A Ball Bearing A 2 -oz steel ball bearing is placed on a...

A Ball Bearing A 2 -oz steel ball bearing is placed on a vertical spring whose force constant is $k=18$ lb/ft. The spring is compressed 3 in. and released. About how high does the ball bearing go? (Hint: The kinetic (compression) energy, mgh, of a spring is $\frac{1}{2} k s^{2}$ , where $s$ is the distance the spring is compressed, $m$ is the mass, $g$ is the acceleration of gravity, and $h$ is the height.)

Key Concepts

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring or any elastic object when it is compressed or stretched. It is calculated using the formula (1/2)k(s^2), where k is the spring constant and s is the displacement from the rest position. This concept is essential for understanding how energy is stored during the compression of the spring.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position relative to the Earth. It is given by the formula mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height above a reference point. This energy converts to kinetic energy or other forms as an object moves within a gravitational field.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed but only converted from one form to another. In mechanical systems, this means that the energy stored in the compressed spring (elastic potential energy) will convert entirely into gravitational potential energy (and possibly kinetic energy) as the object moves, allowing us to equate the two forms to solve for unknown quantities like height.

Unit Conversion in Mechanical Problems

Ensuring consistency of units is crucial when solving physics problems. This concept involves converting units appropriately, such as converting mass from ounces to pounds and distances from inches to feet, to maintain consistency in the energy equations and to ensure accurate calculations.

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