Work-Energy Theorem
The Work-Energy Theorem is a fundamental principle in physics that establishes the relationship between work and kinetic energy. It posits that when work is performed on an object, its kinetic energy changes, which in turn causes the object to move. This theorem is primarily associated with kinematics, a branch of classical mechanics that studies motion without considering the underlying forces. The mathematical expression of the theorem equates the total work done on an object to the change in its kinetic energy, highlighting that work can be viewed as a transfer of energy when it results in the displacement of an object.
In practical terms, the theorem can be illustrated through everyday examples, such as a billiards game, where the collision of balls transfers kinetic energy through the work done during the impact. The principle of conservation of energy underpins the theorem, asserting that energy within an isolated system cannot be created or destroyed, only transformed or transferred. Understanding this theorem is vital for various applications, including engineering efficient machines like combustion engines, which convert potential energy from fuel into kinetic energy to perform work. Overall, the Work-Energy Theorem provides a crucial framework for analyzing the motion of objects in relation to energy and work.
Work-Energy Theorem
FIELDS OF STUDY: Classical Mechanics; Thermodynamics
ABSTRACT: The work-energy theorem describes the relationship between work performed on an object and the kinetic energy of that object. It states that when some net amount of work is performed on an object, that object’s kinetic energy will change. This theorem can be written mathematically to relate an object’s mass and the change in its velocity to the amount of work performed—a useful way of connecting the motion and mass of an object to its capacity for work.
PRINCIPAL TERMS
- conservation of energy: the principle that energy in the universe can be neither created nor destroyed, only transformed and transferred.
- displacement: the absolute distance an object moves from its starting point, regardless of the path it travels.
- kinematics: a subfield of classical mechanics that studies the motion of objects without reference to the forces that cause this motion.
- kinetic energy: the energy contained in an object due to its motion.
- net force: the sum of all forces acting on an object.
- potential energy: the energy stored within an object or system due to its position or configuration relative to the forces acting on it.
- total mechanical energy: the sum of the kinetic energy and the potential energy an object possesses as a result of work done on it.
- work: the successful displacement of an object caused by the application of a force.
Work and Kinetic Energy
The work-energy theorem describes the relationship between kinetic energy (the energy of an object in motion) and work (the displacement of an object by a force). It states that when work is performed on an object, the kinetic energy of that object will change. When the kinetic energy of an object changes, it moves. So, in simple terms, performing work on an object causes it to move.
Because the work-energy theorem is concerned only with masses and velocities, not with forces, it is considered part of the field of kinematics. Kinematics is a subfield within classical mechanics that studies the motion of objects without regard for the forces causing the motion. Classical mechanics, in turn, is the branch of physics concerned with the physical laws that govern both the motion of objects and the forces that move them. Isaac Newton (1642–1727) laid the foundations for modern classical mechanics with his three laws of motion, published in the late seventeenth century.
Work, Energy, and Force
Mathematically, the work-energy theorem is represented by the following equation, where W is the total work performed on an object, ∆K is the change in the object’s kinetic energy, m is the mass of the object, and vi and vf are its initial and final velocities, respectively:
W = ∆K = 1/2mvf2 – 1/2mvi2
Thus, the total work done is equal to the total change in kinetic energy. In this sense, work can be thought of as the transfer of energy, if that transfer of energy results in displacement. Indeed, in the International System of Units (SI), the unit for both work and energy is the joule (J).
Consider a game of billiards. When the cue ball is in motion, it has kinetic energy. When it collides with another ball, it transfers some of its kinetic energy to the second ball. The force of the collision performs work on the second ball, causing it to move. This interaction underlines the relationship between energy, work, and force. In SI units, one joule represents the amount of work done or energy transferred when one newton (N) of force acts over a distance of one meter (m). In other words, if the cue ball exerts one newton of force on the second ball, causing it to be displaced by one meter, then one joule of energy has been transferred from the cue ball to the second ball, and one joule of work has been performed.
Forces in physics are interactions. According to Newton’s second law, the net force (F; sum of the forces) acting on an object is equal to the object’s mass (m) times its resulting acceleration (a):
F = ma
In turn, the work (W) done by that force is equal to the net force (F) applied times the resulting displacement (s) of the object:
W = Fs
Displacement is the absolute distance and direction an object has moved from its starting position, ignoring the path taken. Therefore, a car that drove in a perfect circle and stopped exactly where it started would have a displacement of zero, no matter how large the circle it traveled. Similarly, a car that drove ten miles east, made a U-turn, and drove back five miles west would only have a displacement of five miles east, even though it traveled fifteen miles total.
The equation for work reveals that in order for a force to have performed work on an object, the displacement of that object must have a nonzero value. In other words, the object has to have moved. If an applied force does not result in displacement, no work has been done.
Conservation of Energy
The law of conservation of energy states that in an isolated system, energy is conserved. An isolated system is one from which neither matter nor energy can escape. The universe is, in theory, the ultimate isolated system. Thus, according to this law, energy in the universe is never created or destroyed; it can only be transformed or transferred. The work-energy theorem is an extension of the law of conservation of energy, rewritten in a usable form.
Not all energy is kinetic energy. Energy can exist in a variety of forms. One such form is potential energy, which is energy that is stored in an object or system until it can be converted to another form of energy to do work. Potential energy itself comes in different forms, such as gravitational potential energy and chemical potential energy. The human body makes use of the chemical potential energy that exists in food due to its molecular configuration. When food is digested, it undergoes chemical reactions that break down its molecules and convert some of this chemical potential energy into the thermal energy of body heat and the kinetic energy of moving limbs and beating hearts. A combustion engine similarly converts the chemical potential energy of the fuel into the kinetic energy of the moving pistons that drive the engine.
The principle of conservation of energy is useful when examining any isolated system. Consider the billiards example again. The billiards table can be treated as an isolated system, because the balls stay on the table and any energy from the environment (heat from overhead lights, the kinetic energy in a gust of air) has such a small effect that it can be ignored. Therefore, when two balls collide, the total amount of energy in the system must remain the same before and after the impact. Kinetic energy is simply transferred from one ball to the other. A miniscule amount might be converted to thermal energy due to friction with the table.
Often, when considering some kinematic interaction, it is useful to know the total mechanical energy of the objects at play. The total mechanical energy of an object or system is simply the sum of its potential and kinetic energies. In the real world, total mechanical energy is not typically conserved, because friction must be taken into account. Consider a driver in a speeding car who suddenly slams on the brakes. As the car’s tires stop rotating and start sliding across the surface of the pavement, they generate friction. Friction converts kinetic energy into thermal energy, which is not a form of mechanical energy. This energy then dissipates away from the tire tracks into the surrounding environment.
Sample Problem
A seventy-kilogram sprinter completes a hundred-meter dash in ten seconds. As she crosses the finish line, her coach uses an infrared speedometer to measures her speed at twelve meters per second. How much work did the sprinter perform during the race?
Answer:
Start by making note of the given information: the sprinter’s mass (70 kg), the distance traveled (100 m), the time taken to travel this distance (10 s), and the sprinter’s velocity when she crossed the finish line (12 m/s). (Although velocity consists of both speed and direction of travel, and the speedometer only measured the sprinter’s speed, it can be assumed that her direction of travel remained constant.) She would have started the race from a dead stop, so her initial velocity was 0 m/s. Recall that the work-energy theorem relates work not only to an object’s kinetic energy but also to its mass and change in velocity:
Because no information is provided about the sprinter’s kinetic energy, the change in kinetic energy (∆K) can be disregarded. Plug in the given information:
The second element of the equation (1/2mvi2) can also be ignored, because the initial velocity is zero, and any value multiplied by zero is zero. Simplify the equation and solve:
The sprinter performed 5,040 kilogram–square meters per second squared (kg·m2/s2) of work. One kg·m2/s2 is simply one joule, expressed in SI base units. Therefore, it took 5,040 J of work for the sprinter to accelerate her body from a standstill at the starting line to the 12 m/s she was traveling when she crossed the finish line.
The Work-Energy Theorem in Everyday Life
The work-energy theorem is useful whenever the effect of work on the motion of an object is of interest. For example, understanding how the chemical potential energy in a fuel source performs work when it is released and converted into kinetic energy is an essential part of engineering efficient combustion engines. The fuel has to contain enough energy to move the pistons without breaking them.
Countless other devices in modern life also convert potential energy into kinetic energy in order to perform work. Everyday examples include vacuum cleaners, clocks, and fans. By expressing the relationship between energy transfer (work) and the motion (kinetic energy) of these objects in easy-to-measure terms (mass and velocity), the work-energy theorem makes engineering these devices possible.
Bibliography
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