WORK,ENERGY and POWER

Work, Energy, and Power

Work

Work is defined as the product of force and displacement, mathematically expressed as:
W = F imes d
Where:

  • W = work done (measured in Joules)

  • F = force applied (measured in Newtons)

  • d = displacement (measured in meters)

Conditions for Work to be Zero

Work done is zero under the following conditions:

  1. If the displacement ($d$) is zero, i.e., no movement occurs, then:
    W = 0

  2. If the force ($F$) is perpendicular to the displacement ($d$), then:
    W = 0
    This indicates that no work is done when the force does not contribute to displacement in the direction of the force.

Example Calculation of Work

Example 1: Given a force of 70 N acting over a distance of 10 m:

  1. First, calculate the weight (force due to gravity) of the object:
    F = 70 imes 9.8 = 686 ext{ N}

  2. Work done is then:
    W = 686 imes 10 = 6860 ext{ J}

  3. If no movement occurs (displacement = 0), then:
    W = 0

Energy

Energy is defined in the context of mechanical energy, which is the sum of kinetic and potential energy.

Kinetic Energy (KE)

The kinetic energy can be calculated using the formula:
KE = rac{1}{2} mv^2
Where:

  • m = mass (in kg)

  • v = velocity (in m/s)

Gravitational Potential Energy (GPE)

Gravitational potential energy is given by the formula:
GPE = mgh
Where:

  • m = mass (in kg)

  • g = acceleration due to gravity (approximately $9.8 m/s^2$)

  • h = height above the reference point (in m)

Implications of Energy

  • Energy cannot be created or destroyed; it can only be transformed from one form to another. This is known as the law of conservation of energy.

  • For a body in free fall, the energy at the top (potential energy) is equal to the kinetic energy at the bottom, neglecting air resistance:
    GPE{ ext{top}} = KE{ ext{bottom}}
    If energy is conserved, then:
    mgh = rac{1}{2} mv^2

Example of Law of Conservation of Energy

In an example scenario:

  1. Height ( ext{h}) = 5 m,

  2. Thus, using gravitational potential and kinetic energy:

    1. Calculate GPE at 5 m height:
      GPE = m imes g imes h = m imes 9.8 imes 5

  3. Solve for velocity ($v$) at the bottom:
    v^2 = 2gh
    Thus,
    v = ext{sqrt}(2 imes 9.8 imes 5) = 9.9 m/s

  4. If some energy is wasted, assume GPE = 100 J and KE = 70 J remaining:

    • Wasted energy = Total energy - KE = 100 J - 70 J = 30 J

    • Work done against friction or resistance is thus 30 J.

Power

Power is defined as the rate of doing work, calculated as:
P = rac{W}{t}
Where:

  • P = power (measured in watts)

  • W = work done (in Joules)

  • t = time taken (in seconds)

Alternative Expressions of Power

  1. As a function of force and velocity:
    P = F imes v
    Where:

  • v = velocity (in m/s)

Efficiency

Efficiency is calculated as:
ext{Efficiency} = rac{ ext{Useful Output}}{ ext{Total Input}}
Example:

  • If 30 J of light energy is produced from 100 J of electrical energy, then:
    ext{Efficiency} = rac{30 J}{100 J} = 0.3 ext{ or } 30\%

Stanky's Diagram (Hypothetical Context)

In a generic energy transfer scenario involving electricity, heat, and light:

  • Total input is shown as 100 J, with specific outputs displayed, such as 30 J useful output and residual heat (70 J), indicating a 30% efficiency in energy use.
    This emphasizes the importance of understanding energy transformations in practical applications such as electrical systems.