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**Dr. Steve Peterson Steve.peterson@uct.ac.za**

Physics 1025F Mechanics ENERGY Dr. Steve Peterson

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**Chapter 6: Work and Energy**

We have been using forces to study the translational motion of objects; Energy (and work) can provide an alternate analysis of this motion

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ENERGY Energy … is an extremely abstract concept and is difficult to define; is a number (a scalar) describing the state of a system of objects (for an isolated system this number remains constant, i.e. the energy of the system is conserved); appears in many different forms, each of which can be converted into another form of energy in one or other of the transformation processes which underlie all activity in the Universe; is all there is! (Even matter is energy: E = mc2)

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Systems and Energy Although energy is hard to define and comes in many different forms, every system in nature has associated with it a quantity we call its total energy. The total energy (E) is the sum of all the different forms of energy present in the system, i.e. Energy transformations can occur within a system.

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**System & Energy Transformation**

A system is what we define it to be. Energy can be transformed within the system without loss. Energy is a property of a system.

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**Environment & Energy Transfers**

An exchange of energy between system and environment is called an energy transfer. Two primary energy-transfer processes: Work & Heat Work is a mechanical transfer of energy to or from a system by pushing or pulling it. Heat is a non-mechanical transfer of energy from the environment to the system (or vice versa) because of a temperature difference between the two.

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**Work-Energy Principle**

Work done on a system represents energy that is transferred into or out of the system. The energy of the system (ΔE) changes by the exact amount of work (W) that was done. Work-Energy Principle: The total energy of the system changes by the amount of work done on it.

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**Conservation of Energy**

Suppose we have an isolated system, separating it from its surroundings in such a way that no energy is transferred into or out of the system. Law of Conservation of Energy: The total energy of an isolated system remains constant.

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Feynman on Energy There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. - Richard Feynman

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How to Calculate Work The work done by a constant force F on an object is equal to the product of the force multiplied by the distance through which the force acts. Therefore if the motion is in the same direction as the applied force the magnitude of the work done W is: Dot Product: Vector Multiplication

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How to Calculate Work If, on the other hand, the applied force F makes an angle θ with the subsequent displacement, d then the work done is Note: Work is a scalar quantity

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**More About Work Work can be positive or negative**

Positive if the force and the displacement are in the same direction (θ = 0°) Negative if the force and the displacement are in the opposite direction (θ = 180°) Work can also be zero If the displacement is perpendicular to the force (θ = 90°)

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Units of Work In the SI system, the units of work are joules:

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**More on Work Work is positive when lifting the box**

Work would be negative if lowering the box The force would still be upward, but the displacement would be downward

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Example: Work A sled loaded with bricks has a total mass of 18.0 kg and is pulled at constant speed by a rope inclined at 20.0° above the horizontal. The sled moves a distance of 20.0 m on a horizontal surface. The coefficient of friction between the sled and surface is (a) What is the tension in the rope? (b) How much work is done by the rope on the sled? (c) What is the mechanical energy lost due to friction? Vuille/Serway 5.7

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Kinetic Energy Kinetic energy is the energy of motion. All moving objects have kinetic energy.

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Potential Energy It is sometimes possible within a system to store energy so that it can be easily recoverable. This sort of stored energy is called potential energy. We will look at gravitational potential energy (due to the force of gravity) and elastic potential energy (due to the force from a spring). Interaction forces that can store useful energy are called conservative forces.

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**Gravitational Potential Energy**

Gravitational potential energy (UG) depends only on the height of the object and not the path the objects took to get to that position. Assuming UG = 0 when y = 0

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**Elastic Potential Energy**

The force exerted by a spring (FS) is called Hooke’s Law. Energy can be stored in a spring as elastic potential energy (US).

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Thermal Energy Thermal energy is related to the microscopic motion of the molecules of an object. The molecule’s motion produces kinetic energy and the spring-like molecular bonds produce potential energy. The sum of these microscopic kinetic and potential energies is what we call thermal energy.

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Work & Thermal Energy If work is done in the presence of friction, then thermal energy (heat) is generated - heat is another form of energy and therefore some of the work has gone into producing the heat.

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**Law of Conservation of Energy**

In general, i.e. the work done on the body is converted into changes in KE and/or changes in PE and/or changes in heat. Any change in the energy of a system is the result of work done on the system

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**Law of Conservation of Energy**

So, if there is no work done on the system? This gives rise to the Law of Conservation of Energy which can be stated as: "Energy can be neither created nor destroyed, but can be converted from one form to another or transferred from one system to another”.

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Mechanical Energy If there is no friction present and no external forces (other than gravity) acting on the system we have or This is a very powerful equation, and we often refer to the sum of KE and PE as "mechanical energy”.

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**Conservation of Mechanical Energy**

What is conservation in Physics? To say a physical quantity is conserved is to say that the numerical value of the quantity remains constant throughout any physical process although the quantities may change form. In Conservation of Energy, the total mechanical energy remains constant In any isolated system of objects interacting only through conservative forces, the total mechanical energy of the system remains constant.

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**Conservative & Nonconservative Forces**

There are two general kinds of forces Conservative Work and energy associated with the force can be recovered Nonconservative The forces are generally dissipative and work done against it cannot easily be recovered Potential energy can only be defined for conservative forces.

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**Example: Energy Conservation**

A stone is dropped from a 60-m high cliff onto the ground below. (a) What is the speed of the stone when it hits the ground? (b) Now, the stone is thrown upwards at 20 m/s from the top of the cliff. What is the speed of the stone when it hits the ground? (c) How would the final speed change if the stone were thrown upward at an angle?

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**How Quickly is Energy Transformed?**

The rate at which energy is transformed is called the power (P) and defined as: Power is also defined as the rate at which work is done. In the SI system, the units of power are measured in joules per second or watts (W):

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**Example: Energy Conservation**

A 2-kg block is pulled up a frictionless incline (30° above horizontal) by a 15 N force. What is the speed of the block after traveling 6-m?

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**Example: Energy Conservation**

1-kg and 2-kg masses hang from opposite ends of a string hanging over a frictionless pulley. The 1-kg mass sits on the ground and the 2-kg mass is 5-m in the air. With what speed will the 2-kg mass hit the ground?

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