a) 1.57 m/s
The sock spins once every 2.0 seconds, so its period is
T = 2.0 s
Therefore, the angular velocity of the sock is
[tex]\omega=\frac{2\pi}{T}=\frac{2\pi}{2.0}=3.14 rad/s[/tex]
The linear speed of the sock is given by
[tex]v=\omega r[/tex]
where
[tex]\omega[/tex] is the angular velocity
r = 0.50 m is the radius of the circular path of the sock
Substituting, we find:
[tex]v=(3.14)(0.50)=1.57 m/s[/tex]
B) Faster
In this case, the drum is twice as wide, so the new radius of the circular path of the sock is twice the previous one:
[tex]r' = 2r = 1.00 m[/tex]
At the same time, the drum spins at the same frequency as before, therefore the angular frequency as not changed:
[tex]\omega' = \omega = 3.14 rad/s[/tex]
Therefore, the new linear speed would be:
[tex]v'=\omega' r' = \omega (2r)[/tex]
And substituting,
[tex]v'=(3.14)(1.00)=3.14 rad/s = 2v[/tex]
So, we see that the linear speed has doubled.
