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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 truck of mass 10,212 kg moving at a speed of \(61.2 \mathrm{mph}\) has lost its brakes. Fortunately, the driver finds a runaway lane, a gravel-covered incline that uses friction to stop a truck in such a situation; see the figure. In this case, the incline makes an angle of \(\theta=40.15^{\circ}\) with the horizontal, and the gravel has a coefficient of friction of 0.634 with the tires of the truck. How far along the incline \((\Delta x)\) does the truck travel before it stops?

A ball is thrown up in the air, reaching a height of \(5.00 \mathrm{~m}\). Using energy conservation considerations, determine its initial speed.

A large air-filled 0.100 -kg plastic ball is thrown up into the air with an initial speed of \(10.0 \mathrm{~m} / \mathrm{s}\). At a height of \(3.00 \mathrm{~m}\) the ball's speed is \(3.00 \mathrm{~m} / \mathrm{s}\). What fraction of its original energy has been lost to air friction?

A runner reaches the top of a hill with a speed of \(6.50 \mathrm{~m} / \mathrm{s}\) He descends \(50.0 \mathrm{~m}\) and then ascends \(28.0 \mathrm{~m}\) to the top of the next hill. His speed is now \(4.50 \mathrm{~m} / \mathrm{s}\). The runner has a mass of \(83.0 \mathrm{~kg} .\) The total distance that the runner covers is \(400 . \mathrm{m}\) and there is a constant resistance to motion of \(9.00 \mathrm{~N}\). Use energy considerations to find the work done by the runner over the total distance.

In 1896 in Waco, Texas, William George Crush, owner of the K-T (or "Katy") Railroad, parked two locomotives at opposite ends of a 6.4 -km-long track, fired them up, tied their throttles open, and then allowed them to crash head- on at full speed in front of 30,000 spectators. Hundreds of people were hurt by flying debris; several were killed. Assuming that each locomotive weighed \(1.2 \cdot 10^{6} \mathrm{~N}\) and its acceleration along the track was a constant \(0.26 \mathrm{~m} / \mathrm{s}^{2},\) what was the total kinetic energy of the two locomotives just before the collision?
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