Problem 17 - University Physics with Modern Physics

Question

Can the potential energy of a spring be negative?

Step-by-Step Solution

Answer

Answer: No, the potential energy of a spring cannot be negative.

Step 1: Understand the formula for potential energy of a spring

The potential energy of a spring, also called elastic potential energy, is given by the formula: U = (1/2) * k * x^2 where: U is the potential energy, k is the spring constant (a measure of the stiffness of the spring; always positive), x is the displacement from the equilibrium position (it can be a positive or negative value, depending on whether the spring is stretched or compressed).

Step 2: Analyze the properties of the elements in the formula

In the formula, k is always positive because it represents the 'stiffness' of the spring, and it only makes sense to have a positive value for stiffness. This means that the product of (1/2) and k should also be positive. The displacement x can be positive if the spring is stretched (elongated from its equilibrium position) and negative if the spring is compressed (shortened from its equilibrium position). Importantly, when squaring the displacement x in the formula, x^2 is always positive or zero, since any value (negative or positive) when squared results in a positive value (or zero, if x=0).

Step 3: Evaluate the potential energy formula in light of the properties observed

Now that we understand that both (1/2) * k and x^2 are always positive or zero, let's examine the potential energy formula under these constraints: U = (1/2) * k * x^2 Since both factors (1/2) * k and x^2 are positive or zero, their product can only be positive or zero. A positive value multiplied by a positive value (or zero) cannot result in a negative value.

Step 4: Answer the question

Given that the formula for potential energy of a spring consists of positive or zero factors, it is not possible for the potential energy of a spring to be negative.