Chapter 6: Problem 9
A 0.800-kg ball is tied to the end of a string 1.60 m long and swung in a vertical circle. (a) During one complete circle, starting anywhere, calculate the total work done on the ball by (i) the tension in the string and (ii) gravity. (b) Repeat part (a) for motion along the semicircle from the lowest to the highest point on the path.
Short Answer
Step by step solution
Understand the Problem
Calculate Work Done by Tension in Complete Circle
Calculate Work Done by Gravity in Complete Circle
Calculate Work Done by Tension for Semicircle
Calculate Work Done by Gravity for Semicircle
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Circular Motion
In a circular motion, the velocity of the object is always tangential to the circle's path. This means the direction of the velocity vector changes as the object moves, even if the speed (magnitude of the velocity) remains constant.
To maintain this motion and change direction continuously, a centripetal force is needed, which always points toward the center of the circle.
- The centripetal force is not a separate force but the resultant of other forces acting on the object.
- In the case of a ball tied to a string swung in a circle, the tension in the string contributes to this centripetal force.
Vertical Circle
One key factor in vertical circular motion is how gravitational force interacts with other forces like tension. Unlike horizontal motion, gravity contributes to the dynamics by accelerating or decelerating the object as it rises and falls.
When the object is at the highest point in the circle, gravity acts downward, potentially decreasing tension in the string. Conversely, when the object is at the lowest point, gravity increases tension since it acts in the same direction.
- Gravitational force is crucial because it can lead to changing tension forces throughout the motion.
- Understanding these dynamics enables the calculation of forces and predictability in motion outcomes, especially when calculating work done by gravity and tension.
Work Done by Gravity
In the context of a ball in circular motion, the work done by gravity depends on the vertical displacement of the ball. When the ball returns to its starting height, the work done by gravity over the complete circle is zero, as there is no net change in height.
- Formula: The work done by gravity is given by \[ W = mgh \] where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the change in height.
- During vertical motion through a semicircle, however, there is a net vertical displacement, hence work done is not zero.
- This calculation helps determine how potential energy converts to kinetic energy and vice-versa through motion.
Tension in Strings
When an object moves in a circular path, the tension helps keep the object in motion by providing the necessary centripetal force toward the center of the circle.
In our scenario with the ball on a string, tension remains perpendicular to the displacement because the ball moves along the arc of the circle. As a result, the work done by tension is zero when the object completes a full cycle or semicircle.
- The tension force acts radially, maintaining the grip needed for circular motion.
- Its value changes depending on the ball's position in the circle, especially impacted by the gravitational pull.
- The perpendicular nature of tension regarding motion direction means that it does no work (i.e., transfers no energy over a distance).