Work-Energy Relationship Already understand that lifting an object will increase the object’s gravitational potential energy. W=ΔPE= mgΔh No friction

Conservative Forces – When work is done against a force is independent of the path taken. – Examples: force of gravity and elastic force of a spring. – Only look at the initial position and then the final position. – Potential energy only has meaning in relation to work done against conservative forces.

Nonconservative Forces – When work is done against a force is dependent of the path taken. – Examples: air resistance and friction Look at Figure Below: Because of friction, more work is required to move a box on incline B than on incline A. Why? The force of friction is greater for smaller angles (incline B) because the normal force is greater (F f = μF N ). The force of friction acts over a greater distance on incline B. B A

Conservation of Energy In a closed system, the sum of the potential energies (PE g and/or PE s ), kinetic energy, and internal energy remains constant. E T = PE + KE +Q Problem: What is the internal energy, if the total energy of the closed system is 25 J, the kinetic energy is 10 J and the potential energy is 12 J?

Law of Conservation of Energy Energy cannot be created or destroyed. The sum of the changes in energy within a closed system is zero.  PE g +  PE s +  KE +  Q = 0  = final – initial PE gi + PE si + KE i + Q i = PE gf + PE sf + KE f + Q f

Apply the Law of Conservation of Energy PE gi + PE si + KE i + Q i = PE gf + PE sf + KE f + Q f Initial position A: What do you know? Final position B: What do you know? Final position C: What do you know ?

Ideal Mechanical Systems Total Mechanical Energy (TME) is the sum of the kinetic and potential energies. An Ideal Mechanical System is a closed system with no friction or other nonconservative forces acting on an object. TME = constant = KE + PE g + PE s  KE +  PE g +  PE s = 0 Note: Q, Internal Energy is not considered

Examples of Ideal Mechanical System Simple Pendulum consists of a mass (bob) attached to a string that is attached at the other end to a pivot point. When KE is max, PE is zero. When PE is max, KE is zero. PE=KE when mass is ½ h. The TME is constant at every position.

Examples of Ideal Mechanical System Free Falling Object in a vacuum As the object falls, its gravitational potential energy decreases, but because its speed increases the kinetic energy increases.  KE +  PE g = 0 If KE increases, then  KE is + If PE g decreases, then  PE g is - ½ mv 2 = mgh If h is given, then speed can be determined Speed is independent of mass!

Nonideal Mechanical Systems When a system is acted upon by a nonconservative force (friction). Friction converts some or all of the kinetic energy to internal energy. Recognize by: Increases the temperature of the objects in contact. Pendulum does not swing back to its original position. Dropped object has less KE when it hits the ground than its initial PE g. Dropped object hits ground, but does not bounce (inelastic collision), energy converted to internal energy. E T = PE + KE +Q

Roller Coaster Type Example Problem If a 53 kg cart is moving at 3 m/s at the top of a hill that is 140 meters above the ground, how fast is the cart moving when it is 95 meters above the ground some time later? ½ mv A 2 + mgh A = ½ mv B 2 + mgh B