Chapter 6: Work Energy and power

Work

Work is the product of the component of the force in the direction of the displacement and the magnitude of this displacement.

work, in physics measure of energy transfer that occurs when an object is moved over a distance by an external force at least part of which is applied in the direction of the displacement. If the force is constant, work may be computed by multiplying the length of the path by the component of the force acting along the path.

To express this concept mathematically, the work W is equal to the product of force and the distance d,

work done= f∙d= fdcosθ, If the force is being exerted at an angle θ to the displacement.

Work done on a body is equal to the increase in the energy of the body, for work transfers energy to the body. If, however, the applied force is opposite to the motion of the object, the work is considered to be negative, implying that energy is taken from the object.

Positive, negative and zero work done

The work done by a force on an object can be positive, negative, or zero, depending upon the direction of displacement of the object with respect to the force. For an object moving in the opposite direction to the direction of force, such as friction acting on an object moving in the forward direction, the work done due to the force of friction is negative.

Positive work: when the force is along the displacement or angle θ between force and displacement is acute.

W= fd cosθ >0  , when cosθ >0 , θ< 90.

Example of positive work:

1. Players kicking the football in the direction of motion.
2. A nurse moves the patient into a wheelchair.
3. A person riding a skateboard.
4. Vehicles on the road, moving forward.
5. Cutting the vegetables using the knife.
6. Lifting the chair and moving it in another direction.
7. Moving a box across the table.
8. Two children throwing a ball at each other.

Negative work: when the force is opposite to the displacement or angle θ  between force and displacement is obtuse.

W= fd cosθ < 0 , when cosθ  <0 ,  θ >90.

Example of negative work

(i) When a body is thrown upwards, gravity does the negative work. Since the gravitational force acts downwards but the displacement is upwards.

(ii) When we walk frictional force does the negative work since frictional force acts opposite to displacement.

(iii) For a liquid flowing, viscous force does the negative work since it acts opposite to the direction of the force.

(iv) On a see-saw, negative work is done since we apply the force downwards but the person sitting opposite to us is displaced upwards.

Zero work done:

An object experiences zero work when

• The angle of displacement is perpendicular to the direction of the force
•  when the force applied couldn’t produce motion.

Consider an example of a coolie lifting a mass on his head moving at an angle of 90˚ with respect to the force of gravity. Here, the work done by gravity on the object is zero.

w= fd cosθ= fd cos 90= 0

Work done by constant force

Work done by a constant force is defined as the distance moved

multiplied by the component of force in the direction of displacement.

The area under the graph of force and displacement gives the value of work done by the force.

Example of work done by a constant force

• Work done by gravity
• When we apply a constant force F on a book and it moves.
• Motion of ball falling toward the ground.

Work done by a variable force

Variable force occurs when the direction and amount of a force vary throughout the motion of a body. Magnetic force, spring force, and electrostatic force are examples of variable forces. The majority of the forces we experience in our daily lives are variable forces. By splitting displacement into tiny intervals, the work done by a variable force may be computed.

A force is said to perform work on a system if there is displacement in the system upon application of the force in the direction of the force. In the case of a variable force, integration is necessary to calculate the work done.

The work done by a constant force of magnitude F, as we know, that displaces an object by Δx can be given asL:

W = F.Δx

In the case of a variable force, work is calculated with the help of integration.

The work done by force can also be calculated from the graphical method. The area under the curve in the graph of force vs displacement will give the magnitude of work done by the force.

For example, in the case of a spring, the force acting upon any object attached to a horizontal spring can be given as:

Fs = -kx

Where,

• k is the spring constant
• x is the displacement of the object attached

We can see that this force is proportional to the displacement of the object from the equilibrium position, hence the force acting at each instant during the compression and extension of the spring will be different. Thus, the infinitesimally small contributions of work done during each instant are to be counted in order to calculate the total work done.

The integral is evaluated as:

Mechanical energy

An object that possesses mechanical energy is able to do work. In fact, mechanical energy is often defined as the ability to do work. Any object that possesses mechanical energy - whether it is in the form of potential energy or kinetic energy is able to do work.

Example :

A hammer is a tool that utilizes mechanical energy to do work. The mechanical energy of a hammer gives the hammer its ability to apply a force to a nail in order to cause it to be displaced. Because the hammer has mechanical energy (in the form of kinetic energy) it is able to do work on the nail. Mechanical energy is the ability to do work.

The mechanical energy of a bowling ball gives the ball the ability to apply a force to a bowling pin in order to cause it to be displaced. Because the massive ball has mechanical energy (in the form of kinetic energy), it is able to do work on the pin. Mechanical energy is the ability to do work.

Kinetic energy

kinetic energy is a form of energy that an object or a particle has by reason of its motion. If work, which transfers energy, is done on an object by applying a net force, the object speeds up and thereby gains kinetic energy.

Kinetic energy is a property of a moving object or particle and depends not only on its motion but also on its mass.

Kinetic energy  K.E= 1/2 mv^2

Potential Energy

The energy possessed by the object due to its position and configuration is called Potential Energy.

Potential energy is defined with the conservative force. For example, for the gravitational force, we have gravitational potential energy and for the electric force, we have electrostatic potential energy.

• When a man picks an object of mass ‘m’ from the ground and puts that at height ‘h’. As gravity is pulling the object downward so the man has to do some work to raise the object to a height ‘h’. The work done by the man gets converted into the Potential energy of the object.

gravitational potential energy = mgh

• To compress a spring an external force needs to do work. The work done by external force gets converted into the potential energy of the spring.

Elastic  potential energy=1/2 kx^2, where K is the spring constant of the spring and x= compression/ elongation

Conservation of total mechanical energy.

For motion under conservative force, we have a conservation law of mechanical energy. When the motion is subjected to only conservative force, the total mechanical energy ( kinetic + potential) of the system remains conserved.

Potential energy and Kinetic energy gets converted into each other such that total mechanical energy remains conserved.

In the first figure given above, the Kinetic energy of the bicycle at the ground gets converted into potential energy at the top and then again gets converted into Kinetic energy at the ground.

In the second figure, a girl at some height holds a ball and then drops it. The potential energy of the ball at height gets converted into kinetic energy as it falls under the effect of gravity.

In the above figure, the potential stored in the string of the bow due to stretching gets converted into the Kinetic energy of the arrow

In the case of a pendulum, the motion of the bob is under the force of gravity. At the extreme position on either side Potential energy of the bob is maximum and at the mean position, Kinetic energy is maximum. Total mechanical energy at any instant is constant in the case of Simple harmonic motion.

Work-Energy Theorem

Statement: The work-energy theorem states that the net work done by the forces on an object equals the change in its kinetic energy.

Net work done= change in Kinetic energy= K.E_f-K.E_i

Sometimes people forget that the work-energy theorem only applies to the network, not the work done by a single force.

Derivation of the work-energy theorem for the constant force

we have Work done = F∙d= Fd cosθ, where F is force, d is displacement and θ is the angle between force and displacement.

When F || d , θ=0  then   W= Fd= (ma)d  …(1)

From the third equation of motion we have

v^2= u^2  + 2 a d  ;     2ad= v^2-u^2   ;  d=(v^2-u^2)/ 2a

So now we put the value of d in equation 1

W= ma ( (v^2-u^2)/2a) = m/2(v^2-u^2)= K.E_f-K.E_i  

Derivation of the work-energy theorem for variable force

Work done by small displacement  dW= F∙ds= ma ds=m(dv/dt)ds=m(ds/dt)dv=m v dv

Total work done will be the integration of the dW

Work done in changing the velocity from u to v

W=∫ dW=∫_u^v mv dv= m∫_u^v v dv= m/2  [v^2-u^2 ]= Δ K.E

Conservative  forces

 According to the law of conservation of energy in a closed system, i.e., a system that is isolated from its surroundings, the total energy of the system is conserved”. Conservative force abides by the law of conservation of energy. A conservative force is a force that does zero work done in a closed path. If only these forces act then the mechanical energy of the system remains conserved.

Work done by conservative force= -(change of potential Energy)

Wcons. = -(P.Ef- P.Ei)

Non-conservative forces

Non-conservative forces are dissipative forces such as friction or air resistance. These forces take energy away from the system as the system progresses, the energy that you can’t get back. These forces are path-dependent; therefore it matters where the object starts and stops.

Mechanical energy is not conserved in non-conservative forces.

Work done by non conservative force = change in total Energy

W_(non cons. )= T.E_f- T.E_i=(K.E+P.E)_f-(K.E+P.E)_i

Work done by all forces ( conservative+ non conservative)

W_net= K.E_f- K.E_i

Various form of Energy

In addition to mechanical energy, there are various other forms of energy.

Chemical Energy: Energy stored in the bonds of chemical compounds. Chemical energy may be released during a chemical reaction, often in the form of heat; such reactions are called exothermic. Reactions that require an input of heat to proceed may store some of that energy as chemical energy in newly formed bonds.

 Heat: Heat is the form of energy that is transferred between systems or objects with different temperatures (flowing from the high-temperature system to the low-temperature system). Also referred to as heat energy or thermal energy. Heat is typically measured in Btu, calories, or joules.

Nuclear Energy: Nuclear energy comes from splitting atoms in a reactor to heat water into steam, turn a turbine and generate electricity. Ninety-three nuclear reactors in 28 states generate nearly 20 percent of the nation’s electricity, all without carbon emissions because reactors use uranium, not fossil fuels. These plants are always on: well-operated to avoid interruptions and built to withstand extreme weather, supporting the grid 24/7.

Electrical Energy:  Energy is the ability to do work, where work is done when a force moves an object. We need and we use energy every day, and energy is available in all different forms. Electrical energy is energy that's stored in charged particles within an electric field. Electric fields are simply areas surrounding a charged particle. In other words, charged particles create electric fields that exert force on other charged particles within the field. The electric field applies the force to the charged particle, causing it to move - in other words, to do work.

Principle of conservation of Energy

We have discussed that the total mechanical energy of the system is conserved if the force doing work on it is conservative.

If some of the forces are non-conservative then mechanical energy conservation does not hold. Some energy of the system gets converted into some other form like heat, light and sound.

But the total energy of an isolated system does not change, as long as we can account for all forms of energy.

Energy may be transformed from one form to another but the energy of an isolated system remains conserved.

There is no violation of this principle. Since the universe as a whole may be viewed as an isolated system, the total energy of the universe remains the same. If one part of the universe loses energy, another part must gain an equal amount of energy.

Power

We can define power as the rate of doing work, it is the work done in unit time. The SI unit of power is Watt (W) which is joules per second (J/s). Power is a time-based quantity. Which is related to how fast a job is done.

Power= Energy/ time

The SI unit of power is Watt (W) which is joules per second (J/s).

• The power of motor vehicles and other machines is given in terms of Horsepower (hp), which is approximately equal to 745.7 watts.
• The commercial unit of energy is 1 kWh. One kilowatt-hour is defined as the amount of energy consumed by a device in one working hour at a constant rate of one kilowatt.

The SI unit of energy is Joule.

• Therefore, the relationship between commercial and SI unit of energy is: 1 kWh = 1kW x 1h = 1000W x 1h = 1000(J/s) x 3600 s = 3.6 x10^6 J

Vertical circle

Let an object of mass ‘m’ move in a circle that lines in the vertical plane. This is not a uniform circular motion as the velocity of the object in the vertical circle is not constant.

Let T₁ be the tension in the string at the lowest point of the vertical circle and T₂ be the tension in the string at the highest point of the vertical circle

There is a requirement of centripetal force mv2/r  to make the object move in a circular path.

Let v1 and v2 be the velocities of the object at the lowest and highest points of the vertical circle.

At lowest point    mg-T1= -mv1^2/r

T1= mg +m v1^2/r

At highest point  -T2-mg=-mv2^2/r

T2= m v2^2/r - mg

• Work done by the Tension force is zero and tension is perpendicular to the displacement direction always.
• We can use the conservation of energy in the case of a vertical circle.

Important results for Vertical circle

•  The difference in the tension in the string at the lowest and highest point is 6mg. It is independent of the velocity.   T1-T2=6mg
• The difference in the square of the velocities at the lowest and highest points is 4gr.    v1^2-v2^2= 4gr
• The minimum velocity at the highest point is √rg for the object to move in a complete vertical circle.
• The minimum velocity at the lowest point is √5rg for the object to move in a complete vertical circle.

Collision

Collision, also called impact, in physics, is the sudden, forceful coming together in direct contact of two bodies, such as, for example, two billiard balls, a golf club and a ball, a hammer and a nail head, two railroad cars when being coupled together, or a falling object and a floor.

• In all of the examples of colliding bodies here referred to, the time of contact is extremely short and the force of contact extremely large.
• there is an instantaneous change in the velocity of a body but no change in its position during the period of contact.
• Forces of this nature are known as impulsive forces and, being difficult to measure or estimate, their effects are measured by the change in the momentum  (mass times velocity) of the body.

Two types of collision:

Elastic collision: Collision between two bodies is said to be elastic when there is no energy loss during the collision

Inelastic collision: Collision between two bodies is said to be inelastic when there is some loss of kinetic energy during the collision. In the case of perfectly inelastic collisions bodies stick together after collision and move with a common velocity.

• Conservation of momentum holds for both case elastic collision as well as inelastic collision.
• Conservation of energy holds only in the case of elastic collision.

Coefficient of restitution   e= (velocity of separation )/(velocity of approach )= (v2-v1)/(u2-u1)

For perfectly elastic collision e=1 and for perfectly inelastic collision  e=0

Elastic collision in one Dimension.

Consider two bodies of masses m1 and m2 moving with initial velocities u1 and u2 along the same direction. After the collision, their final velocities are v1 and v2 respectively.

Since it is an elastic collision so there will be no energy loss. So we can apply conservation of energy and momentum both

By conservation of momentum we have Pi= Pf

m1 u1+m2 u2= m1 v1+ m2 v2

Rearranging we have  m1 (u1-v1)=m2 (v2-u2) .. (1)

By conservation of energy we have  K.Ei= K.Ef

1/2 m1 u1^(2 )+1/2 m2 u2^(2 )= 1/2 m1 v1^2+1/2 m2 v2^2

So we have  m1( u1^2-v1^2)=m2 ( v2^2-u2^2)   ..(2)

Divide equation 2 by 1 we will get 

u1+v1= v2+u2

rearranging we get ,   u1-u2= v2-v1

Thus v1=v2+u2-u1    , v2=v1+u1-u2    ,

When we put the value of v1 and v2 separately in the equation of momentum conservation we get the values of final velocities of the object  v1 and v2  in terms of their masses m1 and m2 and initial velocities  u1  and u2.

v1= ((m1-m2))/((m1+m2)) u1 +  2m2/((m1+m2)) u2

v2= 2m1/((m1+m2)) u1 +  ((m1-m2))/((m1+m2)) u2

Special case 1 :  when the target is at rest  u2=0

v1= ((m1-m2))/((m1+m2)) u1

v2= 2m1/((m1+m2)) u1

Special case 2: when the masses of the objects are equal and both objects are moving.

v1=u2  , v2= u1

When the masses of the objects are the same, the velocities of objects get interchanged after the collision.

Special case 3: when the target is at rest and the masses of both the objects are equal.

v1=0 , v2=u1

When the target is at rest and the masses of objects are the same, the first object becomes stationary and the second object starts moving with the initial velocity of the first object.

Elastic Collision in two dimensions.

Suppose we have two objects, one is at rest and the other is moving with initial velocity v1i, after collision object m1 moves with final velocity v1f making an angle θ with the horizontal direction and mass m2 moves with final velocity v2f making an angle Φ with the horizontal direction.

• First, we will resolve the components of v1f and v2f along a horizontal and vertical direction which is the x and y direction respectively. 

(v1f)x= v1f cosθ ,  (v1f) y= v1f sinθ.

(v2f)x= v2f cosΦ  , (v2f)y= v2f sinΦ

• Then we will apply conservation of momentum separately along x and y-direction.px i= px f  ;  py i= py f.

Putting the values we get

• From the conservation of energy, we have K.E i= K.Ff

1/2 m1 v1i^(2 )+0 =1/2 m1 v1f^2  +  1/2 m2 v2f^2

Analyzing the above equations reveals that finding values for four unknown quantities v₁, v₂, θ₁, and θ₂ using the above three equations is not possible. As a result, it cannot predict the variable because there are four of them. However, if we measure any one variable, we can uniquely determine the other variable using the above equation.

Inelastic collision

Suppose we have two objects of mass m1 and m2  moving with initial speeds v1i and v2i respectively. After collision, they stick together and start moving together with final velocity vf.  This is the case of a perfectly inelastic collision.

We cannot use conservation of energy in inelastic collisions. But we can use conservation of momentum to find the final velocity vf

Initial momentum = final momentum

 m1 v1i + m2 v2i =(m1+m2) vf

vf= (m1 v1i+m2 v2i) /(m1+m2)

Loss of kinetic energy in case of inelastic collision