College Physics II – Mechanics, Sound, Oscillations, and Waves
Definition
The term W = -½k(Δx)² represents the work done by a conservative force, specifically a spring force, when a system undergoes a displacement Δx. This expression is derived from the potential energy formula for a spring, and it describes the amount of work required to compress or extend a spring from its equilibrium position.
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The negative sign in the expression W = -½k(Δx)² indicates that the work done by the spring force is negative, meaning the force opposes the displacement of the object.
The work done by a conservative force like a spring is equal to the negative change in the potential energy of the system, which is given by the formula ΔU = -W.
The spring constant, k, is a measure of the stiffness of the spring. A higher spring constant means the spring is stiffer and requires more force to compress or extend it.
The term (Δx)² represents the square of the displacement of the object from its equilibrium position, which is directly proportional to the potential energy stored in the spring.
The expression W = -½k(Δx)² is only valid for small displacements, where the spring force can be considered linear and the spring behaves according to Hooke's law.
Review Questions
Explain how the term W = -½k(Δx)² is related to the concept of conservative forces.
The term W = -½k(Δx)² is directly related to the concept of conservative forces because it represents the work done by a conservative force, specifically a spring force. Conservative forces are path-independent, meaning the work done by the force depends only on the initial and final positions of the object, not the path taken. The negative sign in the expression indicates that the work done by the spring force opposes the displacement of the object, which is a characteristic of conservative forces.
Describe how the spring constant, k, and the displacement, Δx, influence the work done by a spring force.
The spring constant, k, and the displacement, Δx, both play a significant role in determining the work done by a spring force. The spring constant, k, is a measure of the stiffness of the spring, with a higher value indicating a stiffer spring that requires more force to compress or extend. The displacement, Δx, represents the distance the spring is compressed or extended from its equilibrium position. The term (Δx)² in the expression shows that the work done is proportional to the square of the displacement, meaning that as the displacement increases, the work done by the spring force increases exponentially.
Analyze the relationship between the work done by a spring force and the potential energy stored in the spring.
The work done by a spring force is directly related to the change in potential energy of the spring. The expression W = -½k(Δx)² shows that the work done by the spring force is equal to the negative change in the potential energy of the system, which is given by the formula ΔU = -W. This relationship demonstrates the principle of conservation of energy, where the work done by the spring force is equal to the change in the potential energy stored in the spring. As the spring is compressed or extended, the potential energy of the system changes, and this change in potential energy is directly reflected in the work done by the spring force.
Related terms
Conservative Force: A conservative force is a force that does not depend on the path taken by the object, but only on the initial and final positions. The work done by a conservative force is path-independent.
Potential Energy: Potential energy is the energy stored in an object due to its position or configuration. For a spring, the potential energy is given by the formula U = ½k(Δx)².
Hooke's Law: Hooke's law states that the force required to extend or compress a spring is proportional to the distance of the extension or compression. The constant of proportionality is the spring constant, k.