Potential Energy

Potential Energy Diagrams

Example: The Potential Energy U(x) is plotted as a function of the position x. The total mechanical energy E=16.0 J. Determine

1. How much is the Kinetic Energy at x=0.0 m
2. How much is the Kinetic Energy at x=3.0 m
3. What is the maximum distance from x=0.0 the particle can reach

Exact Gravitational Potential Energy

,

where  is the mass of the first object,  is the mass of the second object, r is the distance between the two, and  is the universal gravitational constant.

Example: A 10.0-kg rocket in the gravitational field of the Earth is launched directly vertically with an initial kinetic energy . Determine the maximum height the rocket will reach

Ans. 51.4 km above the surface

Question: What initial velocity and kinetic energy will the rocket need at launch to escape the Earth’s gravitational pull (that is to be able to reach infinite distances from Earth)

k - spring constant, stiffness

x - displacement from equilibrium

Elastic Force of a Spring

Example: The graph gives the function for the elastic force of a spring as a function of displacement from equilibrium. Determine the stiffness of the spring.        Ans. 50 N/m

Elastic Potential Energy

Example. A 1.5-kg object attached to a horizontally placed elastic spring with stiffness of 400 N/m has a total mechanical energy of 50.0 J. What is the maximum distance from equilibrium that the object can reach (ignore resistance)?                        Ans. 0.5 m

Example. A 1.5-kg object attached to a vertically placed elastic spring with stiffness of 400 N/m. At position y=0, the spring is in equilibrium and the object  has a kinetic energy of 50.0 J. What is the maximum distance from equilibrium that the object can reach (ignore resistance)?                        Ans. 0.54 m

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Conservative vs. Non-conservative Forces

Definition The work of a conservative force is zero for a closed path (that is the initial and final positions are the same).

Examples of conservative forces: gravitation, elastic force of a spring, electric force

Example. Is the following force conservative?

Any force that depends only on the position of an object (and not on its velocity, acceleration or other dynamic variables) will always be conservative.

Definition for Potential Energy

Potential Energy is ALWAYS associated only with conservative forces.

1 Dimension:

Higher Dimensions

Example. A force is given as

Determine the function for the potential energy that can be associated with it.

Ans.

Example. A conservative force is associated with a potential energy given as

Determine the function for force                        Ans.

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