Physics I Class 09 Potential Energy and Conservation of Energy Rev. 09-Feb-06 GB 09-1 Work (Review) Work is a measure of the energy that a force puts into (+) or takes away from (–) an object as it moves. Work is a scalar, calculated with a dot product if force is constant: W F d F d cos( ) Fx d x Fy d y Fz d z For varying forces in one dimension: xf W F dx xi 09-2 Work Integral in Multiple Dimensions In multiple dimensions, the work integral looks like this: xf W F dx xi What does it mean to “dot” the force with the variable of integration? F dx xi path of integration xf 09-3 Does work depend on the path? Conservative Forces For general forces, the work does depend on the path that we take. However, there are some forces for which work does not depend on the path taken between the beginning and ending points. These are called conservative forces. A mathematically equivalent way to put this is that the work done by a conservative force along any closed path is exactly zero. F d x cons 0 (The funny integral symbol means a path that closes back on itself.) 09-4 Conservative Forces Non-Conservative Forces Examples of Conservative Forces: Gravity Ideal Spring (Hooke’s Law) Electrostatic Force (later in Physics 1) Examples of Non-Conservative Forces: Human Pushes and Pulls Friction 09-5 How Conservative Forces Help Us Calculate Work If a force is conservative, then the work it does on a particle that moves between two points is the same for all paths connecting those points. This is handy to know because it means that we can indirectly calculate the work done along a complicated path by calculating the work done along a simple (for example, linear) path. Also, if we have a closed path (return to start), then W = 0. 09-6 Conservative Forces and Potential Energy If we are dealing with a conservative force, we can simplify the process of calculating work by introducing potential energy. 1. Define a point where the potential energy is zero (our choice). 2. Find the work done from that point to any other point in space. (This is not too hard for most conservative forces.) 3. Define the potential energy at each point as negative the work done from the reference point to there. Call this function U. 4 The work done by the conservative force from any point A to any point B is then simply W = U(A)–U(B). 09-7 Conservative Forces and Potential Energy Point B: U(B) F dx 0 U(A) F dx A B Point A: 0 W (0 A ) W (0 B) Point 0: U = 0 (defined) W (A B) W (A 0) W (0 B) W (0 A ) W (0 B) U (A ) U (B) 09-8 Two Common Potential Energy Functions in Physics 1 Gravitational Potential Energy U g m g (y y0 ) m g h (y0 is our choice to make the problem easier) Spring Potential Energy U s 12 k ( x x 0 ) 2 (x0 is the equilibrium position and k is the spring constant) 09-9 Potential Energy, Kinetic Energy, and Conservation of Energy Recall the Work-Kinetic Energy Theorem: K Wnet And for conservative forces we have Wcons U If the non-conservative forces are zero or negligible, then Wnet Wcons Putting it together, K U or | K U 0| Another way to say this is the total energy, K+U, is conserved. 09-10 Example Problem Skateboarder Going Up a Ramp K0 U mgh K 12 m v 2 U0 d v h m v2 0 0 m g h v2 h v2 h d 2g sin( ) 2 g sin( ) 1 2 09-11 Class #09 Take-Away Concepts 1. Multi-dimensional form of work integral: xf W F dx xi 2. 3. Conservative force = work doesn’t depend on path. Potential Energy defined for a conservative force: U ( A ) F dx A 0 Ug m g ( y y0 ) m g h U s 12 k ( x x 0 ) 2 4. Gravity: 5. 6. Spring: Conservation of energy if only conservative forces operate: K U or | K U 0 | 09-12 Class #09 Problems of the Day ___1. The sum of potential and kinetic energy for a system of moving objects is conserved A) only when no net external force acts on the objects. B) only when the objects move along closed paths. C) only when the work done by the resultant external force is zero. D) only when non-conservative forces do no work on the system. E) none of the above 09-13 Class #09 Problems of the Day 2. A marble with mass 5 g is placed on a vertical spring that has been compressed 5 cm from its equilibrium position. The spring constant is 196 N/m. Neglecting friction, air resistance, and the mass of the spring, how high does the marble go when the spring is released, counting the marble’s position on the compressed spring as zero height? Use g = 9.8 N/kg. Hint: Make sure you use consistent units. 09-14 Activity #09 Conservation of Energy Objective of the Activity: 1. 2. Use LoggerPro to study mechanical energy in a simple system. Consider how kinetic energy, potential energy, and total mechanical energy vary with position. 09-15