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### Vector Mechanics for Engineers: Dynamics MECN 3010

Department of Mechanical Engineering

Inter American University of Puerto Rico

Bayamon Campus

Dr. Omar E. Meza Castillo

http://www.bc.inter.edu/facultad/omeza

"Lo peor es educar por métodos basados en el temor, la fuerza, la autoridad, porque se destruye la sinceridad y la confianza, y sólo se consigue una falsa sumisión”

Einstein Albert

Topic 3: Kinetics of a particle

Work and EnergyChapter Objectives

- To develop the principle of work and energy and apply it to solve problems that involve force, velocity, and displacement.
- To study problems that involve power and efficiency.
- To introduce the concept of a conservative force and apply the theorem of conservation of energy to solve kinetic problems.

Work of a Force

Work of a Constant Force Moving Along a Straight Line

Work of a Force

Work of a Weight

Work of a Force

Work of a Spring

Principle of Work and Energy for a System of Particles

Work of Friction Caused by Sliding

Power and Efficiency

Power

Conservative Forces and Potential Energy

- Conservative Force:
- It is defined by the work done in moving a particle from one point to another that is independent of the path followed by the particle.
- Two examples are weight of the particle and elastic force of the spring.
- Potential Energy:
- It is the measure of the amount of work a conservative force will do when it moves from a given position to the datum.
- Gravitational Potential Energy:
- If a particle is located a distance y above an arbitrary selected datum, the particle’s weight W has positive gravitational potential energy Vg.

Conservative Forces and Potential Energy

- W has the capacity of doing positive work when the particle is moved back down to the datum.
- The particle is located a distance y below the datum, Vgis negative since the weight does negative work when the particle is moved back up to the datum.
- If y is positive upward, gravitational potential energy of the particle of weight W is

Conservative Forces and Potential Energy

- Elastic Potential Energy:
- When an elastic spring is elongated or compressed a distance s from its unstretched position, the elastic potential energy Ve can be expressed.
- Ve is always positive since, in the deformed position, the force of the spring has the capacity for always doing positive work on the particle when the spring is returned to its unstretched position.

Conservative Forces and Potential Energy

- Potential Function:
- If a particle is subjected to both gravitational and elastic forces, the particle’s potential energy can be expressed as a potential function.

Conservative of Energy

- Potential Function:
- When a particle is acted upon by a system of both conservative and non-conservative forces, the portion of the work done by the conservative forces can be written in terms of the difference in their potential energies using.
- As a result, the principle of work and energy can be written as
- represent the work of the nonconservative forces acting on the particles.

Conservative of Energy

- If only conservative forces are applied to the body, this term is zero and we have
- This equation referred to as the conservation of mechanical energy or simply the conservation of energy.
- It states that during the motion the sum of the particle’s kinetic and potential energies remain constant.

Conservative of Energy

- System of Particles:
- If a system of particles is subjected only to conservative forces, then an equation can be written.
- The sum of the particle’s initial kinetic and potential energies is equal to the sum of the particle’s final kinetic and potential energies.

Omar E. Meza Castillo Ph.D.

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