Ball on a String: Tension and Speed Formulas for Circular Motion

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The discussion focuses on deriving formulas for tension and speed in a tether ball system. The tension T in the rope can be expressed as Tcos(θ) = mg, which accounts for vertical forces. For the horizontal forces, the centripetal force equation F = mv²/r is utilized to find the speed v of the ball. Participants emphasize the need to separate vertical and horizontal forces to solve the problem accurately. Overall, the thread provides insights into the physics of circular motion involving tension and speed calculations.
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A tether ball of mass m is suspended by a rope of length L from the top of a pole. A youngster gives it a whack so that it moves with some speed v in a circle of radius r = L sin(θ) < L around the pole.
a) Find an expression for the tension T in the rope as a function of m, g, and θ.
b) Find an expression for the speed v of the ball as a function of θ.



F = mv^2 / r



I do not know what to do. All I have is Tcosx = mv^2/r - mg
any help please??
 
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+ mg sorry
 
That's not right. There are essentially two equations. One for the vertical forces and one for the horizontal forces.

You've written the equation for the vertical forces, but the centripetal force shouldn't be in it. You should have: Tcos(\theta) = mg For the vertical forces.
From this you get the tension in the rope.

Then you can use the equation for the horizontal forces to get the speed of the ball.
 
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