Energy

Analyzing the motion of an object can often get to be very complicated and tedious – requiring detailed knowledge of the path, frictional forces, etc. There has to be an easier way… It turns out that there is – it is done by analyzing the object’s energy.

Energy The “something” that enables an object to do work is energy. Energy is measured in Joules (J). Forms of Energy:  Mechanical (kinetic and potential)  Thermal (heat)  Electromagnetic (light)  Nuclear  Chemical

Mechanical Energy Mechanical energy is the form of energy due to the position or the movement of a mass.

Kinetic Energy Kinetic energy is the energy of motion – it is associated with the state of motion of an object. The faster an object is moving, the greater it’s kinetic energy; an object at rest has zero kinetic energy. For an object of mass m, we will define kinetic energy as: The SI unit of kinetic energy is the Joule (J).

Kinetic Energy If we do positive work on an object by pushing on it with some force, we can increase the object’s kinetic energy (and thereby increasing it’s speed). We can account for the change in kinetic energy by saying that the force transferred energy from you to the object. If we do negative work on an object by pushing on it with some force in the direction opposite to the direction of motion, we can decrease the object’s kinetic energy (and decrease it’s speed). We can account for the change in kinetic energy by saying that the force transferred energy from the object to you.

Kinetic Energy Whenever we have a transfer of energy via a force, we say that work is done on the object by the force. –Work W is energy transferred to or from an object by means of a force acting on that object. –Energy transferred to the object is positive work. –Energy transferred from the object is negative work. Work is nothing more than transferred energy – it therefore has the same units as energy and is also a scalar quantity. Note that nothing material is transferred. –Think of it like the balance in two bank accounts: when money is transferred the number for one account goes down by some amount and the number for the other account goes up by the same amount.

Work-Energy Theorem Suppose we have an bead which is constrained to move only along the length of a frictionless wire. We then supply a constant force F on the bead at some angle  to the wire. Because the force is constant, we know that the acceleration will also be constant.

Work-Energy Theorem But because of the constraint, only the force in the x direction matters, thus F x = m·a x where m is the bead’s mass. We can relate the bead’s velocity at some distance down the wire to the acceleration using:

Work-Energy Theorem Solving for a x, substituting into the F x equation, multiplying both sides by d, and distributing the ½· m throughout the equation:

Work-Energy Theorem But we can see that the right side of the equation is no more than the kinetic energy after the force has been applied minus the kinetic energy before the force was applied: and that – by definition – is the work done

Work-Energy Theorem When calculating the work done on an object by a force during a displacement, use only the component of the force that is parallel to the object’s displacement. –where  is the angle between the force F and the horizontal. The force component perpendicular to the displacement does no work.

Work-Energy Theorem Work-Energy Theorem: the net work done on an object is equal to the change in kinetic energy of the object. W = K f – K i ; F·d = 0.5·m·(v f 2 - v i 2 ) A net force causes an object to change its kinetic energy because a net force causes an object to accelerate, and acceleration means a change in velocity, and if velocity changes, kinetic energy changes.

Gravitational Potential Energy Potential energy (U): an object may store energy because of its position. Energy that is stored is called potential energy because in the stored state it has the potential to do work. Work is required to lift objects against Earth’s gravity. Potential energy due to elevated positions is gravitational potential energy. The amount of gravitational potential energy possessed by an elevated object is equal to the work done against gravity in lifting it. U g = F w ·h = m·g·h

Work and Potential Energy When we throw a tomato up in the air, negative work is being done on the tomato which causes it to slow down during it’s ascent. As a result, the kinetic energy of the tomato is reduced – eventually to zero at the highest point. But where did that energy go???

Work and Potential Energy Where it went was into an increase in the gravitational potential energy of the tomato. The reverse happens when the tomato begins to fall down. Now the positive work done by the gravitational force causes the gravitational potential energy to be reduced and the tomato’s kinetic energy to increase.

Work and Potential Energy From this we can see that for either the rise or fall of the tomato, the change Δ U in the gravitational potential energy is the negative of the work done on the tomato by the gravitational force In equation form we get:

Only changes in potential energy have meaning; it is important that all heights be measured from the same origin. In many problems, the ground is chosen as the zero level for the determination of the height. As the ball falls from A to B, the potential energy at A is converted to kinetic energy at B. The amount of potential energy of the ball at point A will equal the amount of kinetic energy of the ball at point B.

Elastic Potential Energy Stretching or compressing an elastic object requires energy and this energy is stored in the elastic object as elastic potential energy. The work required to stretch or compress a spring is dependent on the force constant k. The force constant will not change for a particular spring as long as the spring is not permanently distorted (which occurs when the elastic limit of the spring is exceeded). F = k·x

Elastic Potential Energy The force required to stretch/compress an elastic object is not a constant force. The work needed varies with the amount of stretch/compression. W = 0.5·k·x 2 The potential energy of an elastic object is equal to the work done on the elastic object. U e = 0.5·k·x 2 In actual practice, a small fraction of the work in stretching/compressing an elastic object is converted into heat energy in the spring.

Work and Elastic Potential Energy If we give the block a shove to the right, the kinetic energy of the block is transferred into elastic potential energy as the spring compresses. The work done in compressing the spring is the negative of the change in the block’s kinetic energy. And of course the reverse happens when the spring stretches back out – potential energy gets transformed back into kinetic energy.

Conservation of Mechanical Energy Conservative forces: all the work done is stored as energy and is available to do work later. Example: gravitational forces, elastic forces. Nonconservative (dissipative) forces: the force generally produces a form of energy that is not mechanical. Friction is a nonconservative (dissipative) force because it produces heat (thermal energy, not mechanical). The total amount of energy in any closed system remains constant.

Conservation of Mechanical Energy The sum of the potential and kinetic energy of a system remains constant when no dissipative forces (like friction) act on the system. Law of Conservation of Energy: energy cannot be created or destroyed; it may be changed from one form to another or transferred from one object to another, but the total amount of energy never changes.

Conservation of Mechanical Energy In a closed system in which gravitational potential energy and kinetic energy are involved, the potential energy at the highest point is equal to the kinetic energy at the lowest point. m·g·h = 0.5·m·v 2

Notice that the sum of the potential energy (PE) and kinetic energy (KE) at every point is J. Energy is conserved.

The velocity at the lowest point can be determined by the height:

Conservation of Mechanical Energy The change in velocity due to a change in height can also be determined: The height can be determined by the initial velocity (v f = 0 m/s):

Conservation of Mechanical Energy In a closed system in which elastic potential energy and kinetic energy are involved, the potential energy at the maximum distance of stretch/compression is equal to the kinetic energy at the equilibrium (rest) position.

Conservation of Mechanical Energy Conservation of Energy Equation: W done (by applied force) + U gravitational before + U elastic before + K before = U gravitational after + U elastic after + K after + W done (usually by friction) F applied ·d + m ·g ·h i ·k ·x i ·m ·v i 2 = m ·g ·h f ·k ·x f ·m ·v f 2 + FF·d

If there is no change in height: m ·g ·h i and m ·g ·h f drop out of the equation. If there is no spring or elastic object: 0.5 ·k ·x i 2 and 0.5 ·k ·x f 2 drop out of the equation. If there is no change in velocity: 0.5 ·m ·v i 2 and 0.5 ·m ·v f 2 drop out of the equation. If there is no applied force (a push/pull that you supply): F applied ·d drops out of the equation. If there is no friction: FF·d drops out of the equation.

Helpful Online Links Work – Energy Theorem:  The Work-Energy Theorem The Work-Energy Theorem Elastic Constant k:  Hooke’s Law Applet Hooke’s Law Applet