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Chapter 6: Problem 5

Which of the following is not a valid potential energy function for the spring force \(F=-k x ?\) a) \(\left(\frac{1}{2}\right) k x^{2}\) b) \(\left(\frac{1}{2}\right) k x^{2}+10 \mathrm{~J}\) c) \(\left(\frac{1}{2}\right) k x^{2}-10 \mathrm{~J}\) d) \(-\left(\frac{1}{2}\right) k x^{2}\) e) None of the above is valid.

Short Answer

Expert verified
(a) \((\frac{1}{2}) k x^{2}\) (b) \((\frac{1}{2}) k x^{2} + 10\) (c) \((\frac{1}{2}) k x^{2} - 10\) (d) \(-(\frac{1}{2}) k x^{2}\) Answer: (d) \(-(\frac{1}{2}) k x^{2}\)

Step by step solution

01

Identify the valid potential energy functions

We already know that the standard potential energy function for spring force is \(U(x)=\left(\frac{1}{2}\right) k x^{2}\). Adding or subtracting a constant value, in this case \(10 \mathrm{~J}\), does not change the fact that the potential energy function is still valid. So, we can say that options a), b), and c) are valid potential energy functions.
02

Examine the remaining option

Now, we only have to check option d) to determine whether it is a valid potential energy function for the spring force or not. The given function is \(U(x)=-\left(\frac{1}{2}\right) k x^{2}\). Comparing it to the standard potential energy function, we can see that it has a negative sign in front of the expression. This means that the potential energy will be negative, which is not possible for a spring system, as the potential energy should always be non-negative.
03

Determine the answer

Based on our analysis, we can conclude that option d) is not a valid potential energy function for the spring force. Therefore, the correct answer is: (d) \(-\left(\frac{1}{2}\right) k x^{2}\)

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Most popular questions from this chapter

A 0.100 -kg ball is dropped from a height of \(1.00 \mathrm{~m}\) and lands on a light (approximately massless) cup mounted on top of a light, vertical spring initially at its equilibrium position. The maximum compression of the spring is to be \(10.0 \mathrm{~cm}\). a) What is the required spring constant of the spring? b) Suppose you ignore the change in the gravitational energy of the ball during the 10 -cm compression. What is the percentage difference between the calculated spring constant for this case and the answer obtained in part (a)?

A projectile of mass \(m\) is launched from the ground at \(t=0\) with a speed \(v_{0}\) and at an angle \(\theta_{0}\) above the horizontal. Assuming that air resistance is negligible, write the kinetic, potential, and total energies of the projectile as explicit functions of time.

One end of a rubber band is tied down and you pull on the other end to trace a complicated closed trajectory. If you were to measure the elastic force \(F\) at every point and took its scalar product with the local displacements, \(\vec{F} \cdot \Delta \vec{r},\) and then summed all of these, what would you get?

Can a unique potential energy function be identified with a particular conservative force?

A car of mass \(987 \mathrm{~kg}\) is traveling on a horizontal segment of a freeway with a speed of \(64.5 \mathrm{mph}\). Suddenly, the driver has to hit the brakes hard to try to avoid an accident up ahead. The car does not have an ABS (antilock braking system), and the wheels lock, causing the car to slide some distance before it is brought to a stop by the friction force between the car's tires and the road surface. The coefficient of kinetic friction is \(0.301 .\) How much mechanical energy is lost to heat in this process?
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