Answer:
(a) The turn table velocity
(b) 6.86s
Step-by-step explanation:
(a) The velocity of the turntable would be the one causing centripetal acceleration on the coin and the equation for that is
[tex] a_c = \frac{v^2}{r} = \frac{(\omega r)^2}{r} = \omega^2r = (\alphat)^2r[/tex]
Where [tex]\alpha = 0.84 rad/s^2[/tex] is the angular acceleration and r = 13 cm = 0.13 m is the distance form table origin to the coin
There's also an inertia force causes by the turntable angular acceleration, this force would have its acceleration equal to:
[tex]a_i = \alpha r[/tex]
This acceleration would be perpendicular to the centripetal acceleration. So the total acceleration acting on the coin is:
[tex]\vec{a} = \vec{a_c} + \vec{a_i}[/tex]
[tex]a^2 = a_c^2 + a_i^2[/tex]
[tex]a^2 = \alpha^4t^4r^2 + \alpha^2r^2[/tex]
(b) The coins begins to move when its total acceleration force begin to wins its static friction force, which is generated by gravity
[tex]F_a = F_f[/tex]
[tex]ma = mg\mu[/tex]
[tex]a^2 = (g\mu)^2[/tex]
[tex]\alpha^4t^4r^2 + \alpha^2r^2 = (g\mu)^2[/tex]
We can substitute in the known parameters ([tex]g = 9.81 m/s^2[/tex])
[tex]0.84^4*0.13^2t^4 + 0.84^2*0.13^2 = (9.81*0.44)^2[/tex]
[tex]0.008414t^4 + 0.01192464 = 18.631[/tex]
[tex]t^4 = 18.61938432/0.008414 = 2212.9[/tex]
[tex]t = 2212.9^{1/4} = 6.86 s[/tex]
So the coin would start moving after 6.86s.
