1. Work

In everyday language, "work" refers to any physical or mental effort. However, in science, the definition of work is very specific. Work is said to be done only when a force applied on an object causes a displacement of the object in the direction of the force.

1.1 Scientific Conception of Work

For work to be done in science, two conditions must be satisfied:

1. A force must act on the object.
2. The object must be displaced.

Formula: Work done ($W$) by a constant force is defined as the product of the magnitude of the force ($F$) and the distance ($s$) moved in the direction of the force.

The SI unit of work is the Joule (J). One Joule is defined as the amount of work done when a force of $1 \text{ Newton}$ displaces an object by $1 \text{ metre}$ along the line of action of the force.

1.2 Nature of Work (Positive, Negative, and Zero Work)

• Positive Work: When the force and the displacement are in the same direction, work is positive. E.g., a horse pulling a cart.
• Negative Work: When the force and displacement are in opposite directions, work is negative. E.g., the work done by frictional force when a block slides on a table.
• Zero Work: When the force and displacement are perpendicular to each other ($90^\circ$), no work is done. E.g., a person carrying a heavy load and walking on a horizontal road does zero work against gravity.

2. Energy

Energy is defined as the capacity or ability to do work. The object which does the work loses energy and the object on which the work is done gains energy.

Since energy is the capacity to do work, the SI unit of energy is the same as that of work, which is the Joule (J).

2.1 Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. The kinetic energy of an object increases with its speed.

Formula: The kinetic energy ($E_k$) of an object of mass $m$ moving with a uniform velocity $v$ is:

Derivation: Let an object of mass $m$ start from rest ($u=0$) and attain a velocity $v$ over a displacement $s$ due to a uniform force $F$. Work done $W = F \times s$. From Newton's second law, $F = ma$. From the third equation of motion, $v^2 - u^2 = 2as \implies s = \frac{v^2}{2a}$.

Substituting $F$ and $s$ into the work equation:

Since the work done is stored as kinetic energy, $E_k = \frac{1}{2}mv^2$.

2.2 Potential Energy

Potential energy is the energy possessed by an object due to its position, shape, or configuration.

Gravitational Potential Energy: The work done in raising an object against gravity from the ground to a certain height.

Formula: Let an object of mass $m$ be raised to a height $h$. The minimum force required to lift it is equal to its weight ($mg$).

Work done, $W = \text{Force} \times \text{Displacement} = mg \times h = mgh$

This work is stored as gravitational potential energy ($E_p$):

3. Law of Conservation of Energy

The law of conservation of energy states that energy can neither be created nor destroyed; it can only be transformed from one form to another. The total energy before and after transformation always remains constant.

Example of Free Fall: Consider an object of mass $m$ falling freely from a height $h$.

• At the start (height $h$), $E_p = mgh$ and $E_k = 0$. Total Energy = $mgh$.
• As it falls, its potential energy decreases and kinetic energy increases (since velocity increases).
• Just before hitting the ground, $h=0$, so $E_p = 0$. Its velocity is maximum, so all energy is converted to kinetic energy ($E_k = mgh$).

At any point in its path, the sum of its potential and kinetic energy (Total Mechanical Energy) remains constant: $E_k + E_p = \text{Constant}$.

4. Rate of Doing Work (Power)

Power is defined as the rate of doing work or the rate of transfer of energy.

Formula: If an agent does work $W$ in time $t$, then power $P$ is given by:

Unit: The SI unit of power is the Watt (W). $1 \text{ Watt} = 1 \text{ Joule/second}$.

For larger measurements, we use Kilowatt (kW) where $1 \text{ kW} = 1000 \text{ W}$.

4.1 Commercial Unit of Energy

The Joule is too small for expressing large quantities of energy. We use a bigger unit called kilowatt-hour (kWh).

1 kWh is the energy consumed when 1 kW of power is used for 1 hour.

The electrical energy used in homes is measured in units. $1 \text{ Unit} = 1 \text{ kWh}$.

5. Simple Machines (Comprehensive)

A simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force or to apply force at a convenient point and direction.

5.1 Key Technical Terms

• Load (L): The resistive or opposing force to be overcome by a machine.
• Effort (E): The applied force to overcome the load.
• Mechanical Advantage (MA): The ratio of the load to the effort. It determines the force-multiplying capability of the machine.
  $$ MA = \frac{Load}{Effort} = \frac{L}{E} $$
• Velocity Ratio (VR): The ratio of the velocity of the effort to the velocity of the load (or distance moved by effort to the distance moved by load in the same time).
  $$ VR = \frac{d_E}{d_L} $$
• Efficiency ($\eta$): The ratio of the useful work done by the machine (Work Output) to the work put into the machine (Work Input).
  $$ \eta = \frac{\text{Work Output}}{\text{Work Input}} \times 100\% = \frac{MA}{VR} \times 100\% $$

5.2 Types of Simple Machines

1. Levers

A rigid straight or bent bar which is capable of turning about a fixed axis called the fulcrum (F).

• Class I Lever: Fulcrum is between Load and Effort (e.g., scissors, see-saw). MA can be >, <, or = 1.
• Class II Lever: Load is between Fulcrum and Effort (e.g., wheelbarrow, nutcracker). Effort arm is always longer than load arm. Thus, MA is always > 1 (Force multiplier).
• Class III Lever: Effort is between Fulcrum and Load (e.g., sugar tongs, human forearm). Effort arm is always shorter than load arm. Thus, MA is always < 1 (Speed multiplier).

2. Pulleys

• Single Fixed Pulley: Used only to change the direction of force. $MA = 1, VR = 1$.
• Single Movable Pulley: Acts as a force multiplier. $MA = 2, VR = 2$.
• Block and Tackle System: Combination of fixed and movable pulleys. If total number of pulleys is $n$, then ideally $VR = n$ and $MA = n$.

3. Inclined Plane

A flat surface tilted at an angle ($\theta$) to the horizontal. It acts as a force multiplier to lift heavy loads.