Lecture 11: Potential Energy & Energy Conservation

Questions of Yesterday 1) You slam on your brakes in a panic and skid a certain distance d down a straight and level road before coming to a stop. If you had been traveling twice as fast, what would the skidding distance be? a) 2d b) d/2 c) 4d d) d/4 2) As a pendulum swings back and forth, the forces acting on the pendulum are the force of gravity and tension in the supporting cord. Which of these forces does no work on the pendulum? a) Gravity b) Tension c) neither one does work on the pendulum d) they both do work on the pendulum

Gravitational Potential Energy There is a 10 m high and a 2 m high diving board. You know that the faster you are going when you hit the water the bigger splash you will make. Which board will you jump off of to make the biggest splash? Which board gives you the greatest kinetic energy when you hit the water? Is work done on you as you fall? How do you know? What is doing work on you? The higher up you are the greater the potential you have to gain kinetic energy by gravity doing work on you

Gravitational Potential Energy Potential energy is the property of a system with a set “zero point” or “ground level” Potential Energy always refers to some object at some certain point in space relative to the “ground” Instead of looking at the work done by gravity on an object during any vertical displacement… we can look at GRAVITATIONAL POTENTIAL ENERGY (PE) or the energy associated with the potential for gravity to do work on an object Choosing your reference frame is important!

Gravitational Potential Energy Does the work done on an object by gravity increase or decrease as it falls? What about the potential energy? distance fallen (m) Energy (J) Potential Energy Work done by Gravity

Gravitational Potential Energy Does the work done on an object by gravity increase or decrease as it falls? What about the potential energy? distance fallen (m) Energy (J) Potential Energy Work done by Gravity The change in Potential Energy between two points = The negative of the work done by gravity through that displacement

Gravitational Potential Energy What is the work done by gravity on the book from height y i to y f ? yiyi yfyf W g = (F g cos  y W g = mgcos    y i - y f )  PE = -W g  PE = mg(y f - y i )

Gravitational Potential Energy PE = mgy y = vertical distance (height) of an object from the Earth’s surface or some other “zero” point g = magnitude of acceleration due to gravity Units kg*m 2 /s 2 = Joule (J) Scalar Quantity Always Positive!

Gravitational Potential Energy PE = mgy Recall the work energy theorem… W nc + W c =  KE W g = mg  y i - y f ) W nc =  KE +  PE

Reference Levels, y = 0 PE = mgy When calculating PE you can choose “y = 0” to be at any vertical height in your system Only the CHANGE in PE matters! yiyi yfyf y = 0 W nc =  KE +  PE

Energy Conservation If only conservative forces are acting on a system… Energy is Conserved! The sum of the kinetic energy and potential energy of an isolated system remains CONSTANT at all times! W nc =  KE +  PE KE i + PE i = KE f + PE f when W nc = 0

Mechanical Energy In any isolated system of objects interacting only through conservative forces, the total mechanical energy E = KE + PE of the system remains the same at all times! Kinetic Energy (KE) + Potential Energy (PE) = Mechanical Energy (E) KE i + PE i = KE f + PE f E = KE + PE E i = E f

Energy Conservation Energy conservation when gravity is the only conservative force acting on the system KE i + PE i = KE f + PE f (1/2)mv i 2 + mgy i = (1/2)mv f 2 + mgy f PE = mgy KE = (1/2)mv 2

Practice Problem Assuming the ramps are frictionless and the balls of equal mass start from rest…. Which ball has a greater speed when it reaches the bottom? If the ramps have friction (equal coefficients of friction)… would your answers change? h h 30 o 60 o Which ball reaches the bottom first?

Practice Problem A 70-kg diver steps off a 10-m tower and drops from rest straight down into the water. If he comes to rest 5.0 m beneath the surface, determine the average resistive force exerted on him by the water.

Practice Problem Starting from rest, a 10.0 kg block slides 4.00 m down to the bottom of the frictionless ramp inclined 30.0 o. The block then slides an additional 5.00 m along the floor before coming to a stop. Determine: -The speed of the block at the bottom of the ramp -The coefficient of kinetic friction between the block and floor -The mechanical energy lost due to friction What if the coefficient of friction between the ramp and the block was 0.5?

Questions of the Day 1) A 50-kg student starting from rest slides down a frictionless waterslide of height 10 m while a 100-kg student slides down a similar slide that is only 5 m high. Which student is going faster when they reach the bottom? a) the 50-kg student b) the 100-kg student c) they are going the same speed 2) A women pulls a crate up a rough (with friction) inclined plane at a constant speed. Which statement is NOT true? a) The work done on the crate by the normal force of the inclined plane on the crate is ZERO b) The work done on the crate by gravity is ZERO c) The work done by the net force on the crate is ZERO d) The gravitational PE is increasing