91Ó°ÊÓ

In this exercise, we witness an excellent example of energy conversion. When the 6.8 kg bowling ball is released from rest, it transforms the potential energy it holds due to its elevated position into kinetic energy as it swings downward.
This transformation underscores one of the core principles of physics: the conservation of energy. As the bowling ball descends, the gravitational potential energy it initially possesses decreases, and correspondingly, its kinetic energy increases.
This energy conversion can be mathematically represented as:

• Potential Energy (PE) decreases by: \[ PE = mgh \] where \( m \) is mass, \( g \) is gravitational acceleration, and \( h \) is the height from which it falls.
• At the lowest point in its swing, the kinetic energy (KE) is at its maximum and equal to the lost potential energy: \[ KE = \frac{1}{2}mv^2 = PE \]

This illustrates how these forms of energy are interchangeable in a closed system, permitting us to track one type of energy to determine the characteristics of another.

Potential energy is a form of stored energy that depends on an object's position relative to a zero position. In this case, the bowling ball has gravitational potential energy because it is suspended at a certain height above the ground.
Gravitational potential energy is determined by the formula:
\[ PE = mgh \] where:

• \( m \) is the mass of the object (6.8 kg).
• \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth).
• \( h \) is the height change (1.4 meters in this scenario).

As the bowling ball drops, its height decreases, leading to a reduction in potential energy. This lost potential energy is precisely what gets transformed into kinetic energy, causing the ball to accelerate as it moves downward. Understanding this concept helps in appreciating how objects store and convert energy based on their position and motion.

Hooke's Law is fundamental to understanding how materials respond to force, and it describes the behavior of elastic materials like the nylon cord in this problem. The law is expressed as:
\[ F = k \Delta L \] where:

• \( F \) is the force applied to the material.
• \( k \) is the spring constant, reflecting the stiffness of the material.
• \( \Delta L \) is the change in length of the material due to the applied force, given here as 2.7 x 10^{-3} m.

In this context, Hooke's Law helps us understand how the cord stretches when the bowling ball exerts a force on it during its motion. The spring constant \( k \) can be determined using the material's properties and is key to calculating other quantities such as the unstrained length of the cord by connecting it with concepts like Young's Modulus.

Young's Modulus is a measure of a material's ability to withstand changes in length when under lengthwise tension or compression. It characterizes the stiffness of a given material and is crucial in understanding how materials like the nylon cord behave when forces are applied.
Young's Modulus \( Y \) is used in conjunction with other constants and variables to calculate the spring constant \( k \), essential for applying Hooke's Law. The relationship is given by:
\[ k = \frac{Y A}{L} \] where:

• \( Y \) is Young's Modulus.
• A is the cross-sectional area of the material (3.4 x 10^{-5} m²).
• \( L \) is the unstrained length of the material.

Understanding and applying Young's Modulus allows us to solve for the unstrained length of the cord by rearranging and solving the equation for \( L \). This insight into material behavior under stress reveals the interconnectedness of theoretical physics principles and real-world observations.