Name: finding force from potential energy
Uploaded: 2023-04-30T10:57:22.000Z
Duration: 1 min 13 s
Channel: JoVE

K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Physics

Energy

PS.Phy.6.2

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Force and Potential Energy in One Dimension

Conservative forces can be expressed in terms of potential energy.
Consider a body moving along a straight line due to a conservative force that is a function of position.
Now, the work done by the conservative force is equal to the negative of the change in its potential energy.
The work done is also equal to the product of force and the displacement caused by the force.
Thus, the force on the body is equal to the negative of the change in potential energy divided by the displacement. 
The negative sign indicates that the conservative force drags the system to a lower potential energy.
For very small displacements, the expression for force and potential energy in one dimension can be established.
For verification of the force and potential energy relationship, consider the example of elastic potential energy. When substituted into the force and energy relationship, the formula for Hooke&#39;s law is established.

force and potential energy in one dimension

Force can be calculated from the expression for potential energy, which is a function of position. The component of a conservative force, in a particular direction, equals the negative of the derivative of the corresponding potential energy with respect to the displacement in that direction. For regions where potential energy changes rapidly with displacement, the work done and force is maximum. Also, when force is applied along the positive coordinate axis, the potential energy decreases with an increase in displacement. Hence, force and the derivative of potential energy with respect to displacement have opposite signs.
For instance, if the potential energy of a particle undergoing one-dimensional motion along the x-axis is
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where c = 8 N/m3, what will the force be for the given potential energy? Here, the known quantity is potential energy. The unknown quantity force needs to be calculated.
<img alt="Static equilibrium diagram; &#931;Fx=0, &#931;Fy=0 equations; illustrating forces and conditions of balance." src="/files/ftp_upload/12677/12677_Equation_2.png" style="height: 32px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
Incorporating the value of potential energy in the above expression, the force along the x-direction is calculated to be &#8722;cx3. Substituting the value of c, the force is calculated to be &#8722;8x3.
The potential energy function for the force between two atoms in a diatomic molecule is approximately given by
<img alt="Chemical reaction mechanism diagram, depicting nucleophilic substitution with product formation." src="/files/ftp_upload/12677/12677_Equation_3.png" style="height: 32px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is
<img alt="Protein purification diagram, HPLC setup, stepwise elution process illustration." src="/files/ftp_upload/12677/12677_Equation_4.png" style="height: 14px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
what is the value of D? Here, the known quantity is the potential energy. The unknown quantity is the dissociation energy of the molecule. Substituting the potential energy function, we get the expression for force. Since the net force is zero at equilibrium, we get
<img alt="Catalytic reaction mechanism diagram showing substrate to product transformation and enzyme interaction." src="/files/ftp_upload/12677/12677_Equation_5.png" style="height: 32px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
Now, the value of x6 is substituted in the expression for potential energy. Since potential energy at infinity is zero, the dissociation energy of the molecule is calculated to be b2/4a.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/8-4-potential-energy-diagrams-and-stability">Openstax, University Physics Volume 1, Section 8.4: Potential Energy Diagrams and Stability</a>.

Force can be calculated from the expression for potential energy, which is a function of position. The component of a conservative force, in a particular ...

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Unit 1

Potential Energy and Energy Conservation

KEY TERMS AND CONCEPTS

finding force from potential energy

Force can be found from potential energy when the energy is written as a function of position. For a conservative force, the force component in a given direction equals the negative derivative of the matching potential energy with respect to displacement. This means force and the slope of potential energy have opposite signs.

Where potential energy changes quickly with displacement, the force and the work done are largest. When a force acts along the positive coordinate axis, potential energy decreases as displacement increases. In one-dimensional motion, this relationship is often used to move from a known potential energy function to the force it produces.

For example, if a particle moves along the x-axis and its potential energy is given with c = 8 N/m^3, the force can be calculated from that expression. The potential energy is the known quantity, and the force is the unknown quantity. Differentiating the energy function gives the x-direction force as -cx^3, which becomes -8x^3 after substituting the value of c.

The same idea also applies to the force between two atoms in a diatomic molecule. Their potential energy can be written approximately as a function of distance x, with a and b as constants. If the dissociation energy is needed, the force is first found from the potential energy, and the equilibrium point is used because the net force is zero there. Then the value of x^6 is substituted into the energy expression, and since the potential energy at infinity is zero, the dissociation energy is b^2/4a.

Finding Force from Potential Energy

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Force can be found from potential energy when the energy is written as a function of position. For a conservative force, the force component in a given direction equals the negative derivative of the matching potential energy with respect to displacement. This means force and the slope of potential energy have opposite signs. Where potential energy changes quickly with displacement, the force and the work done are largest. When a force acts along the positive coordinate axis, potential energy...

force and potential energy in three dimensions

force-and-potential-energy-in-three-dimensions

Gradient Force from 3D Potential Energy

Gradient force from 3D potential energy shows how a conservative force changes along the x, y, and z axes. The potential energy function U depends on all three spatial coordinates. Each force component is linked to how U changes with position along one coordinate. In one dimension, force can be written as the negative ratio of the change in potential energy to the displacement along that axis. For very small displacements, these ratios become derivatives. When a function has several variables,...

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energy diagrams - ii

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Potential energy diagrams show how stable a system is at different positions. Their shape, or topology, helps identify equilibrium points. An equilibrium point is a place where the system can stay in balance because the net force is zero. At a local minimum, the potential energy is lowest. The system tends to remain there unless a net force acts on it. The slope at the minimum is zero. The slope is positive on both sides, so the force acts as a restoring force and pushes the system back. A...

key terms and concepts

Learn how to calculate force from potential energy in one dimension, including the sign rule, a particle example, and dissociation energy for atoms.

Finding Force from Potential Energy | JoVE K12

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Finding Force from Potential Energy

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