Physics problem help -- simple centrifugal force problem given distance, weight and speed?

Question

I thought this problem was straight forward and simple, but when I do it I am unable to get the correct answer, no matter what I do. Help?

A child is playing with a ball on a string. The ball has mass 0.065 kg, the string has radius 1.0 m, and assume that the string is massless and neglect air resistance. The child swings the ball on a string in a vertical circle. Halfway up the circle the speed of the ball is measured to be 9.8 m/s. What is the maximum tension in the string?

Picture to go along with problem: http://i284.photobucket.com/albums/ll28/bathtub2008/showmepl.jpg

Tmax = _____ N

Answer 1

First, notice that the tension of the string is at its maximum at the bottom of the vertical circle. So when you draw the free body diagram of the ball, tension is pointing upwards, and force of gravity is pointing downward.

Let upwards be positive

Assuming that the speed of the ball is constant, then:

a = v² / ρ,

where,

a is the acceleration normal to the path of motion of the ball

v is the speed of the ball

ρ is the radius of curvature of the path

=> a = (9.8 m/s)² / (1.0 m) = 96.04 m/s² [upwards]

The sum of all forces normal to the path of motion of the ball = the mass of the ball * the acceleration of the ball. In other words,

T - mg = (0.065 kg)(94.04 m/s²)

=> T - (0.065 kg)(9.81 N/kg) = 6.2426 N

=> T = 6.88 N

Therefore, the maximum tension, Tmax = 6.88 N.