1. How work and kinetic energy notion came – we know in constant accelerationυ2=υ0

2+2. a⃗. (r⃗−r⃗0 )=υ02+2. a⃗ . d⃗

12m.υ2−1

2m.υ0

2=m . a⃗ .d⃗=F⃗ . d⃗

Half the mass times the square of the speed is called kinetic energy (K). product of the displacement and the component of the force along the displacement is called work (W).So K f−K i=W

It is a special case of the work-energy (WE) theorem : The change in kinetic energy of a particle is equal to the work done on it by the net force.

2. Work –W=F⃗ . d⃗=(F .cosθ ) . d=diplacement× force componet∈displacement direction

if there is no displacement, there is no work done even if the force is large. Ex - While we move cycle on road, displacement of road is 0 so work done by cycle on road is 0 (in ground frame)

If the force and displacement are mutually perpendicular, then no work done. Ex- centripetal force.

the work done on A by B is not necessarily equal and opposite to the work done on B by A

Work can be both +ve or –ve. –ve direction of work mean force is in opposite direction of displacement (we always take component of F in displacement direction and multiply it with displacement).

It is scalar.3. Kinetic energy – (scalar)

The kinetic energy of an object is a measure of the work an object can do by the virtue of its motion. Ex- Sailing ships employ the kinetic energy of the wind.

K=12m.υ2

4. Work done by variable force – (consider in 1 dimension Fx –x curve) If the displacement ∆x is small, we can takethe force F (x) as approximately constant and the work done is then ∆W=F (x)∆ x

W ≅∑x i

x f

F (x)∆ x

If the displacements are allowed to approach zero, then

W= lim∆ x→0

∑x i

x f

F (x)∆ x=∫xi

x f

F (x)dx=area under thecurve F ( x )−x

In 3-D

W=∫ri

rf

F⃗ . r⃗

5. Work energy theorem for variable force –(1-D)dKdt

= ddt12.m . υ2=m. dυ

dt. υ=F .υ=F . dx

dtdK=F .dx

By integration - ∫K i

K f

dK=∫xi

x f

F .dx

K f−K i=WIt is an integral form of Newton’s second law. Newton’s second law is a relation

between acceleration and force at any instant of time. Work-energy theorem involves an integral over an interval of time. In this sense, the temporal (time) information contained in the statement of Newton’s second law is ‘integrated over’ and is not available explicitly. Newton’s second law for two or three dimensions is in vector form whereas the work-energy theorem is in scalar form. In the scalar form, information with respect to directions contained in Newton’s second law is not present.Note- Never ever use W-E theorem to calculate F.

6. Potential energy - Word potential suggests capacity for action. potential energy means stored energy by virtue of the position or configuration of a body. The body left to itself releases this stored energy in the form of kinetic energy. Potential energy is negative of work done by the conservative force.

Notion of potential energy is applicable only to the class of forces where work done against the force gets stored up as energy called conservative forces. Mathematically, in 1-D the potential energy V(x) is defined if the force F(x) can be written as

F ( x )=−dVdx

⟹∫x i

x f

F ( x ) . dx=−∫V i

V f

dV=−(V f−V i)⟹W=−(V f−V i)

The work done by a conservative force such as gravity depends on the initial and final positions only.Example- If an object of mass m is released from rest, from the top of a smooth (frictionless) inclined plane of height h, its speed at the bottom is √2gh irrespective of the angle of inclination.Gravitational potential energy - Let us raise the ball up to a height h (keeping v constant i.e. ∆ K=0). The work done by the external agency against the gravitational force is mgh. Thiswork gets stored as potential energy.

Gravitational potential energy of an object V (h )=negativeof work doneby the gravitational force=m .g .h

gravitational force F=−ddh

V (h )=−m. g negative sign indicates that the gravitational force

is downward(i.e. opposite to h)When released, the ball comes down with an increasing speed. υ2=2.g .h

⟹ 12.m . υ2=m.g .h

the gravitational potential energy of the object at height h, when the object is released, manifests itself as kinetic energy of the object on reaching the ground.

Potential energy due to spring –a block attached to a spring and resting on a smooth horizontal surface. The other end of the spring is attached to a rigid wall. (we take stretching as +ve displacement x and compressing as –ve x) By Hook’s law spring force F s=−kx

Let spring is pulled by x distance then work done by spring-

W s=∫0

x

F ( x ) .dx=−12

k . x2

If the block is moved from an initial displacement xi to a final displacement xf, the work done

by the spring force is W s=∫x i

x f

F ( x ) .dx=−12

k .(x f ¿¿2−xi2)¿

Thus spring force does work which only depends on the initial and final positions. So, the spring force is a conservative force.

So potential energy for it by definition of potential energy-

∫0

x

F (x ) . dx=−∫0

V ( x )

dV⟹V ( x )=12k . x2

7. Conservation of mechanical energy- (1-D)Suppose that a body undergoes displacement ∆x under the action of a conservative

force F the change in potential energy, for a conservative force, ∆V is equal to the negative of the work done by the force

∆V=−F (x)∆ x

from work energy theorem∆ K=F( x)∆ x

By these 2 eq.

∆V=−∆ K⟹∆ (V +K )=0

i.e. sum of the kinetic and potential energies of the body is a constant.¿K i+V (x i)=K f +V (x f )

in other words, quantity K +V(x) called the total mechanical energy of the system remain constant.

The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.

When work done by conservative force is require to be calculated we have W-E theorem and –ve of difference of final potential energy and initial potential energy. But if we want to calculate change in velocity then we use mechanical energy conservation.

Conservation of energy for spring- If a block of mass m is extended to xm and released from rest, then its total mechanical energy at any arbitrary point x, where x lies between – xm and + xm, will be given by

12k . xm

2=12k . x2+ 1

2m.υ2⟹υ2= k

m(xm

2−x2)

So block will have maximum speed when x=0 and υm=√ km

xm

Information on time is absent from the above discussions. In the example considered above, we can calculate the compression, but not the time over which the compression occurs. A solution of Newton’s Second Law for this system is required for temporal information.

Not all forces are conservative. Friction, deformative forces for example, is a non-conservative force. The principle of conservation of energy will have to be modified in this case.By W-E theorem W=∆ K⟹F ∆ x=∆ K⟹ (Fc+Fnc )∆ x=∆ K

⟹ Fc∆ x+Fnc ∆ x=∆ K⟹−∆V+Fnc∆ x=∆ K⟹∆ (K+V )=Fnc∆ x⟹∆ (E )=Fnc∆ x⟹ Ef−Ei=Fnc ∆x=W nc

where E is the total mechanical energy. The zero of the potential energy is arbitrary. It is set according to convenience. For the

spring force we took V(x) = 0, at x = 08. Conservative force various definitions – (1-D)

a) A force F(x) is conservative if it can be derived from a scalar quantity V(x) by the relation

F ( x )=−dVdx

b) work done by the conservative force depends only on the end points.

W=K f−K i=V (x¿¿ i)−V (x¿¿ f )¿¿

c) work done by this force in a closed path is zero. Since x i=x f

9. Motion in vertical circle –Length of string =L ; mass of ball=m; initial velocity given at

lowest point A in υ0 so forces on ball are tension and weight. Tension (T) is always

perpendicular to displacement so work done by it is 0. So work is only done by gravitational force which is equal to change in potential energy. Now to calculate minimum υ0 such that ball complete full circular motion.(assuming at A gravitational potential energy=0)By mechanical energy conservation- (at point C)

12.m. υ0

2+0=2m. g .L+ 12m.υC¿

2①

at point A-

T A−m.g=m.υ0

2

L¿②

at point C –

T C+mg=12m.υC

2but TC=0 for lowest υ0

⟹mg=m.υC

2

L¿③

By eq. 1,3 - υ02=5 g .L

Speed at any point at h height from A by mechanical energy conservation12.m. υ0

2+0=m .g .h+ 12m .υB

2⟹υB2=g .(5 L−2h)

At point C, the string becomes slack and the velocity of the ball is horizontal and to the left. If the connecting string is cut at this instant, the bob will execute a projectile motion with horizontal projection akin to a rock kicked horizontally from the edge of a cliff. Otherwise, the ball will continue on its circular path and complete the revolution.

10. Various forms of energy –a) Mechanical energy – Potential energy + Kinetic energyb) Heat – Friction is non conservative force but still a –ve work is associated with it, this

work actually changes the temperature of block and surface between which it work. So work done by friction is not lost but transferred into heat energy i.e. increase in internal energy of block and surface (i.e. increase in speed of molecules forming them)

c) Chemical energy- Chemical energy arises from the fact that the molecules participating in the chemical reaction have different binding energies. A stable chemical compound has less energy than the separated parts. A chemical reaction is basically a rearrangement of atoms. If the total energy of the reactants is more than the products of the reaction, heat is released and the reaction is said to be an exothermic reaction. If the reverse is true, heat is absorbed and the reaction is endothermic.

d) Electrical Energy- There are laws governing the attraction and repulsion of charges and currents. Energy is associated with an electric current.

e) The Equivalence of Mass and Energy and Nuclear Energy – it was believed that in every physical and chemical process, the mass of an isolated system is conserved. Matter might change its phase, e.g. glacial ice could melt into a gushing stream, but matter is neither created nor destroyed. Albert Einstein showed that mass and energy are equivalent and are related by the relation

E=mc2

The most destructive weapons made by man, the fission and fusion bombs are manifestations of the above equivalence of mass and energy. Energy output of the sun is also based on the above equation where four light hydrogen nuclei fuse to form a helium nucleus whose mass is less than the sum of the masses of the reactants.

11. Principle of conservation of energy - We have seen that the total mechanical energy of the system is conserved if the forces doing work on it are conservative. If some of the forces involved are non-conservative, part of the mechanical energy may get transformed into other forms such as heat, light and sound. However, the total energy of an isolated system does not change, as long as one accounts for all forms of energy.

Energy may be transformed from one form to another but the total energy of an isolated system remains constant. Energy can neither be created, nor destroyed.

Since the universe as a whole may be viewed as an isolated system, the total energy of the universe is constant.

The principle of conservation of energy cannot be proved. However, no violation of this principle has been observed.

12. Power - Power is defined as the time rate at which work is done or energy is transferred.

Pav=Wt

instantaneous power P=dWdt

if F⃗ is constant than P=d ( F⃗ .r⃗ )

dt=F⃗ . d r⃗

dt= F⃗ .υ⃗

13. Collision -