,

Acceleration, $a=\dfrac{dv}{dt}$

$a=\dfrac{d(20\ cos2-30\ sin2)}{dt}$

$a=-40\ cos2t-60\ cos2t$

At t = 1 s

$v=-40\ cos2-60\ cos2$

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

(b) For maximum velocity, $\dfrac{dv}{dt}=a=0$

$-40\ cos2t-60\ cos2t=0$

t = 45 seconds

For maximum acceleration, $\dfrac{da}{dt}=0$

$80\ sin2t+120\ cos2t=0$

t = 61.8 seconds

Hence, this is the required solution.