91Ó°ÊÓ

The principle of conservation of energy is vital when studying moving objects like the collar in this problem. This law tells us that energy can change forms but cannot disappear or be created out of nothing.
In the realm of mechanical physics, we often deal with potential and kinetic energy. When a collar moves without friction, as described in our scenario, mechanical energy is conserved. At the start, the collar has zero kinetic energy because it is not moving. When the collar displaces, potential energy stored in the springs changes into kinetic energy.
The equation \[ \text{KE}_\text{initial} + \text{PE}_\text{initial} = \text{KE}_\text{final} + \text{PE}_\text{final} \] tells us that the total energy before and after the collar moves is the same due to energy conservation. Initially, there’s no potential energy because the springs are unstressed. Once the springs are stretched and compressed, they store energy. This energy converts into kinetic energy as the collar gains speed.

Springs hold potential energy through deformation. This energy follows Hooke's Law, which states that the force needed to compress or extend a spring is proportional to its deformation from the natural length.
This force is described with the formula \[ F = k \times \Delta L \] where \( F \) is the force from the spring, \( k \) is the spring constant, and \( \Delta L \) is the change in length.
The springs contribute to the system's potential energy when they are either compressed or extended. In the exercise, both springs are deformed by an inch when the collar shifts. The changes in spring lengths determine the energy stored, calculated using \[ \text{PE} = \frac{1}{2} k (\Delta L)^2 \].
For both springs, substituting \( k = 2800 \) lb/in and \( \Delta L = 1 \text{ inch} \) shows how springs directly relate to the collar's motion and energy change.

Calculating velocity involves understanding how potential energy of the springs turns into kinetic energy of the moving collar. We can find velocity by equating the kinetic energy gained with the total potential energy from the springs: \[ \frac{1}{2} m v^2 = \text{PE}_{\text{total}} \].
This equation allows solving for velocity if mass and energy data are given.
In our scenario, after moving 1 inch, the total potential energy from decompressed springs is 233.33 ft-lb. Accounting for system parameters like the collar's weight of 4 lb (converted into mass using \( m = \frac{4}{32.2} \) slugs), you can solve for \( v \), the velocity.
The formula simplifies to \[ v^2 = \frac{233.33 \times 2 \times 32.2}{4} \], resulting in a calculated velocity of approximately 61.35 ft/s. This velocity indicates how changing potential energy from spring tension accelerates the collar along its path.