WORK POWERAND ENERGY

PREPARED BY :Ashutosh Mishra CE 0 0 7Divakar Singh Jadoun CE 0 1 0Bhoopendra Bansal CE 0 0 9Ashish Bohra CE 0 0 6 Bhaskar Sharma CE 0 0 8

CONTENTS

Principles of Work and Energy Work of Some Typical Forces Conservative Forces Non - Conservative Forces Concept of Energy, Kinetic Energy Gravitation Potential Energy ME of Motion ME due to Position Principle of Work and Energy for a Single Particle,

for a System of Particles Principle of Conservation of Energy of ME

INTRODUCTION

ENERGY

“It is the fundamental property of a particle or a system referring to its potential to influence changes to other particles by imparting work or heat.”

In short, energy is the capacity to do work.

Energy exists in many forms – Mechanical, electromagnetic, electrical, nuclear, chemical and thermal.

Mechanical energy can be Kinetic Or Potential…..

Wo r k

Work is a type of controlled energy transfer when one system is exerting force in a specific direction and thus is making a purposeful change (Displacement) of the other systems.

Work done on a particle or on a body is equal to the product of the force imparted on it and the displacement along the line of action of the active force either forward or reverse direction of motion.

Work = Force x Displacement

Unit of work = 1 Newton x 1 Metre = 1 Joule

It is a scalar quantity.

r

r1

r2

r + dr

FαdW = F . ds

Let the initial position vector of the particle at any time t be r and after a small time interval dt let its position vector be r+dr. Therefore the displacement is dr. During the time interval dt it is assumed that a constant force F, inclined at an angle α with the displacement vector acts.

Then the differential work dW is defined asdW = F . dS =F . dr

1

2

TOTAL AMOUNT OF WORK DONEW1-2 =

K I N E T I C E N E R G Y

The kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes.

The kinetic energy of an object is related to its momentum by the equation:

where:Ek is momentumm is the mass of the body

POTENTIAL ENERGY

Potential energy is energy which results from position or configuration.

The work of an elastic force is called elastic potential energy.

The work of the gravitational force is called gravitational potential energy.

The work of the Coulomb force is called electric potential energy;

The work of the strong nuclear force or weak nuclear force acting on the charge is called nuclear potential energy.

The work of intermolecular forces is called intermolecular potential energy.

Chemical potential energy, such as the energy stored in fossil fuels.

Gravitational Potential EnergyGravitational potential energy is the energy stored in an object as the result of its height. The energy is stored as the result of the gravitational attraction of the Earth for the object.

The gravitational potential energy of the ball (as shown) is dependent on two variables - the mass of the ball (m) and the height (H) to which it is raised.

H

m

Potential Energy = Mass (m) x Height (H) x g ( acc. due to gravity ) = m.g.H

Ram

Elastic Potential Energy

Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy.

EXAMPLES

Q & AQ .1

A cart is loaded with a brick and pulled at constant speed along an inclined plane to the height of a seat-top. If the mass of the loaded cart is 3.0 kg and the height of the seat top is 0.45 meters, then what is the potential energy of the loaded cart at the height of the seat-top?

Ans. 1

PE = m x g x h

PE = (3 kg ) X (9.8 m/s/s) X (0.45 m)

PE = 13.2 J

Q & A

Q . 2

If Ram is pushing a 12 N box at a constant speed of 35 m/s and Shyam is pushing another box weighing 35 N but with the speed 12 m/s. Which of the box has higher Kinetic Energy ?

RAM

SHYAM

ANS .2

RAM’s Box

K.E1 = ½ m v2

= 0.5 x 12 x 35 x 35 = 7350 J

SHYAM’s Box

K.E2 = ½ m v2

= 0.5 x 35 x 12 x 12 = 2520 N

K.E1 > K.E2

CONSERVATIVE FORCES

A conservative force is a force with the property that the work done in moving a particle between two points is independent of the taken path.

Example is shown on the right. If the force is conservative then, we can take any path may be 1 or 2 or any other path to reach b.

Equivalently, if a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) by a conservative force is zero.

= 0

Illustration :

POINTS TO REMEMBER :

1. Conservative Forces are reversible forces, meaning that the work done by a conservative force is recoverable.

2. When an external agent is applied to change the state of a system that is also acted upon by conservative force then the system can be restored to its initial state without using any other external agent.

3. Each type of conservative force is associated with it a potential energy.

4. Conservative forces are path independent.5. The work done by a conservative force can be transformed into

a change in potential energy.

Non-Conservative force

The work done by a non-conservative force does depend upon the path taken.

EXAMPLE : Friction

Conservative force and Potential energyPotential energy is always associated with a conservative force. It is not defined with respect to a non conservativeForce.Potential energy can be expressed as the integral form of a conservative force as U(x) =- F(x)dx + U(x⨜ 0 ) The constant of integration shown in the equation is an arbitrary one showing that any constant can be added to The potential energy . Practically it means that the initial or the reference point of computation of potential energyCan be set at any convenient point.

The potential energy U(x) is equal to the work necessary to move an object from x0 reference point to the positionx in the conservative force field.Differentiating the above equation wrt to x we getF(x) = - dU(x) dxThis means F(x) is the negative of the slope of the potential energy curve.

Work-Energy Principle for particleLet us assume that the velocity of travel of a mass for a differential displacement be v, thendW= F . drIntegrating the equation

2

1Therefore, the principle of work-energy can be stated as : the work done due to movement is equal with the change in Kinetic energy.

Work-Energy Principle for Rigid BodyTotal work done by a rigid body will be the summation of work done by the net forces acting at the mass centre and the work done by the net moment about the mass centre and is represented by the equation.

W= [ ½ m (VC2)2 - ½ m (VC

2)1 ] + [ ½ IC w22 – ½ IC w1

2 ]

Similarly the change in kinetic energy of a rigid body is the summation of change in kinetic energy due to translationand due to rotation.

ΔΚE = [ ½ m (VC2)2 - ½ m (VC

2)1 ] + [ ½ IC w22 – ½ IC w1

2 ]

Principle of Conservation of Energy

The law of conservation of energy states that the total energy of an isolated system cannot change—it is said to be conserved over time.

“ Energy can be neither created nor destroyed, but can be changed into one form to another form. ”

Principle of Conservation of Energy

Let us assume that a particle subjected to a system of conservative forces onlyThen,WC = -Δ P.E Applying Work-Energy PrincipleWC = Δ K.EEquating the two equations,Δ K.E = -Δ P.E Δ K.E + Δ P.E = 0

Einitial = Efinal

i.e. the mechanical energy of the system remains constant if only conservative forces act on the system.

Q. 1

A sphere of mass m and radius r allow to roll down without slipping along an α inclined as shown in the figure.Derive the expressions for the velocity for rolling down a distance s. α

Ans. 1 Initially the linear velocity and the angular velocity both are zero and hence the initial kinetic energy Kei = 0, Final Kinetic energy of the bodyKef = ½ mV2 + ½ I w2 = ½ mV2 + ½ I v2/r2

Work done on the body by the gravity to move a distance S,W= mg sinα S

Using work energy theorem

mg sinα S= ½ mV2 + ½ I v2/r2

V= =

Mg sin αMg cos α

Normal Reaction force

Static friction

Power

Power is defined as the rate of change of work done per unit time by the body .If the total work done during a differential time interval Δt is ΔW, then the average powercan be expressed as,Pavg = ΔW/ Δt

Pinstantaneous = dW/dt = (F . dS)/dt = F . V

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