# Work, energy and power

### Definition of work

Work is done by forces. If a force acting on an object displaces the object, then it is said that a work is done on the object by the force. In the figure below, an object is pulled by a constant force $\vec F_p$ that results in the displacement of the object, $\vec d$. Work done by the constant force is defined as

$W=Fd\cos\theta$

where $F$ is the magnitude of the force, $d$ is the magnitude of the object's displacement and $\theta$ is the angle between the force and the displacement. This equation is valid for any constant force. From the equation, you can realize that there is no work is done even though there is a force acting on, when (i) there is no displacement and (ii) the force and the displacement are perpendicular to each other as $\theta=90^\circ$ and $\cos90=0$.

Work can be positive or negative. If a force is acting opposite to the direction of motion ($\theta=180^\circ$ and $\cos180=-1$), then the work done by that force is negative. Since friction force acts in the direction opposite to the direction of motion, work done by friction is always negative.

SI unit of work is joule (J), it is same as $N.m$.

#### Work done by gravity and work against gravity

If you drop an object, the object makes a displacement in the direction of the force of gravity. If the object moves a distance $h$ downwards, then the work done by gravity over that distance is

$W=F_g h\:\cos0=mgh$.

But when you lift an object, the object's displacement is against gravity (i.e., opposite to the force of gravity), so gravity does a negative work on the object.

When an object is lifted, the lifting force does a work on the object against gravity. In this case, the force and the object's displacement are in the same direction, so the work by the force is positive. If the lift is being done by a rope, then the force of tension on the rope does the work. But if you put an object on your hand and raise the object, the normal force on the object by the hand does the work.

How much work is required to lift an object without acceleration?

Assume you are moving an object vertically upward to a height $h$ with a constant velocity ($a=0$). Since the velocity is constant, the net force on the object is zero. So the net work done on the object is zero because the net force is zero. Since gravity does negative work on the object, a positive work of same size needs to be done on the object to lift the object. Therefore, work needs to be done to lift an object to a height, $h$ is $mgh$.

#### Work done when you climb a stair or a ramp

When you climb a stair, you are doing a work against gravity. Who is doing that work? the stair, no?. It is done by your body muscle. As you see in the picture, the angle of your right leg increases as you lift your body, which is done by the body muscle. Since, you need to do a positive work while lifting your body against gravity, work required to climb a a stair of height $h$ is
$W=mgh$

Note that, whether it is the work by gravity, work against gravity or the work required to climb a stair or a ramp depends only on the height. The length of the ramp do not affect the work. The reason is simple. If you make a displacement on a ramp, you have two components of displacements as displacement is a vector: a vertical and a horizontal displacement. Work for the horizontal displacement is zero, as the force and this displacement are perpendicular to each other. So work results only from the vertical displacement. Therefore, the work by gravity or against gravity is independent of the path, it depends on the displacement, not on the distance.

#### Energy

Energy is the ability to do work. If an object has an ability to do some work, then we say that it possess an energy. There are different forms of energy, such as mechanical energy, thermal energy, electrical energy and radiant energy. Kinetic and potential energy are called mechanical energy.

### Kinetic energy

When an object is in motion, it has an energy called kinetic energy. Kinetic energy, $KE$ of an object of mass, $m$ moving with a velocity, $v$ is defined as,
$KE=\dfrac{1}{2}mv^2$

#### Work-Energy Principle

It states that "net work done on an object is equal to the change in kinetic energy of the object."

$W_{net}=\Delta KE$.

We can derive this principle by using Newton's second law and the kinematics.
Assume that an object is in motion with some velocity. Let this velocity be the initial velocity, $v_i$. Now, assume you are applying a constant force, $F$ in the direction of motion of the object (i.e.,$\theta=0$). Assume, this is the only force acting on the object, so the $F$ is the net force on the object. You keep applying the force for some time period and over that time, the object makes a displacement, $d$.

Now, the work done by the force is, $W_{net}=Fd\cos0$ or

$W_{net}=Fd$

Since, there is a net force on the object, the object accelerates. From Newton's second law, we have

$F=ma$

where $a$ is the acceleration of the object and $m$ is its mass.

Substituting this $F$, in the net work equation,

$W_{net}=mad$.

From kinematics, you can find the acceleration of the object. If $v_f$ is the final velocity, i.e., the velocity of the object at the end of the displacement $d$, then

$v_f^2=v_i^2+2ad$.

Solving for $a$,

$a=\dfrac{v_f^2-v_i^2}{2d}$

Substituting this $a$ in the work equation above, we get,

$W_{net}=m\bigg(\dfrac{v_f^2-v_i^2}{2d}\bigg)d$.

After canceling $d$ and removing the parenthesis,

$W_{net}=\dfrac{1}{2}mv_f^2-\dfrac{1}{2}mv_i^2$.

Right hand side is the final kinetic energy minus initial kinetic energy, that is just the change in kinetic energy of the object. i.e.,

$W_{net}=\Delta KE$

This is the work-energy principle.

Note that, for simplicity, we assumed that force and the displacement are in the same direction. But if you take they are not in the same direction, still you will get the same equation.

### Potential energy

If you lift an object to some height, it acquires an ability to do some work. This is because, if you release the object, its kinetic energy changes (due to the increase of speed) and the change in kinetic energy results in a work according to the work-energy principle. Also, a stretched or compressed spring gains ability to do some work. The energy an object possess because of its position or state is called potential energy.

#### Gravitational potential energy

You have learned that, to lift an object to some height, a work needs to be done on the object. The work that was done is not wasted instead it is stored as the potential energy of the object. The potential energy is exactly same as the work done on the object to raise it. We call this potential energy of a lifted object as the gravitational potential energy. Gravitational potential energy, $PE_g$, of an object at a height $h$ from the ground (or some reference) is defined as

$PE_g=mgh$.

where $m$ is the mass of the object.

#### Elastic or spring potential energy:

To stretch or compress an elastic spring, a force needs to be applied on the spring. That force does some work on the spring. When a spring is compressed or stretched it gains some energy. The acquired energy comes from the work done on the spring. The magnitude of the force required to stretch or compress a spring by an amount, $x$ is given by Hooke's law:

$F=kx$, where $k$ is the spring constant or force constant of the spring.

This force is not constant as it varies with $x$. Since the force is not constant, we cannot use the equation, $W=Fdcos\theta$ to calculate the work. Instead, you need to use the general work equation (that is out of scope for an algebra based physics) that is valid for any type of force whether it is constant or not. But it requires calculus to do that. By using calculus, you can find that the work required to stretch or compress a spring by an amount $x$ is

$W=\dfrac{1}{2}kx^2$

Due to this work, the spring gains some energy, and that energy is called, spring potential energy or elastic potential energy.Therefore, elastic potential energy of a spring is

$PE_{spring}=\dfrac{1}{2}kx^2$.

### Mechanical energy

Potential and kinetic energies are called mechanical energy. If you add the potential and the kinetic energy of an object, you will get the total mechanical energy of the object.

$Total \:Mechanical \: energy = PE + KE$

#### Conservative and nonconservative forces

You saw that the work done by gravity is independent of the path. If the work done by a force is independent of the path of the object, then the force is called, a conservative force. Therefore, gravity is a conservative force. Another example of conservative force is the force applied to stretch or compress a spring. But if a work depends on the path of the object, then the force that did that work is called a nonconservative force. Friction is a non-conservative force.

#### Work and potential energy with conservative force

Work done by a conservative force is

$W_c=-\Delta PE$

You can prove this by considering the work done by gravity. When gravity does a work on an object, the potential energy of the object decreases as $h$ decreases. Therefore, the change in potential energy is negative, and since the work by gravity is positive, we have a minus sign in the equation. The above equation is true for gravitational potential energy as well as spring potential energy.

#### Energy dissipation by friction and thermal energy

Force of friction does negative work on an object as friction force acts in a direction opposite to the object's displacement (motion). Friction reduces the mechanical energy of an object as it decreases the objects speed and hence the kinetic energy. So friction force is called, a dissipative force.

Work done by friction is $W_{fr}=F_{fr}d\:\cos(180)$

or

$W_{fr}=-F_{fr}d$

where $d$ is the distance the object traveled under friction. Note that in friction, $d$ is the distance not the displacement as work by friction is path dependent.

When friction does work on an object, heat is generated. Heat is a form of energy, so we call the generated heat thermal energy. Thermal energy is $=F_{fr}d$.

#### Total work on an object or a system of objects

An object can have both conservative and non-conservative force acting on. So, when calculating the work on an object we need to consider the work by both. Net work on an object is therefore, sum of the work by conservative forces and that by nonconservative forces:

$W_{net}=W_c+W_{nc}$

#### Total work with friction

When friction is the only non-conservative force acting on a system, then

$W_{nc}=W_{fr}$.

Substituting, $W_{net}=\Delta KE$ (work-energy theorem) and $W_c=-\Delta PE$,

$\Delta KE=-\Delta PE+W_{fr}$.

$\Delta (KE+PE)=W_{Fr}$.

$\Delta E=W_{Fr}$.

where $E=KE+PE$, is the total mechanical energy of the system.

Since, $\Delta E=E_{final}-E_{initial}$, we have

$E_{initial}+W_{fr}=E_{final}$

#### Conservation of mechanical energy

When there is no non-conservative force (such as friction) acting on a system, $W_{fr}=0$, and we have

$E_{initial}=E_{final}$

That is total mechanical energy of the system is constant., i.e., initial mechanical energy is same as the final mechanical energy. So there is no loss in mechanical energy. We call this the law of conservation of mechanical energy.

#### Law of conservation of energy

Mechanical energy does not include friction. Substituting, $W_{fr}=-F_{fr}d$

$E_{initial}-F_{fr}d=E_{final}$

or

$E_{initial}=E_{final}+F_{fr}d$

On the left hand side of the equation, we have the initial mechanical energy of the object. On the right hand side, the second term $F_{fr}d$, is actually the thermal energy. So, the part of the initial energy has now been transformed into thermal energy and the final mechanical energy is now lower than the initial mechanical energy. So, there is a transformation of energy but nothing is lost. That is the total energy is always conserved. This is called the law of conservation of energy, an important law of physics.

### Power

Power is the rate at which work is done. If you have done a work, $W$ in a time, $t$, then your power rating is

$P=\dfrac{W}{t}$

Since, when work is done, energy is transformed, power is also the rate at which energy is transformed,i.e.,

$P=\dfrac{E}{t}$

SI Unit of power is watt $(W)$ that is same as $J/s$

If you look at the electric bulbs or other electric equipments, you can see its power rating. A 100 W bulb uses 100 J of energy in 1s. Electric utility services charge us for how much energy was used by our electric equipments.

There is another common unit of power, that is $hp$ or horse power. This is not an SI unit, but it is used when mentioning the power of electric motors. The unit of $hp$ for power was first adapted by James Watt, a Scottish engineer in the 18th century. It is just the average power of a horse, i.e., the average work a horse can do per second. The conversion factor from $hp$ to $watt$ is

$1hp=746W$.