Answer:
a)[tex]\omega=1.36rad/s[/tex]
b)[tex]\omega=12.99rpm[/tex]
c)[tex]F=705.6N[/tex]
Explanation:
a) The angular velocity is related to the centripetal acceleration by the formula [tex]a_{cp}=\omega^2r[/tex], which for our purposes we will write as:
[tex]\omega=\sqrt{\frac{a_{cp}}{r}}[/tex]
Since we want this acceleration to be 1.5 times that due to gravity, for our values we will have:
[tex]\omega=\sqrt{\frac{1.5g}{r}}=\sqrt{\frac{(1.5)(9.8m/s^2)}{(8m)}}=1.36rad/s[/tex]
b) 1 rpm (revolution per minute) is equivalent to an angle of [tex]2\pi[/tex] radians in 60 seconds:
[tex]1\ rpm=\frac{2\pi rad}{60s} =\frac{\pi}{30}rad/s[/tex]
Which means we can use the conversion factor:
[tex]\frac{1\ rpm}{\frac{\pi}{30}rad/s}=1[/tex]
So we have (multiplying by the conversion factor, which is 1, not affecting anything but transforming our units):
[tex]\omega=1.36rad/s=1.36rad/s(\frac{1\ rpm}{\frac{\pi}{30}rad/s})=12.99rpm[/tex]
c) The centripetal force will be given by Newton's 2nd Law F=ma, so on the centripetal direction for our values we have:
[tex]F=ma=(48kg)(1.5)(9.8m/s^2)=705.6N[/tex]
