Chapter 02: Work, Energy and Power - Vardaan Master Notes

(A) WORK, POWER AND ENERGY

2.1 WORK

In Physics, the term 'work' is used only when a force produces displacement in a body.

Definition: Work is said to be done only when the force applied on a body makes the body move.

Fig 2.1 A man pushing a car

If there is no displacement (e.g., pushing a rigid wall or holding a heavy load while standing still), the work done is zero.

2.2 MEASUREMENT OF WORK

GENERAL FORMULA

$$W = F \times S \cos \theta$$

Where $F$ is force, $S$ is displacement, and $\theta$ is the angle between them.

Special Cases ($\theta$):

(i) $\theta = 0^\circ$ (Positive Work): Force & displacement in same direction. $W = F \times S$.

(ii) $\theta = 90^\circ$ (Zero Work): Force & displacement are perpendicular. $W = 0$ (e.g., Circular orbit).

(iii) $\theta = 180^\circ$ (Negative Work): Displacement opposite to force. $W = -F \times S$ (e.g., Friction).

Work Done by a Variable Force

The area under a Force-Displacement graph gives the work done.

2.3 WORK DONE BY FORCE OF GRAVITY

GRAVITATIONAL WORK

$$W = mgh$$

where $h$ is the vertical height climbed or descended.

ZERO WORK ALERT If a coolie moves horizontally, work done by gravity is ZERO ($\theta = 90^\circ$). Work is only done when lifting the load ($mgh$).

2.4 UNITS OF WORK

SI Unit: joule (J). $1 \text{ J} = 1 \text{ N} \times 1 \text{ m}$.
CGS Unit: erg. $1 \text{ erg} = 1 \text{ dyne} \times 1 \text{ cm}$.

RELATIONSHIP J vs ERG

$$1 \text{ joule} = 10^7 \text{ erg}$$

2.5 POWER ($P = W/t$)

Power is the rate of doing work.

POWER FORMULAS

$$P = \frac{W}{t} \quad \text{or} \quad P = F \times v$$

Units of Power:

SI Unit: watt (W) = $1 \text{ J/s}$.
Bigger Units: 1 kW = $10^3 \text{ W}$, 1 MW = $10^6 \text{ W}$.
Mechanical Unit: 1 Horse power (hp) = $746 \text{ W} = 0.746 \text{ kW}$.

Difference between Work and Power

Work	Power
Force $\times$ Displacement.	Rate of doing work.
Independent of time.	Depends on time.

NUMERICAL Power Calculation (Crane)

A crane pulls up a car of mass $500 \text{ kg}$ to a vertical height of $4 \text{ m}$ in $20 \text{ s}$. Calculate power. ($g = 9.8$)

Solution:
$W = F \times h = (500 \times 9.8) \times 4 = 19600 \text{ J}$.
$P = W/t = 19600 / 20 = 980 \text{ W}$.

2.7 & 2.8 ENERGY AND ITS UNITS

Definition: Energy is the capacity to do work. It is a scalar quantity with the same units as work (Joules/Ergs).

COMMERCIAL & OTHER UNITS

kilowatt-hour (kWh): Energy spent by $1 \text{ kW}$ in $1 \text{ h}$. $1 \text{ kWh} = 3.6 \times 10^6 \text{ J} = 3.6 \text{ MJ}$.
Calorie: $1 \text{ calorie} = 4.18 \text{ J}$.
Electron-volt (eV): $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$.

Energy Transformations

In various devices, energy is converted from one form to another. Examples:

Motor: Electric $\to$ Mechanical
Loudspeaker: Electric $\to$ Sound
Cell: Chemical $\to$ Electric
Photosynthesis: Light $\to$ Chemical

DEGRADED ENERGY During conversion, part of the energy is lost as non-useful forms (like heat due to friction). This is called dissipation of energy or degraded energy.

(B) DIFFERENT FORMS OF ENERGY

2.9 - 2.11 POTENTIAL ENERGY (U)

The energy possessed by a body due to its changed position or configuration.

Gravitational PE: Due to position above earth ($U = mgh$).
Elastic PE: Due to deformed state (e.g., stretched spring, bent bow).

2.12 KINETIC ENERGY (K)

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

KINETIC ENERGY & MOMENTUM

$$K = \frac{1}{2} mv^2 \quad | \quad p = \sqrt{2mK}$$

BOARD FAVORITE If two bodies (light and heavy) have the same momentum, the lighter body will have more Kinetic Energy (since $K \propto 1/m$).

2.13 WORK-ENERGY THEOREM

Work done by a force on a moving body equals the increase in its kinetic energy: $W = K_f - K_i = \Delta K$.

(C) CONSERVATION OF ENERGY

2.16 PRINCIPLE OF CONSERVATION OF ENERGY

Definition: Energy can neither be created nor destroyed; it only changes form. Total mechanical energy ($K + U$) remains constant in absence of friction.

2.17 THEORETICAL VERIFICATION (FREE FALL)

For a freely falling body, total energy $E = mgh$ at all points.

Position	Kinetic (K)	Potential (U)	Total (E)
A (Highest)	0	$mgh$	$mgh$
B (Middle)	$\frac{1}{2}mgh$	$\frac{1}{2}mgh$	$mgh$
C (Ground)	$mgh$	0	$mgh$

2.18 SIMPLE PENDULUM

Mean Position (A): $U=0$. Velocity is max $\implies$ KE is Maximum.
Extreme Positions (B & C): $K=0$. Height is max $\implies$ PE is Maximum ($mgh$).

MASTER NUMERICAL SET (Top 5 Essential Qs)

Q1 Basic Power & Work

A boy of mass $40 \text{ kg}$ climbs $8 \text{ m}$ in $5 \text{ s}$. Power? ($g=10$)

Ans: $P = \frac{mgh}{t} = \frac{40 \times 10 \times 8}{5} = 640 \text{ W}$.

Q2 Efficiency (Hydro Dam)

In a dam, water falls at $1000 \text{ kg/s}$ from $100 \text{ m}$. Calculate power if efficiency is $60\%$.

Ans: 1s Potential Energy = $mgh = 1000 \times 10 \times 100 = 10^6 \text{ J}$.
Power Generated = $60\% \text{ of } 10^6 = 600,000 \text{ W} = 600 \text{ kW}$.

Q3 Inclined Plane Logic Fig 2.12

A block of mass $30 \text{ kg}$ is pulled up a slope of length $3 \text{ m}$ and height $1.5 \text{ m}$ with force $200 \text{ N}$. Calculate: (i) Work done, (ii) Potential energy gain.

Ans: (i) $W = F \times \text{length} = 200 \times 3 = 600 \text{ J}$.
(ii) $U = mgh = 30 \times 10 \times 1.5 = 450 \text{ J}$. (The $150 \text{ J}$ difference is lost to friction).

Q4 Rebound & Energy Loss

A $10\text{ g}$ ball falls from $5\text{ m}$ and rebounds to $4\text{ m}$. Find loss in energy. ($g=9.8$)

Ans: Initial PE = $0.01 \times 9.8 \times 5 = 0.49\text{ J}$.
Final PE = $0.01 \times 9.8 \times 4 = 0.392\text{ J}$.
$\text{Loss} = 0.49 - 0.392 = 0.098\text{ J}$ (Lost as heat/sound).

Q5 Ski Jump Conservation Fig 2.18

A skier ($60 \text{ kg}$) takes off from A ($75 \text{ m}$) to B ($15 \text{ m}$). Calculate speed at B if $75\%$ energy becomes KE.

Ans: $\Delta U = mg(h_1 - h_2) = 60 \times 10 \times (60) = 36000 \text{ J}$.
$K = 0.75 \times 36000 = 27000 \text{ J}$.
$\frac{1}{2}mv^2 = 27000 \implies \frac{1}{2}(60)v^2 = 27000 \implies v^2 = 900 \implies v = 30 \text{ m/s}$.

End of Chapter 02

Practice makes perfect. Review the numericals thrice.

Work, Energy & Power