General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor, M.Sc., Ph.D.

Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294

Work and Kinetic Energy


In this chapter we will learn the following concepts:

Energy is a scalar quantity.

Energy is conserved: The change in energy of a system plus the change in energy of surroundings is zero.

A single particle is the simplest possible system.

Single particle energy has two parts:

Rest energy (energy an object has even when at rest)

Kinetic energy (energy associated with motion)

The energy of the system can be changed only by external forces.

Work is energy transfer into or out of a system.

Work involves external force acting through a distance.

Work can be positive or negative.

Computation of work due to a variable force.

Work done by a spring force.

Power is the time rate of energy transfer into or out of a system

The Energy Principle

In everyday life we see flow of energy in our daily life.

We ingest chemical energy in the form of food and use this energy in our activities.

We put chemical energy in the form of gasoline into our car, and car uses up this energy to run the engine.

You compress a spring, and the mechanical energy stored in the spring can launch a ball.

Conservation of energy

Energy can neither be created nor be destroyed, but it can change form. The only way for system to gain or loose energy is if the surroundings loose or gain the same amount of energy.


ΔE is a change in energy, which is final energy minus initial energy,


The energy principle is fundamental principle.

The validity of the energy principle has been verified through a wide variety of observations and experiments.

A Single Particle (The Simplest System)

The simplest particle system consists of a single particle.

"A particle" could be a proton or an electron, a baseball or even a planet.

According to Einstein the energy WorkAndEnergy_3.gif of a particle moving with speed v is given by the expression:


The unit of energy is joule, abbreviated "J".


WorkAndEnergy_6.gif is the speed of light.

What is energy?

When a particle is at rest, what is its energy?

When a particle is at rest, its speed v=0, therefore



That means when the particle is at rest, its energy is given as


We call WorkAndEnergy_10.gif the "rest energy"  of a particle at rest.

Einstein realized that the mass of an object is its energy content divided by a constant: WorkAndEnergy_11.gif.
In real sense, mass and energy is same thing, although for historical reasons we use different units for them: mass in Kg and energy in joules.

For example a hot object, with more thermal energy, has slightly more mass than a cold object.

Kinetic Energy

Let us plot energy of a particle in the units of its "rest energy" as a function of v/c.


We can see that energy of the particle increases with increase in speed.

As consequence of its motion, the particle has additional energy, known as "kinetic Energy" K.


The kinetic energy K of a particle is the energy, a moving particle has in addition to its rest energy.

Let us use the "binomial expansion" and we get the expression of the energy as




It can be seen that energy of a particle can be written as


Where the expression for Kinetic Energy K can be written as


Example 1: Energy of a fast moving proton

A proton in a particle accelerator has a speed of WorkAndEnergy_18.gif.(a) What is the energy of the proton? (b) What would the energy of the proton be if it were sitting still? (c) What is the kinetic energy of the moving proton? WorkAndEnergy_19.gif.

Click to see Solution :

Kinetic Energy at Low - Speed

Let us consider a particle moving with a speed v that is much small compared to the speed of light (v<<c).

As we have seen  with "binomial expansion", the expression for the kinetic energy K of a particle can be written as


When v<<c, then v/c << 1, therefore higher powers of v/c will be further small and can be neglected.

We can say that at (v<<c) Low-Speed Kinetic Energy can be given as


Since we will be dealing with low speed (v<<c) problems, therefore whenever we use the term kinetic energy, we mean approximate kinetic energy.

Example 2: Kinetic Energy of a fast moving proton with approximate formula

Calculate the Kinetic energy of the proton in a particle accelerator given in example 1 with approximate formula. Proton speed was WorkAndEnergy_31.gif. WorkAndEnergy_32.gif.

Click to see Solution :

Example 3: Approximate Kinetic Energy

The solar wind consists of charged particles streaming away from sun. The speed of a proton in the solar wind can be as high as WorkAndEnergy_36.gif. What is the kinetic energy of such a proton? WorkAndEnergy_37.gif.

Is it appropriate to use approximate formula for Kinetic Energy in this case?

Click to see Solution :


Work is defined as energy transferred into or from a system.

Consider a bead of mass m moving with initial velocity WorkAndEnergy_43.gif along a frictionless wire.

What is its (initial) kinetic energy?


On the way, (Click Play to move) we apply force  WorkAndEnergy_45.gif, directed at an angle φ to the wire. After a displacement WorkAndEnergy_46.gif along x-axis the velocity of bead has changed to WorkAndEnergy_47.gif.

Now what is its (final) kinetic energy?


How much energy is transferred to the bead? Or how much is the change in energy of bead?  


As per definition of work this change in energy of the bead, produced by the applied force is called work W.


Mathematically we can write it as


This is also known as work-kinetic energy theorem.

Relation between work and applied force.

If force is a constant (constant acceleration) according to kinematic equation the final velocity WorkAndEnergy_53.gif is related to displacement WorkAndEnergy_54.gif and acceleration along x-axis (direction of motion).


Multiply both sides by WorkAndEnergy_56.gif.


As per above equation change in kinetic energy can be given as


Or work is related to the displacement and acceleration as



Which component of force will cause the change in magnitude of bead velocity?

As per Newton's law, the product WorkAndEnergy_61.gif is nothing but the x-component WorkAndEnergy_62.gif of the applied force WorkAndEnergy_63.gif.


By substituting WorkAndEnergy_65.gif we get


Term d F cos φ is the dot product  WorkAndEnergy_67.gif of displacement WorkAndEnergy_68.gif and applied force WorkAndEnergy_69.gif, therefore we can write work W as


The SI unit of Work is Joule (J) same as units of energy.


Work W is energy transferred to or from an object by means of a force acting on the object.
Energy transferred to the object is positive work.
Energy transferred from the object is negative work.

Although this expression is derived from change in energy of a single particle but for any force acting on a system, work W is defined as the dot product WorkAndEnergy_72.gif of the force WorkAndEnergy_73.gif and the displacement WorkAndEnergy_74.gif caused by that force. Since dot product of two vectors is a scalar, therefore work W is a scalar quantity.

Checkpoint - 1

A particle moves along an axis. Does the kinetic energy of the particle increase, decrease, or remain the same if the particle’s velocity changes
(a) from −3 m/s to −2 m/s and
(b) from −2 m/s to 2 m/s?
(c) In each situation, is the work done on the particle positive, negative, or zero?

Example 4: Work

A paper airplane flies from location  WorkAndEnergy_75.gif to location WorkAndEnergy_76.gif.
The net force acting on it during this flight, due to the earth and the air, is nearly constant at WorkAndEnergy_77.gif.
What is the total work done on the paper airplane by the earth and the air?

Click to see Solution :

Example 5: Work

A coin slides over a frictionless plane and across an xy coordinate system from the origin to a point with x y coordinates (3.0 m, 4.0 m) while a constant force acts on it. The force has magnitude 2.0 N and is directed at a counterclockwise angle of 100° from the positive direction of the x axis.

How much work is done by the force on the coin during the displacement?


Click to see Solution :

Example 6: Work and Kinetic Energy

An 8.0 kg object is moving in the positive direction of x axis. When it passes through x=0 m, a constant force directed along the axis begins to act on it. Following figure gives its kinetic energy K versus position x as it moves from x = 0 m to x = 5 m ; WorkAndEnergy_100.gif. The force continues to act.
What is v when the object moves back through x = -3 m?


Click to see Solution :

Work Done by the Gravitational Force

Consider a bouncing ball (click play to throw ) of mass m that is thrown upward from point, with initial speed WorkAndEnergy_113.gif.

What is its initial kinetic energy ?


As the ball rises, its speed v (red arrow) reduces, that means, the ball’s kinetic energy decreases to?


Which force is responsible for decreasing the kinetic energy of ball?

Gravitational force WorkAndEnergy_117.gif.

Gravitation force is doing a negative work  WorkAndEnergy_118.gif when the ball is moving up with positive displacement WorkAndEnergy_119.gif.

The work  WorkAndEnergy_120.gif done by the gravitational force on the ball for upward displacement  WorkAndEnergy_121.gif is:      


Minus sign means gravitational force takes away energy from the object when it moves up.

What will be the work done WorkAndEnergy_124.gif by gravitational force WorkAndEnergy_125.gif when ball moves down?

When ball moves downward, the angle between WorkAndEnergy_126.gif and downward displacement WorkAndEnergy_127.gif is 0°, the work WorkAndEnergy_128.gif done by the gravitational is:


Plus sign means gravitational force gives energy back to the object when it moves down.

Checkpoint - 2

A ball is moving upwards, acted on by a downwards gravitational force.
Is the kinetic energy of the ball increasing or decreasing?
Is the work done by the gravitational force positive or negative?

Checkpoint - 3

A greased pig has a choice of three frictionless slides along which to slide to the ground. Rank the slides according to how much work the gravitational force does on the pig during the descent, greatest first.


How to compute Work if force is Variable?

Work done by a constant force is given as the product of displacement and the component of the force along the displacement.

What about variable force?

Consider a variable force acting on an object along x-axis.

Work done along x-axis, needs only x-component of force F(x).

Plot the magnitude of x-component of force F(x) as a function of position x.

Let us calculate the work W that F does on an object to move it from WorkAndEnergy_132.gif to WorkAndEnergy_133.gif.

First divide the interval WorkAndEnergy_134.gif into N “elements” of length Δx (select Show) such that we can assume that force is constant for each small displacement Δx.

Since force is assumed to be constant in the interval Δx therefore if  WorkAndEnergy_136.gif  is the average force in the 4th interval, the work done WorkAndEnergy_137.gif in 4th interval is


Now the total work done can be computed by adding work done over all the intervals.


From knowledge of calculus we know if we take limit N or Δx0, we get


We can see for very small value of Δx (take Δx slider to minimum) , geometrically, W is area of the curve from WorkAndEnergy_142.gif to WorkAndEnergy_143.gif.

Checkpoint - 4

Following figure gives the x component WorkAndEnergy_144.gif of a force that can act on a particle. If the particle begins at x = 0, with some initial kinetic energy, what is its coordinate when it has
(a) maximum kinetic energy and (b) maximum speed?
(c) How much work is done by the force, by the time particle reaches x = 6 m?


Click to see Hint :

Example of variable Force

Spring Force

Spring force is an example of variable force.

When  we pull  (click-hold and drag the block to right) one end of the spring and stretch it by amount  WorkAndEnergy_148.gif.

Spring resists by exerting a force WorkAndEnergy_149.gif in opposite direction.

When we push (click-hold and drag the block to left) one end of the spring and compress it by amount WorkAndEnergy_151.gif.

Again the spring resists by exerting a force WorkAndEnergy_152.gif in opposite direction.

This force is called spring force or restoring force, and is given as


This is also known as Hook's law and k is called spring's constant.

In one dimension we can write it as


Work done by a Spring Force

Spring force is a variable force, work done by any variable force in one dimension is given as


In one dimension spring force is written as



If spring moves from WorkAndEnergy_158.gif to WorkAndEnergy_159.gif, work WorkAndEnergy_160.gif done by spring force can be computed as




If initial position WorkAndEnergy_164.gif(equilibrium), work WorkAndEnergy_165.gif done by spring force can written as


Example 7: Work Done by Spring Force

Fig. 1 gives spring force versus position x for the spring–block arrangement of Fig. 2. After pulling, the block was released from x = 12 cm.
How much work does the spring do on the block when the block moves from  x = 8.0 cm to (a) x = 5.0 cm, (b) x = -8.0 cm?

WorkAndEnergy_167.gif WorkAndEnergy_168.gif
Fig. 1 Fig. 2
Click to see Solution :


The time rate of energy transfer into or out of a system is called power.

If change in kinetic energy ΔK of a system takes place in time Δt, the average power WorkAndEnergy_183.gif is defined as


The instantaneous power P is the instantaneous time rate of change of energy


The SI unit of power is the joule per second. This unit is called watt (W).


Power is a scalar quantity

A commonly used non SI unit is horsepower

The kilowatt-hour (k Wh) is a unit of work (power multiplied by time). It is used by electrical utility companies.


Instantaneous power and Velocity

If the work is done by a force WorkAndEnergy_188.gif through a displacement of WorkAndEnergy_189.gif, the average power can be written as


If the work is done by a constant force, average power will be


The instantaneous power can now be given as


Here WorkAndEnergy_193.gif is the instantaneous velocity of the object.

Therefore if force is constant, instantaneous power can be given as


Checkpoint 3

A block moves with uniform circular motion because a cord tied to the block is anchored at the center of a circle. Is the power due to the force on the block from the cord positive, negative, or zero?

Example 8: Power

The loaded cab of an elevator has a mass of WorkAndEnergy_195.gif and moves 210 m up the shaft in 23 s at constant speed.
At what average rate does the force from the cable do work on the cab?

Click to see Solution :