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kolezko [41]
3 years ago
15

) The angular acceleration of the disk is defined by ???? = 3t 2 + 12 rad/s, where t is in seconds. If the disk is originally ro

tating 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.
Physics
1 answer:
pshichka [43]3 years ago
6 0

Answer:

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

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 = \frac{1}{4}\cdot t^{4} + 6\cdot t^{2} + 12\cdot t

The angular motion when t = 2 s. is:

\theta = \frac{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\,\frac{rad}{s}

Finally, the magnitude of the velocity is:

v_{A} = 44\cdot r \,\frac{m}{s}

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

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

\alpha = 24\,\frac{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\,\frac{m}{s^{2}}

a_{A,t} = \alpha \cdot r

a_{A,t} = 24\cdot r\,\frac{m}{s^{2}}

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