Need help calculating tangential speed.?

Question

A 0.600 kg ball is on the end of a rope that is 1.77m in length. the ball and rope are attached to the pole and the entire apparatus, including the pole, rotates about the pole's symmetry axis. The rope makes an angle of 68 degrees with respect to the vertical. WHAT IS THE TANGENTIAL SPEED OF THE BALL?

Answer 1

Consider the forces on the ball:

The force of gravity downward
=.6*9.8

The centrifugal force outward
.5*m*w^2^r
And the tension on the rope

I solved the problem by considering the horizontal and vertical components of the rope tension:

The vertical component must offset the force of gravity:

sin(22)*T=m*g
T=m*g/sin(22)

Then the horizontal component of the tension equal to the centrifugal force
.5*m*w^2*r=T*sin(68)
r=sin(68)*1.77
.5*m*w^2*sin(68)*1.77=
m*g*sin(68)/sin(22)
The mass divides out
sin(68) divides out
.5*w^2*1.77=g/sin(22)
w=sqrt(9.8/(sin(22)*.5*1.77)
to compute tangential velocity, multiply w by r
v=sin(68)*1.77*sqrt(9.8/(sin(22)*.5*1.77)
v=8.9 m/s

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