10.6 - Elastic Potential Energy
Hooke's law and its application to springs
Hooke's law describes the relationship between the force applied to an elastic object, such as a spring, and the resulting extension. Extension (e) is the change in length of the object from its original, unstretched position. This law applies as long as the object remains within its elastic limit, meaning it returns to its original shape after the force is removed.
Key components of Hooke's law
Formula for Hooke's law:
Where:
- F = Applied force (N)
- k = Spring constant (N/m), which measures the stiffness of the spring - a higher value means the spring is harder to stretch
- e = Extension (m), calculated as the stretched length minus the original length
This equation shows that the force is directly proportional to the extension. For example, doubling the extension requires double the force, assuming the spring constant remains the same.
The equation for elastic potential energy in stretched objects
When a force stretches an elastic object like a spring, work is done, and energy is stored as elastic potential energy (Ee). This energy is released when the object returns to its original shape, potentially converting to other forms such as kinetic energy.
Understanding elastic potential energy
The amount of stored energy depends on both the spring constant and the extension. Greater stiffness or larger extension results in more stored energy because more work is required to achieve the stretch.
Formula for elastic potential energy:
Where:
- Ee = Elastic potential energy (J)
- k = Spring constant (N/m)
- e = Extension (m)
This formula derives from the work done during stretching, which is the average force multiplied by the distance stretched. Since force increases linearly with extension per Hooke's law, the formula uses half the product of maximum force and extension.
Using force-extension graphs to visualize energy storage
Force-extension graphs provide a visual way to analyze the behavior of elastic objects. These graphs plot applied force on the vertical axis against extension on the horizontal axis, typically showing a straight line for elastic behavior.
Interpreting force-extension graphs
- Gradient of the line - This equals the spring constant (k), as it represents the change in force per unit change in extension, directly from Hooke's law
- Area under the curve - This represents the elastic potential energy stored in the object, calculated as the area of the triangle formed by the line from the origin to the point of maximum extension
For elastic materials, the graph is linear up to the elastic limit. Beyond this point, the object deforms permanently, and the graph becomes non-linear.
Applying conservation of energy in spring systems
The principle of conservation of energy states that energy cannot be created or destroyed, only transferred between forms. In spring systems, this means the total energy remains constant as it shifts between elastic potential energy and kinetic energy, assuming no energy losses to friction or other factors.
Energy transfers in spring systems
When a stretched spring is released, its elastic potential energy converts to kinetic energy as the spring contracts. At maximum extension, all energy is elastic potential. As it moves toward equilibrium, elastic potential decreases while kinetic energy increases. At the equilibrium position (original length), all energy is kinetic, before converting back to elastic potential if the system oscillates.
This conservation allows predictions of system behavior, such as maximum speed or extension in oscillating springs.
Worked example - Calculating elastic potential energy
A spring with a spring constant of 200 N/m is stretched by 0.15 m. Calculate the elastic potential energy stored in the spring.
Step 1: Identify the values
- Spring constant (k) = 200 N/m
- Extension (e) = 0.15 m
Step 2: Apply the formula
Step 3: Substitution and calculation
The spring stores 2.25 J of elastic potential energy.
Worked example - Energy transfer in a spring system
A mass attached to a spring is stretched to store 4.5 J of elastic potential energy. When released, all this energy converts to kinetic energy at the equilibrium position. Calculate the maximum kinetic energy of the mass, assuming conservation of energy.
Step 1: Understand the conservation
Total energy remains constant, so elastic potential energy at maximum extension equals kinetic energy at equilibrium.
Step 2: Apply conservation principle
Maximum kinetic energy = Initial elastic potential energy = 4.5 J
Step 3: Interpretation
The mass reaches a maximum kinetic energy of 4.5 J at the equilibrium position, with no elastic potential energy at that point.