Calculating Angular Velocity of Rock Swinging in a Circle

[TonkaQD4]

A boy ties a 500kg rock to a 1.0m long string and swings it around his head in a horizontal circle. At what angular velocity, in rpm, does the string tilt down at a 10 degree angle?

sintheta = mg / T

sin10 = 0.5kg(9.8m/s^2) / T

T = 4.9N / sin10

T = 28.22N

THEN...

v = SqRt [(rTcostheta) / m] ---> However the radius of the circle is not the length (L) so, r = Lcostheta

v = SqRt [(LTcos^2 theta) / m]

v = SqRt [(1.0m)(28.22N)(cos10^2) / 0.5kg]

v = 7.4m/s

FINALLY...

OMEGA = v / r
= v / Lcostheta

7.4rad / 1sec x 1rev / (2pi)rad x 60sec / 1 min --> rad and sec cancel out, leaving...

= 70.7 rpm


This was my initial attempt to the problem but I am not sure if everything is correct?

 

[Astronuc]

Looks fine through v = 7.4 m/s, but then look at $\omega$ = v/r = v/(L cos (10°).

cos 10° = 0.985, so $\omega$ = (7.4 m/s)/(0.985 m) = 7.51 rad/s

The rest is correct in form.

 

[TonkaQD4]

Thanks for the correction!