Potential Energy and Conservation of Energy Work Done by Gravity If one lifts an object of mass m from the floor yi 0 to a height yf h you have done work on the object W F cos y mg y f y i mgh We have imparted energy to it but it is at rest v 0 So this energy is not kinetic energy It is called Potential Energy PE or U or in this particular case gravitational potential energy U is energy that is stored and which can be converted to another kind of energy K for example U g mgy U is a scalar with units of J in S I h is the height above some reference point e g table floor Conservation of Mechanical Energy Total mechanical energy is constant within some specified system total energy is conserved conservation principles are very important in physics we will see many others later E K U E is always constant but K and U can change If U and K change they must change in such a way as to keep E constant Example Consider the 1D free fall of an object of mass m from a height of yi h y m y h t 0 v 0 i i i y h t v 0 system The initial energy of the system defines the total energy U i mgh K i 0 E K U mgh System the collection of objects being study to the exclusion of all other objects in the surroundings in this example we consider the object of mass m only Some time later K and U have changed but E has not E K U mgh 12 mv 2 mgy Energy E U mgy K mgh mgy mg h y What is K U and v when y h 2 K 0 h h h U mgy mg K mg h mg 2 2 2 h y h 1 K mg 2 mv 2 2 v gh 9 80 sm2 1 00 m 3 13 ms E mgh 1 00 kg 9 80 sm2 1 00 m 9 80 J h E U K mg 4 90 J 2 2 For y 0 just before the object hits the ground U mgy 0 E K mgh 9 80 J 1 2 2 mv mgh v 2 gh 2 9 80 sm2 1 00 m 4 43 ms Note we have neglected air resistance and what happens when the object hits the ground Example Problem A particle starting from point A is projected down the curved runway Upon leaving the runway at point B the particle is traveling straight upward and reaches a height of 4 00 m above the floor before falling back down Ignoring friction and air resistance find the speed of the particle at point A 4 00 m B vi A 3 00 m Example Problem A grappling hook attached to a 1 5 m rope is whirled in a circle that lies in the vertical plane The lowest point on this circle is at ground level The hook is whirled at a constant rate of three revolutions per second In the absence of air resistance to what maximum height can the hook be cast Method use concepts of conservation of mechanical energy and uniform circular motion
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