Question

A ball of mass m=8 kg is suspended from a string of length L=5m the ball revovles in constant speed of horizontal circle of radius r=3m calculate the speed

Answer 1

Let's apply the Newton's Second Law of Motion in projections on axis $x$ and $y$:

$$Tsin\theta=\dfrac{mv^2}{r},$$ $$Tcos\theta=mg.$$

Let's divide the first equation by the second one:

$$tan\theta=\dfrac{v^2}{gr}.$$

From this formula we can find the speed of the ball:

$$v=\sqrt{rgtan\theta}.$$

We can find angle $\theta$ from the geometry:

$$sin\theta=\dfrac{r}{L},$$ $$\theta=sin^{-1}(\dfrac{r}{L})=sin^{-1}(\dfrac{3\ m}{5\ m})=36.8^{\circ}.$$

Finally, we can calculate the speed of the ball:

$$v=\sqrt{3\ m\cdot9.8\ \dfrac{m}{s^2}\cdot tan36.8^{\circ}}=4.69\ \dfrac{m}{s}.$$