5-3: Conservation of Energy Objectives: Identify situations in which conservation of mechanical energy is valid Recognize the forms that conserved energy can take Solve problems using conservation of mechanical energy

Conserved Quantities If we have a certain amount of conserved quantity at some instant of time, we will have that same amount at a later time. This does not mean that it cannot change form during that time There are many forms of energy that it can change into. This is also true with mass.

Mechanical Energy is often conserved If we have a 75 g egg on a counter 1.0 meters above the ground, and it is knocked off we can use our equations from chapter 2 to solve for speed and acceleration at any given time. From there we can find the height When we have the height we can find PE g Knowing this we can find the KE Finally we can find the ME

Egg Falling Time (s)Height (m)Speed (m/s)PE g (j)KE g (j)ME (j) EquationΔy= Vf 2 / 2g 1.0- Δy V=Δtgmgh½mv 2 PE+KE This shows the conservation of energy. Energy has changed from potential to kinetic but remains at a constant of 0.74j Consider if the egg is thrown into the air. The kinetic energy will change to potential energy and then back to kinetic energy.

Conservation of Mechanical Energy Initial Mechanical Energy= Final Mechanical Energy ME i = ME f Alternate form: – 1/2mv i 2 + mgh i = 1/2mv f 2 + mgh f If other forces are present other than friction just add those potential energy terms associated with the force. – For instance if the egg were to stretch a spring as it falls we would add an elastic potential energy term to each side of the equation.

Starting from rest, a child zooms down a frictionless slide with an initial height of 3.00 m. What is her speed at the bottom of the slide? She has a mass of 25.0 kg.

PE=KE The initial potential energy will equal the final kinetic energy PE i = mgh PE i = (25.0 kg)(9.81 m/s 2 )(3.00 m) PE i = 736j KE f = 736j ½ mv f 2 = 736j ½ (25.0 kg)(v f 2 ) = 736j v f 2 = 58.9 v f = 7.67 m/s

V f 2 = 2mgh/ m The masses will cancel, which makes sense because acceleration does not depend on mass. So the final v f 2 = 2gh

Energy Conservation occurs even when acceleration varies If we ignore friction the acceleration does not matter. For example if a slide had different slope throughout, and we ignored friction, the shape of the slide has no affect on the problems ME i = ME f

Mechanical Energy is not conserved in the presence of friction When friction is present energy is lost in a non- mechanical form. Consider rubbing sandpaper on wood. You need to push with a high force to move the sandpaper against the kinetic friction. This energy is not “lost” it transforms to another form, heat. Total energy is always conserved. It may just turn to a form that we cannot measure as easily.

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