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Chapter 6: Problem 17

Can the potential energy of a spring be negative?

Short Answer

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

Step by step solution

01

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).
02

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).
03

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.
04

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.

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Most popular questions from this chapter

A cannonball of mass \(5.99 \mathrm{~kg}\) is shot from a cannon at an angle of \(50.21^{\circ}\) relative to the horizontal and with an initial speed of \(52.61 \mathrm{~m} / \mathrm{s}\). As the cannonball reaches the highest point of its trajectory, what is the gain in its potential energy relative to the point from which it was shot?

A body of mass \(m\) moves in one dimension under the influence of a force, \(F(x)\), which depends only on the body's position. a) Prove that Newton's Second Law and the law of conservation of energy for this body are exactly equivalent. b) Explain, then, why the law of conservation of energy is considered to be of greater significance than Newton's Second Law.

A 0.500 -kg mass is attached to a horizontal spring with \(k=100 . \mathrm{N} / \mathrm{m}\). The mass slides across a frictionless surface. The spring is stretched \(25.0 \mathrm{~cm}\) from equilibrium, and then the mass is released from rest. a) Find the mechanical energy of the system. b) Find the speed of the mass when it has moved \(5.00 \mathrm{~cm}\). c) Find the maximum speed of the mass.

The energy height, \(H\), of an aircraft of mass \(m\) at altitude \(h\) and with speed \(v\) is defined as its total energy (with the zero of the potential energy taken at ground level) divided by its weight. Thus, the energy height is a quantity with units of length. a) Derive an expression for the energy height, \(H\), in terms of the quantities \(m, h\), and \(v\). b) A Boeing 747 jet with mass \(3.5 \cdot 10^{5} \mathrm{~kg}\) is cruising in level flight at \(250.0 \mathrm{~m} / \mathrm{s}\) at an altitude of \(10.0 \mathrm{~km} .\) Calculate the value of its energy height. Note: The energy height is the maximum altitude an aircraft can reach by "zooming" (pulling into a vertical climb without changing the engine thrust). This maneuver is not recommended for a 747 , however.

A ball of mass \(1.84 \mathrm{~kg}\) is dropped from a height \(y_{1}=$$1.49 \mathrm{~m}\) and then bounces back up to a height of \(y_{2}=0.87 \mathrm{~m}\) How much mechanical energy is lost in the bounce? The effect of air resistance has been experimentally found to be negligible in this case, and you can ignore it.
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