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calculate-work-done-by-variable-force

🚀 The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. When dealing with variable forces, the work done can be calculated by integrating the force over the distance moved. This is particularly important in physics as it allows us to analyze systems where forces change with position, such as springs or objects moving through a fluid.

Theory Explanation

Understanding Work Done by a Variable Force

When a force varies with position, the work done by that force can be calculated using the integral of the force function over the distance. If F(x) is the force as a function of position x, and the object moves from position x1 to x2, the work done W is given by the integral W = ∫(x1 to x2) F(x) dx.

\[ W = \int_{x_1}^{x_2} F(x) \, dx \]
Calculating Work Done

To calculate the work done by a variable force, follow these steps: 1. Identify the force function F(x). 2. Determine the limits of integration (the initial and final positions). 3. Integrate the force function over the specified limits to find the work done.

Example of Work Done by a Variable Force

Consider a spring where the force exerted by the spring is given by Hooke's Law, F(x) = kx, where k is the spring constant. The work done in stretching the spring from x1 to x2 can be calculated using the integral of kx from x1 to x2.

\[ W = \int_{x_1}^{x_2} kx \, dx \]

Key Points

  • 🎯 Work done is the integral of force over distance for variable forces.
  • 🎯 The limits of integration are crucial in determining the work done.
  • 🎯 Understanding the force function is essential for accurate calculations.

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Examples:💡

Calculate the work done by a variable force F(x) = 3x^2 from x = 1 to x = 3.

Solution:

Step 1: Identify the force function F(x) = 3x^2 and the limits of integration x1 = 1, x2 = 3.

Step 2: Set up the integral for work done: W = ∫(1 to 3) 3x^2 dx.

\[ W = \int_{1}^{3} 3x^2 \, dx \]

Step 3: Calculate the integral: W = [x^3] from 1 to 3 = (3^3) - (1^3) = 27 - 1 = 26.

Common Mistakes

  • Mistake: Students often forget to set the correct limits of integration when calculating work done.

    Correction: Always double-check the initial and final positions before integrating.

  • Mistake: Confusing the force function with the work done; they are not the same.

    Correction: Remember that the work done is the integral of the force function, not the force itself.

  • Mistake: Not applying the correct units when calculating work done.

    Correction: Ensure that all quantities are in consistent units (e.g., Newtons for force, meters for distance).

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