Question

) The angular acceleration of the disk is defined by ???? = 3t 2 + 12 rad/s, where t is in seconds. If the disk is originally rotating at ????0 = 12 rad/s. a) Determine the angular motion when t = 2 s. b) Determine the magnitude of the velocity of point A on the disk when t = 2 s. c) Determine the normal and tangential components of acceleration of point A on the disk when t = 2 s.

Answer 1

a)
`\theta = 52\,rad`, b)
`v_(A) = 44\cdot r \,(m)/(s)`, c)
`a_(A,n) = 1936\cdot r\,(m)/(s^(2))`,
`a_(A,t) = 24\cdot r\,(m)/(s^(2))`

Step-by-step explanation:

a) The angular motion is obtained by integrating the angular acceleration function twice:

`\alpha = 3\cdot t^(2) + 12`

`\omega = t^(3) + 12\cdot t + 12`

`\theta = (1)/(4)\cdot t^(4) + 6\cdot t^(2) + 12\cdot t`

The angular motion when t = 2 s. is:

`\theta = (1)/(4)\cdot (2\,s)^(4) + 6\cdot (2\,s)^(2) + 12\cdot (2\,s)`

`\theta = 52\,rad`

b) Let be
`r` the distance between A and the rotation axis, measured in meters. The magnitude of the angular velocity when t = 2 s. is:

`\omega = (2\,s)^(3) + 12\cdot (2\,s) + 12`

`\omega = 44\,(rad)/(s)`

Finally, the magnitude of the velocity is:

`v_(A) = 44\cdot r \,(m)/(s)`

c) The angular acceleration of the disk when t = 2 s. is:

`\alpha = 3\cdot (2\,s)^(2) + 12`

`\alpha = 24\,(rad)/(s^(2))`

Lastly, the normal and tangential components at point A are, respectively:

`a_(A,n) = \omega^(2)\cdot r`

`a_(A,n) = \left(44\right)^(2)\cdot r`

`a_(A,n) = 1936\cdot r\,(m)/(s^(2))`

`a_(A,t) = \alpha \cdot r`

`a_(A,t) = 24\cdot r\,(m)/(s^(2))`