MInd map :: Chapter 11 : Work and Energy

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Scientific Conception of Work

Definition of Work in Science

• Work is said to be done when two conditions are satisfied:
  1. A force is applied on an object.
  2. The object is displaced in the direction of the applied force.
• If any of these conditions are not met, no work is done in the scientific sense.

Work Done by a Constant Force

• Formula: $$W = F \times s$$ where,
  • $W$ = Work done (Joule, J)
  • $F$ = Force applied (Newton, N)
  • $s$ = Displacement (meter, m)
• Work is a scalar quantity (has magnitude but no direction).

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SI Unit of Work

• The unit of work is Newton meter (N·m) or Joule (J).
• 1 Joule (J) = Work done when 1 Newton (N) force moves an object 1 meter (m) in the direction of the force. $$1J = 1N \times 1m$$

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Positive and Negative Work

• Positive Work:

  • When force and displacement are in the same direction.
  • Example:
    • Pulling a toy car parallel to the ground.
    • Lifting an object upwards (force exerted by hands).
    • A horse pulling a cart forward.

• Negative Work:

  • When force and displacement are in opposite directions.
  • Example:
    • A retarding force (friction) slowing down an object.
    • Gravity acting on an object being lifted.
    • A person pushing a box up a slope while friction resists it.
  • Work done is calculated as: $$W=-F\times s$$

Definition of Energy

• Energy is the capacity to do work.
• Any object that possesses energy can exert a force on another object and transfer energy to it.
• The object that loses energy does work, and the object that gains energy receives work.

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Sources of Energy

Natural Sources:

• The Sun 🌞 – The largest natural energy source.
• Nuclear Energy – Released from atomic nuclei.
• Geothermal Energy – Heat from the Earth's interior.
• Tidal Energy – Energy from ocean tides.

Unit of Energy

• Energy is measured in Joules (J), the same unit as work.
• 1 Joule (J) = Energy required to do 1 Joule of work.
• Larger unit: Kilojoule (kJ) = 1000 Joules (J).

• Full Name: James Prescott Joule

• Nationality: British

• Profession: Physicist

• Notable Contributions:

  1. Research in Electricity & Thermodynamics: Joule made significant contributions to understanding electricity and heat.
  2. Heating Effect of Electric Current: He formulated a law explaining how electric current produces heat.
  3. Law of Conservation of Energy: Joule experimentally verified this fundamental law, showing that energy cannot be created or destroyed, only transformed.
  4. Mechanical Equivalent of Heat: He discovered that mechanical energy can be converted into heat energy, leading to the concept of the mechanical equivalent of heat.

• Legacy: The unit of energy and work is named the joule (J) in his honor

Definition:

• Kinetic Energy (KE) is the energy possessed by an object due to its motion.
• It depends on the mass of the object and its velocity.
• The faster an object moves, the more kinetic energy it has.

Derivation of Kinetic Energy Formula

1. Work Done on the Object:

   • Consider an object of mass (m) moving with an initial velocity u.
   • The object is displaced by a distance s, and a force (F) acts on the object in the direction of displacement.

   The work done (W) is given by:

   $$W = F \times s \quad \text{(Equation 11.1)}$$

2. Equations of Motion:

   • The relationship between the initial velocity u, final velocity v, acceleration a, and displacement s is given by:
   $$v^2 - u^2 = 2as \quad \text{(Equation 8.7)}$$
   • Rearranging for s:
   $$s = \frac{v^2 - u^2}{2a} \quad \text{(Equation 11.2)}$$

3. Force and Acceleration:

   • From Newton's second law of motion, the force F acting on the object is:
   $$F = ma \quad \text{(Equation 9.4)}$$

4. Substitute for Acceleration (a):

   • From Equation (11.2), substitute s in the work equation (Equation 11.1):
   $$W = F \times s = ma \times \frac{v^2 - u^2}{2a}$$
   • Simplifying:
   $$W = \frac{m}{2} (v^2 - u^2) \quad \text{(Equation 11.3)}$$

5. Final Work Done if Object Starts from Rest:

   • If the object starts from rest, then u = 0:
   $$W=\frac{1}{2}m{v}^2\text{(Equation 11.4)}$$

Conclusion:

• The kinetic energy (KE) of an object is given by:

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

where:

• m is the mass of the object,
• v is the velocity of the object.

This shows that kinetic energy is directly proportional to both the mass and the square of the velocity.

Potential Energy: Notes

Definition:

• Potential Energy (PE) is the energy possessed by an object due to its position or configuration.
• This energy is stored in the object and can be released to do work.
• Common forms of potential energy include gravitational potential energy and elastic potential energy.

Gravitational Potential Energy:

• Definition: The gravitational potential energy of an object at a height is the energy it possesses due to its position relative to the Earth's surface.

• When an object is raised to a height, work is done against gravity. This work is stored as potential energy in the object.

Derivation of Gravitational Potential Energy Formula:

1. Work Done Against Gravity:

   • Consider an object of mass (m).
   • The object is raised to a height h above the ground.
   • The force required to raise the object is equal to its weight, which is mg (where g is the acceleration due to gravity).

   The work done to raise the object is:

   $$\text{Work done (W)} = \text{Force} \times \text{Displacement}$$

   Since the force required is equal to the weight (mg) and the displacement is the height (h), we have:

   $$W = mg \times h$$

2. Energy Gained by the Object:

   • The object gains energy equal to the work done on it.
   • Therefore, the energy gained is:
   $$E_{\text{p}} = mg \times h$$

   where Eₚ is the potential energy of the object.

3. Final Formula:
   The gravitational potential energy (Eₚ) is given by:

   $$E_{\text{p}} = mgh \quad \text{(Equation 11.7)}$$

   where:

   • m = mass of the object,
   • g = acceleration due to gravity,
   • h = height from the ground.

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Key Points:

• Potential Energy Depends on Height: The potential energy of an object depends on its height above the reference point (ground level).
• Reference Level: The potential energy of an object is relative to the reference level or the chosen ground level.
• Path Independence: The work done by gravity depends only on the vertical displacement (height difference), not the path taken.

Definition:

• Law of Conservation of Energy states that energy cannot be created or destroyed. It can only be converted from one form to another. The total energy of an isolated system remains constant during any transformation.

1. Transformation of Energy:

   • Whenever energy changes from one form to another (e.g., from potential energy to kinetic energy), the total amount of energy remains the same.
   • For example, when an object falls, its potential energy (due to height) gets converted into kinetic energy (due to motion).

2. Example (Free Fall of an Object):

   • At the start:
     • The object has potential energy equal to mgh (where m is mass and h is the height).
     • The object has zero kinetic energy because its initial velocity is zero.
   • As the object falls:
     • The potential energy decreases while the kinetic energy increases.
     • At any point, the sum of kinetic energy (½mv²) and potential energy (mgh) remains constant.
   • At the ground:
     • The potential energy becomes zero because the height h is zero.
     • The kinetic energy is maximum because the velocity is the highest.
   • The total mechanical energy is constant throughout the fall:
   $$\text{Potential Energy} + \text{Kinetic Energy} = \text{Constant}$$
   • This can be written as:
   $$mgh+\frac{1}{2}m{v}^2=\text{Constant}\text{(Equation 11.7)}$$

Mechanical Energy:

• The sum of kinetic energy (K.E.) and potential energy (P.E.) is called the total mechanical energy.
• As the object falls, the decrease in potential energy is equal to the increase in kinetic energy.
• Mechanical Energy is conserved during the free fall of an object (ignoring air resistance).

Definition of Power:
Power is the rate at which work is done or energy is transferred. It measures how fast or slow work is done. Power is defined as:

$$\text{Power} = \frac{\text{Work}}{\text{Time}} = \frac{W}{t}$$

Unit of Power:
The unit of power is watt (W).

• 1 watt = 1 joule/second (1 W = 1 J/s)
• 1 kilowatt (kW) = 1000 watts = 1000 J/s

Average Power:

If the power of an agent changes over time, average power is the total energy consumed divided by the total time taken.

$$\text{Average Power} = \frac{\text{Total Energy Consumed}}{\text{Total Time Taken}}$$