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When an elastic object like a spring or an elastic cord is stretched or compressed, it stores energy known as elastic potential energy. This stored energy is determined by the object's ability to return to its original shape. In the given example, the elastic potential energy is stored in the cord when it is stretched 1 foot beyond its original 2-foot length. The formula used to calculate this energy is:

• Elastic Potential Energy (PE) = \( \frac{1}{2} k x^2 \)
• Here, \( k \) is the spring constant and \( x \) is the amount of stretch or compression

For the ball and elastic cord in this problem, by stretching it 1 foot with a spring constant \( k = 1 \) lb/ft, the work done is transformed into 0.5 ft-lb of elastic potential energy. This energy represents the potential work the cord can do when it returns to its original length. Understanding this concept helps us see how potential energy transforms into kinetic energy as the object moves.

Kinetic energy is the energy an object possesses due to its motion. It depends on the mass and velocity of the object. In problems involving conservation of energy, kinetic energy can be converted from potential energy as in this case where the stretched elastic cord’s energy converts into kinetic energy when it becomes slack. The formula to determine kinetic energy is:

• Kinetic Energy (KE) = \( \frac{1}{2} mv^2 \)
• \( m \) is the mass of the object
• \( v \) is the velocity of the object

In the scenario of the sliding ball, when it is closest to the fixed point O, all the stored potential energy transitions into kinetic energy. At this point, the ball's velocity is no longer influenced by the elastic potential energy because the cord is slack, thus its kinetic energy can be computed by setting it equal to the initial elastic potential energy. This method is helpful in determining various states of motion and speed during the system's operation.

Mechanical energy is a combination of potential energy and kinetic energy within a system. Conservation of energy principles state that the total mechanical energy remains constant if the system is not affected by external forces like friction.

• The principle of conservation of mechanical energy: Initial Total Energy = Final Total Energy

In this ball and elastic cord example, the system initially has both potential and kinetic energy. As the ball moves closer to point O, the potential energy from the stretched cord converts entirely into kinetic energy, with the mechanical energy remaining constant throughout because of the absence of friction. This initially stored energy is why the system can predict the speed of the ball despite not observing it directly. Examining mechanical energy transformations allow us to solve many physics problems by equating conditions at different points and can demonstrate energy flow within dynamic systems.