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Chapter 6: Problem 16

Can a potential energy function be defined for the force of friction?

Short Answer

Expert verified
Answer: No, a potential energy function cannot be defined for the force of friction, as it is a non-conservative force and its work done is path-dependent. Potential energy functions are only associated with conservative forces, where the work done depends solely on the initial and final positions, not on the path taken between them.

Step by step solution

01

Understand conservative forces and potential energy functions

A conservative force is a force that has a potential energy function associated with it. Mathematically, a force is conservative if its work done on an object moving around any closed path is zero. In other words, the work done depends only on the initial and final positions, and not on the path taken between the two points. Examples of conservative forces include gravitational force, spring force, and electrostatic force. For a conservative force, we can define a potential energy function, such that: ∇ × F = 0 Here, F is the force vector and ∇ × is the curl operator.
02

Examine the properties of friction

Friction is a contact force that opposes the relative motion between two surfaces in contact. The friction force can be either static or kinetic. The force of friction, F_f, is proportional to the normal force, N, acting on the object: F_f = μ * N Here, μ is the coefficient of friction. The work done by the force of friction is path-dependent and is generally different for different paths taken between two points.
03

Determine whether the force of friction is conservative

As the force of friction is path-dependent, the work done by friction around any closed path is not always zero. This means that the force of friction is a non-conservative force.
04

Answer the question: Can a potential energy function be defined for the force of friction?

Since the force of friction is a non-conservative force, a potential energy function cannot be defined for it. The potential energy functions are only associated with conservative forces, where the work done depends only on the initial and final positions and not on the particular path taken between these points.

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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 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?

A father exerts a \(2.40 \cdot 10^{2} \mathrm{~N}\) force to pull a sled with his daughter on it (combined mass of \(85.0 \mathrm{~kg}\) ) across a horizontal surface. The rope with which he pulls the sled makes an angle of \(20.0^{\circ}\) with the horizontal. The coefficient of kinetic friction is \(0.200,\) and the sled moves a distance of \(8.00 \mathrm{~m}\). Find a) the work done by the father, b) the work done by the friction force, and c) the total work done by all the forces.

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?

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