Lecture on Calculating Instantaneous Power

Key Concepts

• Instantaneous Power: The power of a machine is determined by how fast work is done over a period of time.
• Work Done: The work done by a machine is time-dependent and not a constant value.
• Power Definition: Power is defined as the rate of change of work done.

Formula for Power

• Power is calculated using the formula: P = W/t
  • This is the first derivative of the work done with respect to time.

Example Explained

• Scenario: Calculating the power when time = 5 seconds.
• Comparison Example: Two people running from point A to B:
  • One takes 2 minutes (less power)
  • One takes 20 seconds (more power)
  • Despite the same work done, the power differs due to time variation.

Calculation Steps

1. Take the Derivative:
   • Given work done: Work = 2t^3 + 4t^2 + 5t
   • Derivative formula used:[ frac{d(x^n)}{dx} = nx^{n-1} ]
2. Derivative Calculation:
   • frac{d(2t^3 + 4t^2 + 5t)}{dt} = 6t^2 + 8t + 5
   • Constants are handled separately.
3. Calculate Instantaneous Power at t = 5 seconds:
   • Substitute t = 5 into the derivative:
     - 6(5)^2 + 8(5) + 5
     - Result: 195 watts
4. Power at Different Times:
   • At t = 0 seconds, substitute t = 0:
     - Result: 5 watts
   • Shows power is not constant over time.

Summary

• Instantaneous power varies with time and is not a constant.
• By calculating the derivative of work done over time and substituting specific time values, the power at those moments can be determined.

Additional Notes

• If there are questions, viewers are encouraged to comment.
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