Final answer:
The difference between the tension in the string at the bottom of the circle and the top (T_B - T_T) is equal to the weight of the ball.
Step-by-step explanation:
To find the difference between the magnitude of the tension in the string at the bottom relative to that at the top of the circle (TB - TT), we will use Newton's laws and the concept of centripetal force in a vertical circle motion.
At the top of the circle, the tension (TT) in the string must provide enough force to keep the ball moving in a circle, plus counteract the ball's weight (mg). Hence,
TT + mg = m v2/r,
where m is the mass of the ball, v is the tangential velocity, and r is the radius of the circular path.
At the bottom of the circle, the tension (TB) still provides the centripetal force, but now the weight of the ball works in the same direction as the tension. Thus,
TB = m v2/r + mg.
Subtracting the tension at the top (TT) from the tension at the bottom (TB), we get:
TB - TT = mg.
The difference between the tensions is equal to the weight of the ball.