Work and Energy

Work

  • Definition: Work is defined as the force ( F ) multiplied by the distance ( d ) moved in the direction of the force.

    • Formula: W = F imes d

  • Unit of Work: Joule (1J)

    • It can also be expressed as: 1J = 1N imes 1m = 1Nm

  • Energy Definition: Energy is the ability to do work.

    • Conceptually, energy can be seen as "stored work".

Work Example

  • Scenario: Calculating the work done in moving a mass ( m ) from the floor to a shelf 2m high.

    • Height increase: h = 2 ext{ m}

    • Force of Gravity: Acting in a downward direction, it's negative relative to the height increase.

    • Force Required: The force to lift the mass must equal the gravitational force acting on it.

    • Work done: W = F imes d = mg imes h

    • Thus, W = mgh

    • This work is stored in the form of Potential Energy.

Potential Energy

  • Definition: Potential Energy (PE) is the energy a body has due to its position in a force field.

    • Example context: From the previous example, the PE gained when moving the mass against gravity is equal to the work done to lift it onto the shelf.

    • Formula: PE = mgh

Kinetic Energy

  • Definition: Kinetic Energy (K.E.) is the energy a body possesses due to its motion.

  • Example Change in Energy: When the mass falls from the shelf:

    • Use the equation of motion: v^2 = u^2 + 2as

    • Initial velocity ($u$) is 0.

    • Rearranging gives acceleration: a = rac{v^2}{2s}

    • Work done during the fall is expressed as: W = mv^2 / 2

    • Thus, the energy due to motion upon falling is K.E.

Conservation of Energy

  • Principle: Energy can neither be created nor destroyed; it only changes forms.

  • Isolated System: Refers to a system where matter does not interact with the outside universe, allowing energy to remain constant within the system.

  • Universal Energy: The total energy in the universe is constant and is transformed among different forms, maintaining the conservation principle.

Conservation of Mechanical Energy

  • Example Scenario: Find the maximum speed of a stone dropped from a height of h = 10m.

    • Potential Energy at the top: PE = mgh

    • Kinetic Energy just before hitting the sea: KE = \frac{1}{2} mv^2

    • Applying conservation of energy: PE{top} = KE{bottom}

    • Therefore: mgh = \frac{1}{2} mv^2

    • Cancel out mass (
      m herm): gh = \frac{1}{2} v^2

    • Solve for velocity (
      v herm): v = \sqrt{2gh} = \sqrt{2 \times 9.8 \frac{m}{s^2} \times 10 m} = 14 m/s

Work Equation

  • The work done by a force is given by the equation: W = F \times d

    • $W$: Work, measured in Joules (J)

    • $F$: Force applied, measured in Newtons (N)

    • $d$: Distance moved in the direction of the force, measured in meters (m)

    This equation implies that to perform work, a force must cause an object to move a certain distance.

Potential Energy Equation

  • Potential Energy is given by: PE = mgh

    • $PE$: Potential Energy, measured in Joules (J)

    • $m$: Mass of the object, measured in kilograms (kg)

    • $g$: Acceleration due to gravity, approximately 9.8 rac{m}{s^2}

    • $h$: Height above a reference point, measured in meters (m)

    This equation indicates that the potential energy increases as either the mass of the object or the height increases.

Kinetic Energy Equation

  • Kinetic Energy is expressed by: KE = \frac{1}{2} mv^2

    • $KE$: Kinetic Energy, measured in Joules (J)

    • $m$: Mass of the object, measured in kilograms (kg)

    • $v$: Velocity of the object, measured in meters per second (m/s)

    This represents how kinetic energy is directly related to the mass of an object and the square of its velocity, indicating that an increase in speed will lead to a rapid increase in kinetic energy.

Conservation of Energy Principle

  • The conservation of mechanical energy equation can be summarized as: PE{top} = KE{bottom}

    • This states that the potential energy at the highest point (e.g., when an object is dropped) is equal to the kinetic energy just before it hits the ground.

Velocity Calculation

  • To find the velocity of the object at the bottom of its drop, the equation derived from the conservation of energy is: gh = \frac{1}{2} v^2

    • Rearranging gives: v = \sqrt{2gh}

    • This shows how the height from which an object falls affects its final velocity just before impact, as it emerges from the balance between potential and kinetic energy.