Answer
The ball's tangential speed is $5.74~m/s$
Work Step by Step
The vertical component of the rope's tension is equal in magnitude to the ball's weight. We can find an expression for the tension $T$:
$T~cos~\theta = mg$
$T = \frac{mg}{cos~\theta}$
The horizontal component of the tension provides the centripetal force to keep the ball moving around in a circle. We can find the ball's tangential speed:
$T~sin~\theta = \frac{mv^2}{r}$
$(\frac{mg}{cos~\theta})~sin~\theta = \frac{mv^2}{r}$
$mg~tan~\theta = \frac{mv^2}{r}$
$g~r~tan~\theta = v^2$
$v= \sqrt{g~r~tan~\theta}$
$v= \sqrt{(9.80~m/s^2)(1.30~m)~sin~70.0^{\circ}~tan~70.0^{\circ}}$
$v = 5.74~m/s$
The ball's tangential speed is $5.74~m/s$.
