Work and Energy Class 9 Notes Science Chapter 10

After Understanding the laws of motion, We need to understand the reaction of forces on motion. Here, we will gain an understanding of terms like Work, Power(Rate of doing work)  and Energy.

Studying Work, Power, and Energy is crucial as it helps us understand the fundamental principles governing the conversion and transfer of energy in various systems. This knowledge is essential for designing efficient machines, optimizing industrial processes, and developing sustainable energy solutions. 

Flowchart

What is Work?

In everyday life, we often use the term "Work" to describe activities that involve physical or mental effort. However, according to the scientific definition of work, it may not always align with our common understanding.

Example: In the context of pushing a rock, even if we exert a lot of effort and get exhausted, if the rock doesn't move, No work is done according to the scientific definition.

Scientific Conception of Work

Scientifically, work is defined as the application of force on an object resulting in its displacement. Let's consider a few situations to better understand this concept:

• Pushing a pebble: When you push a pebble and it moves a certain distance, you have exerted force on the pebble and it has been displaced. Therefore, work is done in this situation.
• Pulling a trolley: If a girl pulls a trolley and it moves, both the force applied by the girl and the displacement of the trolley meets the conditions for work. Hence, work is done in this scenario.
• Lifting a book: When you lift a book by applying a force, the book moves upwards, satisfying both conditions for work. Therefore, work is done in this case.

Work is defined as the product of the constant force applied on an object and the displacement of the object in the direction of the applied force. 

For work to be done, two essential conditions must be met:

1. Application of a force on the object
2. Displacement of the object in the direction of the force.

 Mathematically, Work (W) can be calculated as :

Work Done  =  F x S

 where,

• F represents the constant force applied. 
• S represents the Displacement along the direction of the force.

SI unit of work is called joule (J or Nm). 

Work has only a magnitude and no sense of direction i.e., it is a scalar.

Example 1: A force of 10 Newtons is applied to an object, causing it to be displaced by 5 meters. What is the work done on the object?Solution: 

We can use the formula: W = F x S
Given:
Force (F) = 10 Newtons
Displacement (S) = 5 meters
Putting these values into the equation, we have:
W = (10 N) x (5 m) = 50 Joules

Therefore, the work done on the object is 50 Joules.

Work is said to be 1 joule if under the influence of a force of 1 N the object moves through a distance of 1 m along the direction of applied force.

Example 2:  A porter lifts a luggage of 15 kg from the ground and puts it on his head 1.5 m above the ground. Calculate the work done by him on the luggage. 

Mass of luggage, m = 15 kg and displacement, s = 1.5 m.
Work done, W = F × s = mg × s = 15 kg × 10 m s^-2 × 1.5 m
= 225 kg m s^-2 m
= 225 N m
= 225 J
Work done is 225 J  

Force at an Angle

When a force is applied at an angle to the direction of displacement, only a part of the force causes motion. The formula to calculate work in such cases is:  Work = Force × Distance × cos(θ)

Work Done at an Angle

where:

• Force is the magnitude of the constant force applied
• Distance is the displacement of the object in the direction of the force
• θ is the angle between the force  and the displacement .

If the force and displacement are in the same direction (θ = 0), the formula simplifies to: Work = Force × Distance. This indicates that work done by a constant force is equal to the product of the force applied and the distance over which the force acts.

Example 3: A box is pushed with a force of 50 N at an angle of 30° to the horizontal. If the box moves 10 m, calculate the work done.

Given:

• Force (F) = 50 N
• Angle (θ) = 30°
• Distance (d) = 10 m

Formula: $Work\backslash text\{Work\}\; =\; F\; \backslash times\; d\; \backslash times\; \backslash cos(\backslash theta)$Work=F×d×cos(θ)

Step-by-step solution:

Answer: The work done is approximately 433 J (Joules).

Positive, Negative & Zero Work Done

• Positive Work: Work is considered positive if the displacement of the object is along the direction of force applied.  Example: Work done by a man is taken positive when he moves from the ground floor to the second floor of his house.
• Negative Work: On the other hand, work is taken as negative if the displacement of the object is in a direction opposite to the direction of force applied. Example: But work done by the man is negative when he is descending from second floor of house to ground floor.

Positive and Negative Work Done

• Zero Work Done: If the displacement of an object is in a direction perpendicular to the application of force, work done is zero inspite of the fact that force is acting and there is some displacement too.
  Example: Imagine pushing a lawn roller forward. While gravity pulls it downward, the roller moves horizontally. Since gravity acts perpendicular to the roller's movement, it does no work. This is an example of zero work, where a force doesn't contribute to the displacement. 

What is Energy?

An object having the capability to do work is said to possess energy. Hence, the energy of an object is defined as its capacity of doing work. Energy of an object is measured by the total amount of work done by the object.

Energy Transformation

Unit of energy is same as the unit of work. So, SI unit of energy will be joule (J). Energy too has magnitude only.

Forms of Energy

Energy has many forms. Some important forms of energy are mechanical energy, heat energy, electrical energy, light energy, chemical energy, nuclear energy etc.

Mechanical energy is of two kinds, namely, 

1. Kinetic energy
2. Potential energy

Potential and Kinetic Energy

Kinetic Energy

For an object of mass m and having a speed v, the kinetic energy is given by:

F = ma

Also, W=Fs

From the second equation of motion, we know that

v² - u² = 2as

Rearranging the equation, we get

Substituting equation for work done by a moving body,

Taking initial velocity as zero, we get
where:

• E_k is the kinetic energy.
• m is the mass of the object.
• v is the velocity of the object

Note : When two identical bodies are in motion, the body with a higher velocity has more KE.

Example 4: An object of mass 15 kg is moving with a uniform velocity of 4 m s^-1. What is the kinetic energy possessed by the object?

Solution: The kinetic energy (KE) possessed by an object can be calculated using the formula:

FormulaIn this case, the mass of the object is given as 15 kg and the velocity is given as 4 m/s. Plugging these values into the formula, we can calculate the kinetic energy:

KE = (1/2) * 15 kg * (4 m/s)² = 0.5 * 15 kg * 16 m²/s²

        = 120 Joules (J)

Therefore, the object possesses a kinetic energy of 120 Joules.

Example 5: What is the work required to increase the velocity of a car from 30 km/h to 60 km/h, given the mass of the car is 1500 kg? 

Potential Energy

An object gains energy when it is lifted to a higher position because work is done against the force of gravity. This energy is known as gravitational potential energy. Gravitational potential energy is defined as the work done to raise an object from the ground to a certain height against gravity.

Let's consider an object with a mass m being lifted to a height h above the ground.

The minimum force required to lift the object is equal to its weight, which is mg  (where g is the acceleration due to gravity).

The work done on the object to lift it against gravity is given by the formula:

Work Done (W) = Force × Displacement

W = mg × h = mgh

The energy gained by the object is equal to the work done on it, which is mgh units. 

This energy is the potential energy (E_p) of the object.

E_p = mgh

Note: It's important to note that the work done by gravity depends only on the difference in vertical heights between the initial and final positions of the object, not on the path taken to move the object. For example, if a block is raised from position A to position B by taking two different paths, as long as the vertical height AB is the same (h), the work done on the object is still mgh.

Example 6: An object with a mass of 2 kilograms is lifted to a height of 10 meters. What is its gravitational potential energy?
Solution: 

We can use the equation Ep = mgh, where:

• Ep represents the gravitational potential energy,
• m represents the mass of the object,
• g represents the acceleration due to gravity,
• h represents the height.

Given:
m = 2 kg, h = 10 meters
g = 9.8 m/s² (approximate value for acceleration due to gravity on Earth)
Putting these values into the equation, we have:
E_p = (2 kg) x (9.8 m/s² )x (10 m) = 196 Joules
Therefore, the gravitational potential energy of the object is 196 Joules

Example 7:  Find the energy possessed by a 10 kg object at a height of 6 m above the ground. Given $g\; =\; 9.8\; \backslash ,\; \backslash text\{m/s\}^2$g=9.8m/s². 

Solution:

• Mass (m) = 10 kg
• Height (h) = 6 m
• Acceleration due to gravity (g) = 9.8 m/s²

Using the formula for potential energy:
$Potential energ=mgh\backslash text\{Potential\; energy\}\; =\; mgh$
$=10\times 9.8\times 6=\; 10\; \backslash times\; 9.8\; \backslash times\; 6$
= 588J

Law of Conservation of Energy

According to the law of conservation (transformation) of energy, we can neither create nor destroy energy. 

• Energy can be changed from one form to another. But the total energy remains unchanged. Energy lost in one form exactly reappears in some other form so that total energy remains unchanged.
• The law of conservation of energy is valid in all situations and for all sorts of transformations. Thus, like conservation law of momentum, the conservation law of energy too is a fundamental law of nature that never fails.
• For a freely falling object, there is a continual transformation of gravitational potential energy into kinetic energy such that their sum remains constant at all points during the fall.

Example of Law of Conservation of Energy

Law of Conservation

The above diagram shows a pendulum, which consists of a mass (m) connected to a fixed pivot point via a string of length (L).

Here's a description of the pendulum at different positions to explain the law of conservation of energy:
1. At the highest point (A): At this point, the pendulum is momentarily at rest and all its energy is potential energy (PE). The height (h) of the mass above the lowest point determines the amount of potential energy. PE = m * g * h, where g is the acceleration due to gravity.
2. At the lowest point (B): As the pendulum swings down, its potential energy is converted into kinetic energy (KE). At the lowest point, its height (h) is zero, so it has no potential energy. At this point, all its energy is kinetic energy, and the pendulum is moving at its highest velocity. KE = 0.5 * m * v², where v is the velocity of the mass.
3. At the highest point on the other side (C): As the pendulum swings upward, its kinetic energy is converted back into potential energy.

Rate of doing Work or Power

The rate of doing work or the rate of transfer of energy is known as the Power.
∴  Power = work / time

formula

• Power means speed with which a machine does work or the speed at which a machine supplies energy.
• SI unit of power is a watt. Power is said to be 1 watt (1 W) if rate of doing work is 1 Js^-1.
• Some other common units of power are:
  1 kilowatt = 1 kW = 1000 W = 1000 J s^-1
  1 megawatt = 1 MW - 10⁶ W - 10⁶ J s^-1
  1 horse power = 1 hp = 746 watt.

Example 8: A boy with a mass of 50 kg runs up a staircase with 45 steps in 9 seconds. If each step is 15 cm high, find his power. Given $g\; =\; 10\; \backslash ,\; \backslash text\{m/s\}^2$g = 10m/s². 

Average Power

Understanding Average Power

• Sometimes, an agent (like a machine or a person) does work at different speeds at different times. Because of this, it’s helpful to talk about average power.
• Average power is calculated by taking the total energy used and dividing it by the total time taken. This gives us a single number that represents the overall power used, even if the rate of work varied.

• Average Power = Total energy consumed(or total work done) / Total time

Old NCERT Topic: Commercial Unit of Power

• For electrical energy used in households and commercial/industrial establishments we generally use a bigger unit of energy, which is known as 1 kilowatt hour (1 kW h). 
• One kW h is the total energy consumed in 1 hour at the rate of 1 kW (or 1000 W or 1000 J s^-1).
  1 kWh = 3.6 x 10⁶ J