Topics for Today Post-game analysis of first exam (Kletzing) Potential energy (8-1) Conservation of mechanical energy (8-2)

Work and Potential Energy We’ve seen how work (in the physics sense) can increase the kinetic energy of an object. Work can also be used to put in energy that is stored in a system. This is called “Potential Energy” Energy can be transferred between potential and kinetic. Start with ball at lowest point. Do work to move ball up. Release ball – energy changes from potential to kinetic and back.

Work and Potential Energy Quiz - If the pendulum ball is released from a given height, can it go higher on the return swing? Yes No Do demo 1M40.10 - Large Classroom Pendulum

Work and Potential Energy We use the symbol U for potential energy. The change potential energy ΔU = -W. Let’s see if this makes sense for a mass in a gravitational field. Quiz - When I lift the mass, does the potential energy? Increase Decrease

Work and Potential Energy Potential energy increases when I lift the mass. What is the work done when I lift the mass? The force (gravity) points down, while the displacement is up. Work 𝑊= 𝐹 ∙ 𝐷 =𝐹𝑑 cos 𝜃 where θ is the angle between 𝐹 and 𝑑 . In this case, the angle is 180° , and cos(180 °) = -1. So the work is negative, W < 0. The change potential energy ΔU = -W, is then positive which matches our intuition: the potential energy increases.

Calculating Potential Energy Recall that: 𝑊= 𝑥 𝑖 𝑥 𝑓 𝐹 𝑑𝑥 , so ∆𝑈=−𝑊=− 𝑥 𝑖 𝑥 𝑓 𝐹 𝑑𝑥 Let’s try this for gravity, where F = -mg. We define y so y increases as you go up, gravity then points to –y. So ∆𝑈=− 𝑦 𝑖 𝑦 𝑓 𝐹 𝑑𝑦 =− 𝑦 𝑖 𝑦 𝑓 −𝑚𝑔 𝑑𝑦 =𝑚𝑔 𝑦 𝑖 𝑦 𝑓 𝑑𝑦 ∆𝑈=𝑚𝑔 𝑦 𝑓 − 𝑦 𝑖 =𝑚𝑔∆𝑦 Define a reference point y = 0 and set U = 0 at that point, then 𝑈(𝑦)=𝑚𝑔𝑦 ← Gravitational potential energy

Conservation of Mechanical Energy Mechanical energy of a system is 𝐸=𝐾+𝑈 When a force does work on a system it changes the kinetic energy by ∆𝐾=𝑊 and the potential energy by ∆𝑈=−𝑊. The total mechanical energy changes by ∆𝐸=∆𝐾+∆𝑈=𝑊+ −𝑊 =0. → Mechanical energy is conserved. After releasing the pendulum, gravity does work on the mass, increasing the kinetic energy and decreasing the potential energy. On the upward portion, the work is negative, so the kinetic energy decreases and the potential energy increases.

Conservation of Mechanical Energy If a 10 kg pendulum ball is released at rest from a height of 1.5 meters above its lowest point, what is its maximum speed? Define y = 0 at lowest point. Energy 𝐸=𝑈+𝐾=𝑚𝑔𝑦+ 1 2 𝑚 𝑣 2 At start, y = 1.5 m, v = 0, so 𝐸 𝑖 =𝑚𝑔𝑦 Where does the maximum speed occur? At lowest point, y = 0, 𝐸 𝑓 = 1 2 𝑚 𝑣 2 𝐸 𝑖 = 𝐸 𝑓 → 𝑚𝑔𝑦= 1 2 𝑚 𝑣 2 𝑣= 2𝑔𝑦 = sqrt(2 × 9.8 m2/s2 × 1.5 m) v = 5.4 m/s

Potential Energy Potential energy is defined for a system. For the pendulum, the system is the mass and the Earth. Potential energy depends on the “configuration” of the system, meaning the arrangement of the parts of the system. For the pendulum, the “configuration” is the height of the mass (the distance of the mass from the center of the Earth). Potential energy arises because there are forces acting between the different parts of the system. For the pendulum, gravity between the mass and Earth. As the parts of the system move, the force does work, adding or subtracting potential energy.

Conservative Forces Imagine system starts in some configuration, is changed to another configuration by doing work W1, and is then changed back to the original configuration by doing work W2. Total work W = W1 + W2 Change in potential energy ΔU = 0 since initial and final states are same. Therefore ΔU = -W = W1 + W2 → W2 = -W1 Forces for which this is true are called “conservative forces”. Examples are gravity and spring forces. Potential energy is defined only for conservative forces. Forces for which this is false are called “non-conservative”. An example is kinetic friction. Energy lost to friction cannot be recovered by moving the object in the other direction.

Conservative Forces For conservative forces, the total work done going from (a) to (b) then back to (a) is zero. The work done around any closed path is zero. Therefore the work done going from (a) to (b) is independent of the path. This means you can substitute a simpler path for a more complex one. Even if the particle follows the path shown by the thin line, we can calculate the work using the dashed lines (which is much simpler).

Conservative Forces Quiz – Suppose you want to ride your bike up a steep hill. Two roads lead from the base to the top, one twice as long as the other. Compared to the average force you would exert on the long path, the average force you would exert on the short path is half same twice Ignore energy losses due to friction or drag.

Conservative Forces Quiz is the force conservative? Yes No

Conservative Forces Quiz – Two identical balls roll down two different tracks with the same starting and ending heights. The first track is straight. The second track curves up and down. How does the speed of the ball at the end of the first track compare with the speed of the ball at the end of the second track? faster same slower Ignore energy losses due to friction or drag.