1. Concepts in Work & Energy Part 2 – varying forces (not so basic)

2. Energy and Work Non-Uniform Force W = F d = F d cos θ  is fine for a constant force, but what if the force varies with the displacement? Calculus Notation Consider a bead pushed across a frictionless wire as shown to the right. We will apply a force (F) in the x-direction that will vary with position, x. So as the bead moves (through a displacement = d), the magnitude of the force doing work will change. d F F(x) x x1x1 x2x2 Graph 1 Graph 1 shows a plot of a (one dimensional) variable force applied through some displacement.

3. Energy and Work Non-Uniform Force Calculus Notation d F F(x) x x1x1 x2x2 Graph 1 Graph 1 shows a plot of a (one dimensional) variable force applied through some displacement. We would typically apply W=Fd where F is constant over d…BUT… F is NOT constant so we need to change our approach.

4. Energy and Work Non-Uniform Force Calculus Notation d F F(x) x x1x1 x2x2 Graph 2 On graph 2 we divide the area under the curve into a large number of very narrow strips of width ∆x. Choose ∆x small enough to permit us to take the force f(x) as (approximately) constant over that distance interval ∆x. ∆x This allows us to apply W=F ∆x for each small segment (recalling that F would be constant within that small segment). Allow animation to run.

5. Energy and Work Non-Uniform Force Calculus Notation d F F(x) x x1x1 x2x2 Graph 3 On graph 3 we will focus on just one of these rectangular segments. This will be the n th interval. ∆x If F n is constant over the interval ∆x, then we can find the work done over this (n th ) interval by FnFn

6. Energy and Work Non-Uniform Force Calculus Notation d F F(x) x x1x1 x2x2 Graph 3 ∆x FnFn NOTICE…that you have just multiplied the base and height of this particular rectangle. The product of base x height for a rectangle gives the area of that rectangle.

7. Energy and Work Non-Uniform Force Calculus Notation d F F(x) x x1x1 x2x2 Graph 2 ∆x FnFn We don’t want the work done by just this single interval…. …we want the work done by all of these segments together.

8. Energy and Work Non-Uniform Force Calculus Notation d F F(x) x x1x1 x2x2 Graph 2 ∆x So, to find the work done over the entire interval we reduce the width of the strip (let ∆x → 0) and sum the areas (under the curve). This process of summing the areas is known as integration and gives us: Let the animation run!

9. Energy and Work Non-Uniform Force Calculus Notation Oh my goodness! We have talked about integration (and differentiation) before. If you are feeling a little rusty on these concepts then revisit the derivative and integration PowerPoints. They are posted online. Otherwise…let’s keep going

10. Energy and Work Non-Uniform Force What if you have a non-uniform force that is not parallel to the displacement? Really? That’s just mean! It’s no big deal, just use the idea of integration with the concept of the scalar product of vector multiplication. W =  F dx Calculus Notation We’ll do this together in class

11. Energy and Work Non-Uniform Force – Springs (the perfect example) Calculus Notation If we stretch the spring by applying a force to the right, the spring exerts a restoring force to the left. This restoring force will increase as we continue to stretch the spring. Thus the force will vary. If we compress the spring by applying a force to the left, the spring exerts a restoring force to the right. This restoring force will increase as we continue to compress the spring. Thus the force will vary. We want to look at how much work is done when we stretch or compress a spring. In both cases the force is changing with ∆x. Figure C 1 : Relaxed PositionFigure C 2 : Stretched PositionFigure C 3 : Compressed Position ΔxΔx ΔxΔx F F

12. Energy and Work Non-Uniform Force – Springs (the perfect example) Calculus Notation The magnitude of the restoring force (of the spring) at a given displacement from equilibrium (the relaxed position, x = 0) is: Figure C 1 : Relaxed PositionFigure C 2 : Stretched PositionFigure C 3 : Compressed Position ΔxΔx ΔxΔx F F k = spring constant (N/m) Indicates the stiffness of the spring. The stiffer the spring (the harder to stretch or compress) the higher k. x = the displacement of the spring (from x=0) F = F R = restoring force This is the force that the spring exerts in attempting to return the spring to its equilibrium position (x=0). Hooke’s Law

13. Energy and Work Non-Uniform Force – Springs (the perfect example) Calculus Notation The magnitude of the restoring force (of the spring) at a given displacement from equilibrium (the relaxed position, x = 0) is: Figure C 1 : Relaxed PositionFigure C 2 : Stretched PositionFigure C 3 : Compressed Position ΔxΔx ΔxΔx F F Watch your signs! Consider figures C1 and C2. As the block/spring is pulled to the right (∆x is to the right) the spring is pulling to the left! Hooke’s Law

14. Energy and Work Non-Uniform Force – Springs (the perfect example) Calculus Notation In order to calculate the work done by the spring (force) as it is displaced from equilibrium we: Figure C 1 : Relaxed PositionFigure C 2 : Stretched PositionFigure C 3 : Compressed Position ΔxΔx ΔxΔx F F Let: x i = initial location x f = final location Recall that we would need to SUM all of the small amounts of work done by a (nearly) uniform force over VERY small (∆x → 0) displacements….or... INTEGRATE! Set the limits…it is being stretched from x o to x f. The force is given by Hooke’s Law. Why negative? Because the SPRING force is in the opposite direction of the displacement (x)

15. Energy and Work Non-Uniform Force – Springs (the perfect example) Calculus Notation In order to calculate the work done by the spring (force) as it is displaced from equilibrium we: Figure C 1 : Relaxed PositionFigure C 2 : Stretched PositionFigure C 3 : Compressed Position ΔxΔx ΔxΔx F F If the spring is initially at equilibrium (xi=0) and we displace it away from there, then this simplifies to k is a constant.