## Tebles of contents

**Work****Energy****Law of Conservation of Power****Power****Commercial Unit of Power**

## [1] Work

- Scalar
- SI unit -joule
- CGS unit-erg
- 1 joule=10^7 erg

**(1)**A force act on the body

**(2)**The point of application of the force moves in the direction of the force

**Example:-**Work is done when a horse pulls a cart, an engine pulls a train, a person climbs a hill, etc.

#### Work done by a constant force

**(1) Work:- When constant Forc is acting in the direction of Displacement**

**(2) Work:- When a constant force is acting at an angle to the Displacement**

**W**

**=**

**F⋅s⋅cos(Î¸)**

**Nature of work done in different situations**

#### (1) Positive work

The formula for positive work is:

$$W=F\cdot s\cdot \mathrm{cos}(\mathrm{\xce\xb8})$$For the work to be positive:

- The angle $\theta$ between the force $F$ and the displacement $d$ must be less than 90° ($\theta < 90^\circ$).
- This ensures that $\cos(\theta) > 0$, making the work $W$ positive.

**Example:-**In free fall, gravity does positive work. When a horse pulls a cart or a spring stretches, the applied force and displacement are in the same direction, resulting in positive work.

#### (2) Negative Work

The formula for negative work is:

$$W=F\cdot s\cdot \mathrm{cos}(\mathrm{\xce\xb8})$$For the work to be negative:

- The angle $\theta$ between the force $F$ and the displacement $d$ must be greater than 90° but less than 180° ($90^\circ < \theta < 180^\circ$).
- This makes $\cos(\theta) < 0$, resulting in negative work.

**Example:-**

**(1)**On a rough surface, friction does negative work because its force opposes the displacement.

**(2)**When brakes are applied to a moving vehicle, the work done by the braking force is negative because it acts opposite to the displacement.

#### (3) Zero Work

**Example:**-

## [2] Energy:

**Note:- 1:-**The unit of work and energy is the same. This is because these two physical quantities are inter-convertible.

**2:-** Other units of energy are electron volt(**eV**),calorie(**cal**),kilowatt-hour(**kWh**)

$$1\text{\hspace{0.17em}}\text{kWh}=3.6\times 1{0}^{6}\text{\hspace{0.17em}}\text{J}$$

It exists in various forms such as mechanical, thermal, chemical, electrical, and more. Energy can neither be created nor destroyed but can only be transformed from one form to another, as the Law of Conservation of Energy states.

### Types of Energy:

**1:-Kinetic Energy:**Energy possessed by an object due to its motion. KE is a scalar quantity

**Where:**

- $KE$ = Kinetic Energy
- $m$ = Mass of the object
- $v$ = Velocity of the object

**(a) Derivation of Kinetic Energy Formula:**

**Start with Newton’s Second Law**:
Newton’s second law states that force ($F$) is equal to mass ($m$) times acceleration ($a$):

**Use the Definition of Work**: Work ($W$) is defined as force ($F$) multiplied by the displacement ($d$) in the direction of the force:$$W=F\cdot d$$

**Substitute Force**: From Newton's second law, substitute $F=ma$ into the equation for work:

$$W=ma\cdot d$$

**Relate Displacement to Velocity**: Using the kinematic equation that relates displacement to velocity ($v$) and acceleration ($a$):$${v}^{2}={u}^{2}+2ad$$

where:

- $v$ is the final velocity,
- $u$ is the initial velocity (which can be 0 if starting from rest),
- $a$ is the acceleration,
- $d$ is the displacement.

Rearranging for $d$:

$$d=\frac{{v}^{2}-{u}^{2}}{2\mathrm{a}}$$**Substitute Displacement into Work Equation**: Substitute $d=\frac{{v}^{2}-{u}^{2}}{2a}$ into the equation for work:$$W=ma\cdot \frac{{v}^{2}-{u}^{2}}{2a}$$

Simplify by cancelling $a$:

$$W=\frac{m({v}^{2}-{u}^{2})}{\mathrm{2}}$$**Final Kinetic Energy Formula**: If the object starts from rest ($u=0$), the work done on the object is:

$$W=\frac{1}{2}m{v}^{2}$$

This work done on the object is equal to the change in its kinetic energy. Therefore, the **kinetic energy** ($KE$) is given by:

**Note:-**

$KE$ in terms of momentum $p$ can be written as:

$$KE=\frac{{p}^{2}}{2\mathrm{m}}$$**: Energy an object possesses due to its position or configuration. The most common type is gravitational potential energy, given by:2:-Potential Energy**

Where:

- $PE$ = Potential Energy
- $m$ = Mass of the object
- $g$ = Gravitational acceleration (9.8 m/s²)
- $h$ = Height above the ground

**Derivation:**

**Gravitational Force**: The gravitational force $F$ acting on an object of mass $m$ near the surface of the Earth is given by:

$$F=mg$$

where $g$ is the acceleration due to gravity.

**Work Done**: The work done in lifting the object to a height $h$ against the force of gravity is the product of the force and the distance moved:

$$W=F\cdot h$$

Substituting $F=mg$ into this equation:

**Potential Energy**: This work is stored as the gravitational potential energy $PE$ of the object:

$$PE=mgh$$

Thus, the formula for gravitational potential energy is:

**The sum of kinetic and potential energy in a system.3:-Mechanical Energy:**

**4:-Thermal Energy:**Energy possessed by an object due to the movement of its particles. This is commonly known as heat energy.

**5:-Chemical Energy:**Energy stored in chemical bonds, released during a chemical reaction (e.g., in fuels, food).

**6:-Electrical Energy:**Energy caused by the movement of electric charges (e.g., in circuits, batteries).

## [3]Law of Conservation of Energy

**Mechanical energy:-**The sum of kinetic energy and potential energy is called mechanical energy

**Conservation of Mechanical Energy**

## [4] Power

Pwer is the rate at which work is done or the rate at which energy is transferred. It tells us how fast energy is used or work is done in a system.

**Formula:**

**Where:**

- $P$is power (measured in
**Watts**) - $W$ is work (measured in
**Joules**) - $t$ is time (measured in
**seconds**)

**Unit of Power:**

- The SI unit of power is the
**watt**(W). - 1 watt = 1 joule/second (1 W = 1 J/s)

**Example:**

If a machine does 100 joules of work in 10 seconds, its power output is:

**Relation Between Power and Energy:**

Power is also the rate at which energy is consumed or converted from one form to another. If energy $E$ is consumed in time $t$:

**Types of Power:**

**Mechanical Power**: Related to the work done by a force (like lifting a load or moving an object).**Electrical Power**: Related to the energy consumed by electrical devices (P = VI, where V is voltage and I is current).

## [5]Commercial Unit of Power:

**commercial unit of energy**is the

**kilowatt-hour (kWh)**. It is commonly used by power companies to charge for electricity usage.

**Kilowatt**(kW) is commonly used for larger values of power.- 1 kW = 1000 W
**Kilowatt-hour**(kWh) is used to measure energy consumption by electrical devices.

**What is a Kilowatt-hour?**

**1 kilowatt-hour (kWh)**is the amount of energy consumed by a device with a power rating of 1 kilowatt (1000 watts) when it runs for 1 hour.**1 kWh**=**3.6 million joules**(1 kWh = 3.6 × 10⁶ J).

**Example: **

If a 1000 W (1 kW) appliance runs for 1 hour, it consumes 1 kWh of energy or 1 unt.

**Why kWh?**

The kWh is convenient for measuring household or industrial energy consumption, where the total energy used over time is more practical than using the smaller unit, the joule.

Energy bills are calculated based on the number of kilowatt-hours consumed over a billing period