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S3.5 Potential energy

[A] Introduction

Examples

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Examples

Examples of potential energy (S3.6) include :
  • elastic potential energy stored in a spring;
  • gravitational potential energy resulting from an object’s position;
  • chemical potential energy stored in the form of bonds between molecules, which may be released in a chemical reaction.
 

[B] Conservative forces

Key Point 3.9

If the work done by a force in moving an object between two states is independent of the path taken between those two states, the force is conservative.
Necessarily, the total work done by a conservative force in moving an object around a closed path is zero.
Graphic - No title - PathIndep

Commentary

Commentary

The definition of a conservative force implies that the work done is recoverable. Eg. consider an object thrown vertically upwards in a gravitational field. On the ascent, all its initial kinetic energy is converted into gravitational potential energy. However on descent this potential energy is all converted back into kinetic energy.

Hint for problem solving

Suppose one is interested in the work done by a conservative force in moving an object between two points along some complicated path for which the calculation is difficult. One can replace the actual path taken by some imaginary path for which the calculation is easier. The answer will be the same because for a conservative force the work is independent of the path.

 

Worked example

Worked example

Consider a particle moving from point A to point B on a semicircular track of radius R, as shown. What is the work done by gravity on the particle?
Graphic - No title - imagpath

Solution:

The force due to gravity is Fg=-mg and acts vertically downwards at all points. From the definition of work (Key Point 3.4):

\[ W = \int_{\rm start}^{\rm finish} \vec{F} \cdot d\vec{r} \]
which seems like a hard calculation.

However since gravity is a conservative force, we can (for the purposes of calculating W) replace the actual path by the imaginary path, as shown. Along this path the work done is simply

W=0-mg×(-R)=mgR

 

[C] Potential energy

Commentary

Commentary

Sometimes when analysing the motion of particles it is useful to define a potential energy curve. For instance, for a particle constrained to move along the x-axis, U(x) might look like this.
Graphic - No title - potlsurf
 

[D] Forces from potential energies

Learning Resources

Textbook: HRW Chapter 8.1 - 8.3. (HRW 8th Ed has a new section on Reading a P.E. curve, 8.6)
Course Questions:
Self-Test Questions: