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Aim: How do we explain the work done by a varying force?

Explore how to calculate work done by both constant and varying forces using calculus, integral evaluation, and practical examples. Understand the concept in spring systems and its implications on work calculations. Also, delve into kinetic energy and the work-energy theorem.

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Aim: How do we explain the work done by a varying force?

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  1. Aim: How do we explain the work done by a varying force?

  2. Work done by a constant force Using our definition of work, we can see that the work done by the force can be calculated by finding the area under the curve.

  3. Work done by a varying force How can we find the work done by a force which varies over space and time?

  4. Definition of work (with calculus) W = ʃ F dx Work can be found by evaluating the integral with force over an interval on xf to xi

  5. Example 1- Calculating work from a graph Calculate the work done as the particle shown below moves from xi = 0 m to xf = 5m

  6. Thought Question 1 The four graphs below show the variable force F versus the position x of a particle on which the force acts. Rank the graphs according to the work done by force F on the particle from xi =0 to xf from most positive work to most negative work. B=Most Positive Work A=Zero Work C=Negative Work D=Most Negative Work

  7. Thought Question 2 The graph below shows the force F directed along the x-axis that will act on the particle at corresponding values x. a) What is the particle’s coordinate when it has its greatest kinetic energy? At x =3 m b) Its greatest speed? At x =3 m c) Its speed is zero? At x= 6m d) What is the particle’s direction of travel when it is at x=6m? It comes to a stop and then begins to move backwards

  8. Example 2:Calculating work from a graph A mass of 3 kg is initially compressed 0.2m and then released. a) Calculate the work done as it moves from xi = 0.2 m to xf = 0m. b) Find its speed at x = 0m. Calculate the work done

  9. Spring Systems are examples of systems with varying forces

  10. Spring System F = -kx (Hooke’s Law) Spring systems are physically systems which involve a variable force. Using calculus, how can we determine the work done on a spring? W=ʃ Fdx = ∫(kx)dx = ½ kx 2

  11. 3. Work Required to Stretch a Spring One end of a horizontal spring (k= 80 N/m) is held fixed while an external force is applied to the free end, stretching it slowly from xa= 0 to xb = 4 cm. • Find the work done by the external force on the spring. • Find the additional work done in stretching the spring from xb = 4cm to xc = 7cm

  12. 4. Work Required to Stretch a Spring If an applied force varies with position according to Fx = 3x2 – 5, where x is in meters, how much work is done by this force on an object that moves from x = 4.00 m to x=7.00m?

  13. 5.Work done by a varying force A force F = (4xi + 3yj) acts on an object as the object moves in the x direction from the origin to x=5.00 m. Find the work done on the object by the force.

  14. Work-Kinetic Energy Theorem (Curved Paths) Even if the path is curved, the work-energy theorem holds.

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