Mechanical Energy

Net Work  The work-energy principle is  K = W net.  The work can be divided into parts due to conservative and non-conservative forces. Kinetic energy  K = W con + W nonKinetic energy  K = W con + W non FfFf d FgFg

Kinetic and Potential Energy  Potential energy is the negative of the work done by conservative forces. Potential energy  U =  W conPotential energy  U =  W con  The kinetic energy is related to the potential energy. Kinetic energy  K =  U + W nonKinetic energy  K =  U + W non  The energy of velocity and position make up the mechanical energy. Mechanical energy E mech = K + UMechanical energy E mech = K + U

Conservation of Energy  Certain problems assume only conservative forces. No friction, no air resistanceNo friction, no air resistance The change in energy,  E =  K +  U = 0The change in energy,  E =  K +  U = 0  If the change is zero then the total is constant. Total energy, E = K + U = constantTotal energy, E = K + U = constant  Energy is not created or destroyed – it is conserved.

Springs and Conservation  The spring force is conservative. U = ½ kx 2U = ½ kx 2  The total energy is E = ½ mv 2 + ½ kx 2E = ½ mv 2 + ½ kx 2  A 35 metric ton box car moving at 7.5 m/s is brought to a stop by a bumper.  The bumper has a spring constant of 2.8 MN/m. Initially, there is no bumper E = ½ mv 2 = 980 kJ Afterward, there is no speed E = ½ kx 2 = 980 kJ x = 0.84 m v x

Energy Conversion  A 30 kg child pushes down 15 cm on a trampoline and is launched 1.2 m in the air.  What is the spring constant?  Initially the energy is in the trampoline. U = ½ ky 2  Then the child has all kinetic energy, which becomes gravitational energy. U = mgh  The energy is conserved. ½ ky 2 = mgh k = 3.1 x 10 4 N/m

Solving Problems  There are some general techniques to solve energy conservation problems. Make sure there are only conservative forces and kinetic energy in the problemMake sure there are only conservative forces and kinetic energy in the problem Identify all the potential and kinetic energy at the beginningIdentify all the potential and kinetic energy at the beginning Identify all the potential and kinetic energy at the endIdentify all the potential and kinetic energy at the end Set the initial and final energy equal to one anotherSet the initial and final energy equal to one another next