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

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The discussion focuses on calculating the tension (T) in the rope and the speed (v) of a tether ball in circular motion. The tension is derived from the vertical forces equation Tcos(θ) = mg, where m is the mass of the ball and g is the acceleration due to gravity. The speed of the ball is determined using the centripetal force equation F = mv²/r, where r = L sin(θ). These formulas provide a clear understanding of the dynamics involved in the motion of the tether ball.

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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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