Work and kinetic energy

When the net force acting on an object is zero, the net work done by all the forces acting on the object is zero.  When the net force acting on an object is not zero, then the net work done on the object is W_net = F_net· d.   When a net force acts on an object, then the object accelerates, it changes its velocity.

Assume an object is moving along a straight line, and a constant force F_net = ma is acting on the object.  Then W = ma·d.  The work is proportional to the component of the acceleration a_||, parallel to the direction of the displacement.  This component of the acceleration causes a change in speed, a_|| = (v_f-v_i)/Dt.  Therefore

W = md(v_f-v_i)/Dt.

The distance traveled is the average speed times Dt,

d = Dt(v_f+v_i)/2.

Therefore

W = m(v_f-v_i)(v_f+v_i)/2 = (1/2)m(v_f²-v_i²).

We can express the net wok done on the object in terms of the change in the quantity (1/2)mv².  We define the (translational) kinetic energy of the object as K = (1/2)mv².  The net work done on the object is equal to the change in the kinetic energy of the object.

W_net = K_f - K_i = DK.

This is called the work-kinetic energy theorem.

Example:

Kinetic energy increases with the square of the speed.  Neglecting friction, an engine does four times as much work to make a car reach a speed of 60 miles/h as to make it reach a speed of 30 miles/h.  When the speed of a car is doubled, its kinetic energy increases by a factor of four.

Problem:

The gravitational potential energy U_g is defined as the negative of the work done by the gravitational force, or the work done by an applied force canceling the gravitational force, in displacing an object from a reference position.  If the gravitational force acting on an object is pointing in the -y direction with constant magnitude mg, and the reference position is at y = 0, then U_g = mgy.

The zero of the gravitational potential energy, i.e. the reference position, is chosen arbitrarily.  However a difference in gravitational potential energy

DU_g = mgDy = mg(y_f - y_i)

is uniquely defined.

The gravitational potential energy gained by an object being lifted is equal to the work done on the object by an applied force, which exactly cancels gravity.  The applied force does positive work.  When the object is being lifted, the gravitational force does negative work.  The gravitational potential energy can be converted back into other forms by letting the gravitational force do positive work.  If we let the object fall towards the ground it will loose potential energy and gain kinetic energy.

The gravitational potential energy does not depend on the path.

Example:

The gravitational force near the surface of the earth points vertically downward.  We can approximate any path between two points P₁ and P₂ by arbitrarily small vertical and horizontal segments.

The total horizontal and the total vertical displacement are the same for each path.  The gravitational force does no work along the horizontal segments of the path, since here it is perpendicular to displacement vector.  It does work W_i = -mgDy_i along each vertical segment of length Dy_i.  The total work done by the gravitational force is W = -mgDy_total = -mgh.  The change in potential energy is U_g = mgh.

A potential energy function is a function of the position of an object.  It can be defined only for conservative forces.  A force is conservative if the work it does on an object between any two points is independent of the path taken by the object.  The gravitational force is a conservative force.  The potential energy function associated with the gravitational force near the surface of the earth is U_g = mgy if the reference point is chosen at y = 0.  (U_g only depends on the position of an object, not on how the object reached that position.)

Another example of a conservative force is the force exerted by a spring.  The elastic potential energy function is U_s = (1/2)kx², where x is the displacement from equilibrium, if the reference point is chosen to be x = 0.  (U_s only depends on the position of an object, not on how the object reached that position.)

Example:

For an isolated, conservative system, i.e. an isolated system only acted on by internal, conservative forces, the sum of the kinetic energy and potential energy is constant.

K_i + U_i = K_f + U_f.

We define

E = K + U,

where E = mechanical energy.  The mechanical energy of an isolated conservative system is conserved.  Conservation of mechanical energy is a powerful tool for solving physics problems.

Problem:

For an isolated, conservative system E is constant, dE/dt = 0.  Assume an object is moving along the x-axis.

dK/dt = d(mv²/2)/dt = mv(dv/dt) = mva_x = F_xv.
dU/dt = (dU/dx)(dx/dt) = (dU/dx)v.
dK/dt + dU/dt = 0

implies

F_xv + (dU/dx)v = 0

or

F_x = -dU/dx.

The conservative internal force acting between parts of the system equals the negative derivative of the potential energy associated with that system.

Microscopically all known forces are conservative.  Therefore, microscopically, all energy is either kinetic or potential energy, and, if the system under consideration is the whole universe, the total energy of the universe is conserved.

Work is the conversion of one form of energy into another.  Macroscopically, there are forms of energy that can easily and completely be converted into other forms.  We call these forms ordered energy.  Gravitational potential energy is an example of ordered energy.  The gravitational potential energy stored in a car on top of a hill is converted into kinetic energy as it rolls towards the bottom of the hill.  Hydroelectric plants convert the gravitational potential energy stored in the water in a reservoir into electric energy.  Kinetic energy also can be easily converted.  Kinetic energy is the energy an object has because it is moving.

There is however a form of energy, thermal energy, which cannot easily and completely converted into other forms.  Thermal energy is disordered energy.  The individual atoms and molecules that make up an object have potential and kinetic energy, but they move in a random fashion about their equilibrium position in the object, so that the object as a whole remains at rest.  The more kinetic energy is stored in the random motion of the atoms or molecules, the higher is the temperature of the object.

Macroscopically, forces that do work converting ordered energy into disordered energy are non-conservative forces.  The work done by macroscopically non-conservative forces on an object as it moves from position P₁ to position P₂ depends on the path of the object.  Friction is an example of such a non-conservative force.  The work done by the force of friction converts ordered energy into thermal energy.

When a system is acted on by an external force, then energy can be transferred into or out of the system.  An external force can do work against internal forces and change the potential energy of the system or it can be a net force changing the kinetic energy of the system.  The work W_app done by the external force on a conservative system may be written as W_app = DK + DU.  The applied external force can be a conservative or a non-conservative force.

Problems:

Links:

Work is done by a force on an object when the object moves in the direction of the force.  Power is a measure of how quickly this work is done, it is the rate at which work is done.

P = DW/Dt = average power

P = dW/dt = instantaneous power

Example:

The SI unit of power is Joule/s = Watt. (J/s = W.)  Often used units are kilowatt (kW), megawatt (MW), and milliwatt (mW).

1 kW = 1000 W.
1MW = 1000000 W = 10⁶ W.
One horsepower (hp) = 745.7 W.

When you pay your electric bill, you pay for each kilowatt-hour (kWh) used.  Kilowatt-hour is a unit of energy or work, not power.
1 kWh = 1kW 1hour = 1000 J/s 3600s = 3600000 J = 3.6 ´ 10⁶ J

The instantaneous power is P = dW/dt.  We have

dW = F·dr.

Therefore we may write

P = F·dr/dt = F·v.

Problem:

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