3.3 Potential Energy

Learning objectives

3.3A—Describe the potential energy of a systemDescribe the potential energy of a system.• A system composed of two or more objects has potential energy if the objects within that system only interact with each other through conservative forces.• Potential energy is a scalar quantity associated with the position of objects within a system.• The definition of zero potential energy for a given system is a decision made by the observer considering the situation to simplify or otherwise assist in analysis.• The potential energy of common physical systems can be described using the physical properties of that system.- i. The elastic potential energy of an ideal spring is given by the following equation, where is the distance the spring has been stretched or compressed from its equilibrium length. Relevant equation:- ii. The general form for the gravitational potential energy of a system consisting of two approximately spherical distributions of mass (e.g., moons, planets or stars) is given by the equation =− UGmm rg 12- iii. Because the gravitational field near the surface of a planet is nearly constant, the change in gravitational potential energy in a system consisting of an object with mass m and a planet with gravitational field of magnitude g when the object is near the surface of the planet may be approximated by the equation• The total potential energy of a system containing more than two objects is the sum of the potential energy of each pair of objects within the system. AP Physics 1: Algebra-Based Course and Exam Description Work, Energy, and Power UNIT 3 | AP Physics 1: Algebra-Based Course and Exam Description TOPIC 3.4 Conservation of Energy

Name the interacting system

Potential energy is a scalar associated with the relative positions of objects in a system that interact through conservative forces. It belongs to the interacting system—not to one isolated object. A mass–spring system can store elastic potential energy; a mass–planet system can have gravitational potential energy.

Choose a consistent zero

The observer chooses where $U=0$ to simplify analysis. That choice changes the numerical value of $U$ , but not a physically meaningful change $\Delta U=U_f-U_i$ when one consistent reference is used.

Match the model to its conditions

System and condition	Potential-energy relationship
Ideal spring displaced $\Delta x$ from equilibrium	$U_s=\tfrac12k(\Delta x)^2$
Two approximately spherical masses separated center-to-center by $r$	$U_g=-G\dfrac{m_1m_2}{r}$
Object–planet system near a surface where $g$ is nearly constant	$\Delta U_g=mg\Delta y$

Calculate a change

Near-surface example: A $2.0\,\text{kg}$ object rises $3.0\,\text{m}$ where $g=9.8\,\text{N/kg}$ .

$\Delta U_g=(2.0\,\text{kg})(9.8\,\text{N/kg})(+3.0\,\text{m})=+59\,\text{J}$ .

The object–Earth system gains $59\,\text{J}$ of gravitational potential energy. The result is the same whichever height was labeled zero.

Sum pairs and respect limits

For a system with more than two objects, total potential energy is the sum of the potential energy of each interacting pair. Do not add a separate potential energy for a lone object, and do not mix model conditions: $mg\Delta y$ is a near-surface approximation, whereas $-Gm_1m_2/r$ is the general two-spherical-mass form.

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—3.4 Conservation of EnergyRead