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Consider a circular-motion problem in which tension not only must provide the proper radial acceleration but also must balance gravity. As we shall see, it can do both. A tetherball is attached to a swivel in the ceiling by a light cord of length L, as measured from the ceiling to the center of the tetherball. When the ball is hit by a paddle, it swings in a horizontal circle with constant speed vv, and the cord makes a constant angle β with the vertical direction. The ball goes through one revolution in time T. Assuming that T, mass mm, and length L of the rope are known, derive algebraic expressions for the tension FT in the cord and the angle β. a)What is the period T of the ball’s motion if L=2.0m and the cord makes an angle of β=40∘with the vertical?

Consider a circular-motion problem in which tension not only must provide the proper radial acceleration but also must balance gravity. As we shall see, it can do both. A tetherball is attached to a swivel in the ceiling by a light cord of length L, as measured from the ceiling to the center of the tetherball. When the ball is hit by a paddle, it swings in a horizontal circle with constant speed vv, and the cord makes a constant angle β with the vertical direction. The ball goes through one revolution in time T. Assuming that T, mass mm, and length L of the rope are known, derive algebraic expressions for the tension FT in the cord and the angle β. a)What is the period T of the ball’s motion if L=2.0m and the cord makes an angle of β=40∘with the vertical?

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Consider a circular-motion problem in which tension not only must provide the proper radial acceleration but also must balance gravity. As we shall see, it can do both. A tetherball is attached to a swivel in the ceiling by a light cord of length L, as measured from the ceiling to the center of the tetherball. When the ball is hit by a paddle, it swings in a horizontal circle with constant speed vv, and the cord makes a constant angle β with the vertical direction. The ball goes through one revolution in time T. Assuming that T, mass mm, and length L of the rope are known, derive algebraic expressions for the tension FT in the cord and the angle β.

a)What is the period T of the ball’s motion if L=2.0m and the cord makes an angle of β=40∘with the vertical?

**Figure Explanation for Educational Website** This image set presents diagrams related to the motion of a ball attached to a string in circular motion, often analyzed in a physics context. **(a) The situation:** - The ball is swinging in a horizontal circle, attached to a string of length \( L \). - The angle formed between the string and the vertical line is labeled as \( \beta \). - The blue dashed line indicates the path of the ball's circular motion with a radius \( R \). - The green arrow labeled \( \vec{v} \) represents the velocity of the ball, which is tangent to the circular path. **(b) The forces on the ball:** - The ball experiences multiple forces, shown as vectors. - A tension force \( \vec{F}_T \) acts along the string toward the pivot point. - The gravitational force \( \vec{w} = mg \) acts downward. - The radial acceleration \( \vec{a}_{\text{rad}} \) is directed towards the center of the circular path. **(c) Free-body diagram of the ball:** - This diagram breaks down the forces acting on the ball into components: - \( \vec{F}_T \) is the tension force in the string. - The tension has components \( F_T \cos \beta \) (vertical) and \( F_T \sin \beta \) (horizontal). - The radial acceleration \( \vec{a}_{\text{rad}} \) is horizontal. - The gravitational force \( \vec{w} = mg \) acts vertically downward. These diagrams collectively illustrate the dynamics of a ball moving in circular motion while anchored by a string. The interplay of gravitational and tension forces results in radial acceleration, crucial for maintaining circular motion.
Transcribed Image Text:**Figure Explanation for Educational Website** This image set presents diagrams related to the motion of a ball attached to a string in circular motion, often analyzed in a physics context. **(a) The situation:** - The ball is swinging in a horizontal circle, attached to a string of length \( L \). - The angle formed between the string and the vertical line is labeled as \( \beta \). - The blue dashed line indicates the path of the ball's circular motion with a radius \( R \). - The green arrow labeled \( \vec{v} \) represents the velocity of the ball, which is tangent to the circular path. **(b) The forces on the ball:** - The ball experiences multiple forces, shown as vectors. - A tension force \( \vec{F}_T \) acts along the string toward the pivot point. - The gravitational force \( \vec{w} = mg \) acts downward. - The radial acceleration \( \vec{a}_{\text{rad}} \) is directed towards the center of the circular path. **(c) Free-body diagram of the ball:** - This diagram breaks down the forces acting on the ball into components: - \( \vec{F}_T \) is the tension force in the string. - The tension has components \( F_T \cos \beta \) (vertical) and \( F_T \sin \beta \) (horizontal). - The radial acceleration \( \vec{a}_{\text{rad}} \) is horizontal. - The gravitational force \( \vec{w} = mg \) acts vertically downward. These diagrams collectively illustrate the dynamics of a ball moving in circular motion while anchored by a string. The interplay of gravitational and tension forces results in radial acceleration, crucial for maintaining circular motion.
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