Answer:
[tex]\mu_s=0.54[/tex]
Explanation:
In order for the friction to be sufficient to keep the mass from falling, the force of gravity (mg) must be the same friction force([tex]f_s[/tex]) and the centripetal force ([tex]ma_c[/tex]) must to have the same value of the normal force (N):
[tex]mg=f_s\\N=ma_c[/tex]
Recall that [tex]f_s=\mu_sN[/tex], so we have:
[tex]f_s=mg\\f_s=\mu_sma_c\\\mu_sma_c=mg\\\mu_s=\frac{g}{a_c}[/tex]
Recall that [tex]v=\omega r[/tex]. The centripetal acceleration is given by:
[tex]a_c=\frac{v^2}{r}\\a_c=r\omega^2[/tex]
Finally, replacing [tex]a_c[/tex]:
[tex]\mu_s=\frac{g}{r\omega^2}\\\mu_s=\frac{9.8\frac{m}{s^2}}{0.5m(6\frac{rad}{s})^2}\\\mu_s=0.54[/tex]
