Problem 9 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

(a) Zero for both; (b) Zero by tension, 25.088 J by gravity.

Step 1: Understand the Problem

We have a 0.800-kg ball tied to a 1.60 m long string, swung in a vertical circle. We need to calculate the work done by the tension in the string and the gravitational force in two scenarios: a complete circle (part a) and a semicircle from the lowest to the highest point (part b).

Step 2: Calculate Work Done by Tension in Complete Circle

Work done by tension is calculated with the formula \( W = F \, \cos(\theta) \, d \) where \( F \) is the force, \( \theta \) is the angle between force and displacement, and \( d \) is displacement. Since tension is perpendicular to displacement through the complete circle, \( \theta = 90^\circ \) and \( \cos(90^\circ) = 0 \). Therefore, work done, \( W = 0 \).

Step 3: Calculate Work Done by Gravity in Complete Circle

The gravitational force does work, \( W = mgh \), where \( h \) is the change in height. In a complete circle, the ball returns to the starting point, so the change in height is zero. Consequently, the work done by gravity is zero, \( W = 0 \).

Step 4: Calculate Work Done by Tension for Semicircle

In a vertical motion from the lowest to the highest point of the semicircle, the tension remains perpendicular to the instantaneous displacement at every point. Thus, similar to the full circle, the work done by the tension remains \( W = 0 \).

Step 5: Calculate Work Done by Gravity for Semicircle

From the lowest to the highest point, the vertical height change is twice the radius, \( 2r = 2 \times 1.60 \times \sin(\theta) \). At the top, \( \theta = 90^\circ \), so the height change is 3.20 m. Work done by gravity is \( W = mgh = 0.800 \times 9.8 \times 3.20 \). Calculate \( W = 25.088 \text{ J} \).

Key Concepts

Circular Motion

Circular motion occurs when an object moves in a path in the shape of a circle. This type of motion is characterized by a constant change in direction, even if the speed of the object remains constant.
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.

Understanding circular motion is key to studying how forces interact in a rotating system and how different scenarios, like vertical circles, affect these dynamics.

Vertical Circle

Vertical circular motion involves an object moving in a circle located in a vertical plane. This adds complexity because, as the object moves, the gravitational force affects it differently at varying points.
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.

Vertical circles provide an excellent real-world application, whether in amusement park rides or understanding planetary motion.

Work Done by Gravity

Work done by gravity plays a crucial role in many physics problems involving motion. It is the energy transferred by the gravitational force over a distance.
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.

Understanding work done by gravity aids in analyzing the energy changes and forces at play in cyclic motion.

Tension in Strings

Tension in strings is a vital concept when studying objects in circular motion. It is the force exerted along the string that transmits a pulling force from one end to the other.
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).

Comprehending how tension works in strings forms the basis for solving complex motion and force problems in physics.