Question

A ball is tied to the top of a pole by a string. The ball rotates at 5.0m/s around the pole in a horizontal circle of radius 1.0m. What is the angle between the string and pole?

Answer 1

If you draw a free body diagram, two forces act on the ball. Gravity vertically down and the tension in the rope directed along the rope. The vertical component of the tension must equal the weight of the ball (mg), and the horizontal component must deliver the centripetal force m v^2 /R, where v is the speed of the circular motion and R the radius of the circle. The ratio of these forces equals the tangent of the angle the rope makes with the pole (the rope, radius and pole form a right angled triangle):

tan(angle) = centre force / weight

= mv^2/R / mg

= v^2/(gR)

Angle = arctan(v^2/(gR) )

= arctan( 5.0^2 /(9.81* 1.0) )

= 68.57 degrees