1 Honors Physics 1 Class 10 Fall 2013 Potential Energy Partial Derivatives The Gradient

2 Conservative forces and Potential Energy If the work line integral between two points is independent of the path taken, then the force is conservative.

3 What potential energy tells us about force

4 Honors Physics 2- - Persans Spring The force can be found from the potential energy function: introducing the gradient

5 Honors Physics 2- - Persans Spring The gradient The gradient points in the direction of maximum change. Roller skate and board The funny looking d in the derivative means to take the derivative with respect to that variable only, while treating all other variables as constants. This type of derivative is called the partial derivative.

6 Honors Physics 2- - Persans Spring Maple files showing the gradient Click on each of the images above to open a Maple file showing a contour plot and the gradient of the surface. Note that the gradient always points “uphill”. If we want the downhill direction, we add a minus sign.

7 Gravitational attraction by a point mass.

8 Uniform Gravitational Field

9 Gravitational force due to 2 masses

10 Honors Physics 2- - Persans Spring Useful properties of the gradient If we want to know the rate of change along a particular direction, then we just find the dot product of with a unit vector in the direction of interest: (directional derivative). The gradient of the electric potential is the electric field: The gradient of the potential energy of a particle is the force on a particle The gradient of the temperature is proportional to the heat flow Because the gradient points in the direction of the force, it tells you which way a particle will accelerate!

11 The height of a half pipe as a function of position is given by: Find the force on a snowboarder of mass 50 kg at position x=1, y=2. The snowboard is pointed in a direction with the unit vector. Find the acceleration of the board. Honors Physics 2- - Persans Spring Activity y x z

12 Honors Physics 2- - Persans Spring Useful properties of the gradient Since the gradient has no component in the direction in which the potential is a constant, it must be perpendicular to any constant potential surface. So the gradient is useful for finding the normal to a surface.

13 How to tell if a force is conservative 1)Is the integral around a closed path zero? We’ll do more with this idea next. 2)Is the force the result of taking the gradient of a single function of position?

14 A formalism for testing path independence

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16 A test for conservative force