Revision: Work, Energy, and Power JEE Main Work, Energy, and Power

The rate of doing work, where if an agent does work W in time t its average power is P_av = \[\frac {W}{t}\], which is a scalar quantity with SI unit watt (W) and dimension [ML²T⁻³], is called Power.

The work done when a force acting on a body has a component in the opposite direction of displacement, so that W = F s cos ⁡θ is a negative value, is called Negative Work.

The work done when either F = 0, or s = 0, or θ = 90°, so that the work done equals zero, is called Zero Work.

The capacity or ability of a body to do work, measured by the amount of work the body can perform, which is a scalar quantity with SI unit joule (J) and dimensional formula [ML²T⁻²], is called Energy.

The energy possessed by a body by virtue of its motion, expressed as K E = \[\frac {1}{2}\]mv², is called Kinetic Energy.

The energy stored in a body or a system by virtue of its position in a field of force or by its configuration is called Potential Energy.

The potential energy stored in stretched or compressed materials like springs or rubber bands, expressed as P Eelastic = \[\frac {1}{2}\]kx², is called Elastic Potential Energy.

The limiting value of the average power as the time interval approaches zero, expressed as P = limΔt→0\[\frac {ΔW}{​Δt}\] = \[\frac {dW}{dt}\]​, is called Instantaneous Power.

The potential energy associated with an object's height above the ground, expressed as P E_{gravitational} = mgh, is called Gravitational Potential Energy.

The commercial unit of electrical energy, also known as the Board of Trade (B.O.T.) unit, which is the electrical energy consumed by an appliance of 1000 watt in 1 hour, equal to 3.6 ×10⁶ joules, is called Kilowatt Hour.

The physical quantity said to be done whenever a force acts on a body and the body moves through some distance in the direction of the force, which is the product of magnitude of displacement and the component of force along the displacement, with SI unit joule (J) and dimensional formula [M¹L²T⁻²], is called Work.

The work done when a force acting on a body has a component in the direction of displacement, so that W=Fscos⁡θW=Fscosθ is a positive value, is called Positive Work.

The kinetic energy of the body due to its vibrational motion is called vibrational kinetic energy or simply vibrational energy.

The kinetic energy of the body due to motion in a straight line is called translational kinetic energy.

The motion of a body in a straight line path is called translational motion.

The energy possessed by a body due to its state of motion is called its kinetic energy.

If a body moves to and fro about its mean position, the motion is called vibrational motion.

The kinetic energy of the body due to rotational motion is called rotational kinetic energy or simply rotational energy.

If a body rotates about an axis, the motion is called rotational motion.

The work done by a force on a body is equal to the product of the force applied and the distance moved by the body in the direction of force i.e.,

Work done = Force × distance moved in the direction of force

The SI unit of work is joule.
1 joule of work is said to be done when a force of 1 newton displaces a body through 1 metre in its own direction.

When a force acts on a rigid body which is free to move, the body starts moving in a straight line in the direction of the force. This is called translational motion.

“Capacity of doing work” is called Energy.

Energy, in physics, is the capacity for doing work. It may exist in potential, kinetic, thermal, electrical, chemical, nuclear, or other various forms. There are, moreover, heat and work-i.e., energy in the process of transfer from one body to another.

Work is said to be done only when the force applied on a body makes the body move (i.e., there is a displacement of the body). 

The energy possessed by a body due to its state of rest or of motion, is called mechanical energy.

The energy possessed by a body at rest due to its position or size and shape is called potential energy.

The energy possessed by a body by virtue of its specific position (or changed configuration) is called the potential energy.

The energy radiated out by the Sun is called solar energy.

The rate of doing work is called power.

Power is defined as the rate of doing work or work done per second.

i.e., Power = `("Work done in joule")/("Times in second")`

or,  p = `("W (in joule)")/("t (in second)")`

A collision as a process where "several objects come together, interact (exert forces on each other) and scatter in different directions."

OR

An event where two or more bodies exert forces on each other in a relatively short time is called a collision.

OR

The interaction between two bodies during a very small duration in which they exert relatively large forces on each other, and during which momentum or kinetic energy is transferred from one object to another, is called Collision.

A collision in which both linear momentum and kinetic energy are conserved is called an elastic collision.

A collision in which linear momentum is conserved but kinetic energy is not conserved is called an inelastic collision.

For two colliding bodies, the negative of the ratio of the relative velocity of separation to the relative velocity of approach is called the coefficient of restitution.

The ratio of the work done by the machine to the work done on the machine is called the efficiency of a machine

Efficiency =`"Output energy"/" Input energy"`

(Work done by a machine is called the output energy and the work done on a machine is called the input energy.)

The energy of a body is its capacity to do work.

K = \[\frac {1}{2}\] mv²

Kinetic Energy = \[\frac {1}{2}\] mass × (velocity)²

W = \[\int_{\mathrm{b}}^{\mathrm{a}}\vec{\mathrm{F}}.\overline{\mathrm{ds}}=\int_{\mathrm{b}}^{\mathrm{a}}\mathrm{F}\mathrm{ds}\cos\theta\]

Gravitational Potential Energy U_h = mgh

Power P = \[\frac{\text{Work done }W}{\text{Time taken }t}\]

or

P = \[\frac {W}{t}\]

Statement:

According to the work-energy theorem, the increase in kinetic energy of a moving body is equal to the work done by a force acting in the direction of the moving body.

Proof:

Let a body of mass m be moving with an initial velocity u. When a constant force F is applied to the body along its direction of motion, it produces an acceleration a, and the body's velocity increases from u to v over a distance S.

Force,

F = ma

Work done by the force,

W = F × S

From the equation of motion,

\[v^2=u^2+2aS\Rightarrow S=\frac{v^2-u^2}{2a}\]

Substituting equations (i) and (iii) into (ii):

W = \[ma\times\frac{v^2-u^2}{2a}=\frac{1}{2}m(v^2-u^2)\]

Now,
Initial kinetic energy, K_i = \[\frac {1}{2}\]mu²
Final kinetic energy, K_f = \[\frac {1}{2}\]mv²

Therefore,

W = K_f − K_i

Conclusion:

Work done on the body = Increase in its kinetic energy.
Hence, the work-energy theorem is proved.

Statement: Energy can neither be created nor be destroyed. It can only be converted from one form to another.

Conservation of Mechanical Energy:
When only conservative forces act on a system, the total mechanical energy remains constant:

• This implies that the sum of kinetic and potential energies at any given point of time is equal to the initial total mechanical energy of the system.
• When a body falls freely in air, mechanical energy is conserved as potential energy is lost and an equal amount of kinetic energy is gained.
• If motion involves friction/collision, conservation of mechanical energy is not applicable, as energy is converted to heat and sound.

• The S.I. unit of work is joule (J).
    1 joule = 1 newton × 1 metre, i.e., work done when a force of 1 N moves a body 1 m in its direction.
• The C.G.S. unit of work is erg.
    1 erg = 1 dyne × 1 cm, i.e., work done when a force of 1 dyne moves a body 1 cm in its direction.
• The relation between joule and erg is:
    1 joule = 10⁷ erg

• Work done by a constant force is given by W = \[\vec F\] . \[\vec s\] = F s cos⁡ θ; for infinitesimal displacement, dW = \[\vec F\] . d\[\vec x\].
• For a variable force, the standard formula is not applicable; work done is calculated using W = \[\int_{s_{1}}^{s_{2}}\vec{F}\cdot d\vec{s}.\]
• The area under the force-displacement graph represents the work done; for linearly variable force, W = Area APQB.
• Conservative force (e.g., gravity) — work done is path independent; non-conservative force (e.g., friction) — work done is path dependent.
• Mechanical energy is conserved under conservative forces only; W_conservative = −ΔU and W_{non-conservative} = ΔKE + ΔPE.

• There are two main types of potential energy: gravitational and elastic.
• Gravitational potential energy is due to height and is given by U = mgh.
• It is zero at infinity and becomes less negative as the distance from Earth increases.
• Elastic potential energy is stored when an object is stretched or compressed.
• Lifting a body stores energy as gravitational potential energy by doing work against gravity.

• S.I. unit: If 1 joule of work is done in 1 second, the power spent is said to be 1 watt.
• C.G.S. unit: The C.G.S. unit of power is erg per second (erg s^-1).
• Relationship between S.I. and C.G.S. units:
   1 W = 1 J s^-1 = 10⁷ erg s^-1
• 1 horse power (H.P.) = 746 W = 0.746 kW

• Energy and work are directly related — when work is done, energy is transferred; doing work decreases energy, and receiving work increases it.
• No energy transfer occurs when the applied force is perpendicular to the displacement (e.g., centripetal force in circular motion).
• Units of energy are the same as those of work:
    - S.I. unit: Joule (J)
    - C.G.S. unit: Erg
    - 1 J = 10⁷ erg
• Practical units of energy:
    - 1 Wh = 3600 J = 3.6 kJ
    - 1 kWh = 3.6 × 10⁶ J = 3.6 MJ
    - 1 calorie = 4.18 J, 1 kilocalorie = 4180 J
• Energy in atomic-scale processes is measured in electron volt (eV), where 1 eV = 1.6 × 10⁻¹⁹ J