Chapter 5 Work and Energy

Mechanical Energy  Mechanical Energy is the energy that an object has due to its motion or its position.  Two kinds of mechanical energy: –Kinetic Energy –Potential Energy  Energy is measured in [joules] = [N·m]

Work  Mechanical Work –A transfer of mechanical energy –Measured in [Joules] –In order for mechanical work to be done, a force must move an object through a displacement –Only the component of force in the direction of displacement contributes to work done

Work  Consider pushing a lawn mower with a force at an angle:  Force applied along handle.. Work = F·dcosθ

Work  Work is a scalar quantity measured in [Joules] = [N·m]  Work can be positive or negative (depending on cos θ)  Calculate Work using W = F·d·cosθ  Net Work on an object may be zero

Examples  Examples 5.1, 5.3 and 5.3….

When Force depends on Displacement  Often force is constant regardless of the displacement of an object… notice that Work is the area under a Force vs. displacement graph.  Sometimes the force becomes greater (or weaker) as displacement increases. Then work can only be calculated by finding the area under a Force vs. displacement graph.

Springs!  F r = -kx where k is the spring constant measured in [N/m] measured in [N/m] Since the force is not constant, work must be found using the area under the F vs x curve

Work – Energy Theorem for Kinetic Energy  K.E. = ½ mv 2  Work is a transfer of mechanical energy.  Work – Energy Theorem says Work = Δ KE  If positive work is done to increase an object’s speed, then the work done is equal to the change in kinetic energy of the object.  When negative work is done, an objects speed (kinetic energy) decreases.

Math Derivation  Consider a constant horizontal force moving an object horizontally:  We know that F = ma and a = (v f 2 – v i 2 )/2x So F = m (v f 2 – v i 2 )/2x So F = m (v f 2 – v i 2 )/2x  We calculate Work using Work = F·x·cosθ W = (m (v f 2 – v i 2 )/2x)·x W = (m (v f 2 – v i 2 )/2x)·x W = ½ mv f 2 – ½ m v i 2 W = ½ mv f 2 – ½ m v i 2

Potential Energy  Potential Energy is stored energy.  An object with potential energy has the potential to do work (potential to apply force).  Many objects have stored energy due to their position.  Work Done = ΔPE = PE f – PE i [Joules]

Gravitational Potential Energy  Consider doing work on an object by lifting it within the Earth’s gravitational field: –Work = F·d = mg(y f – y i ) –Work = mgy f – mgy i –Work= PE f – PE i

Potential Energy Formulas  Gravitational: PE = mgy g = 9.8 m/s 2  Spring:PE = ½kx 2 k = spring constant [N/m 2 ] [N/m 2 ] x is displacement

Examples  If a pendulum of mass 1.5kg swings to a height of 2.0 meters, what is the change in PE of the pendulum as it swings from its lowest point to its highest point?  A spring with a spring constant of 3.45 N/m 2 is compressed by 1.5 meters. How much work has been done on the spring?

Conservative vs. Non- Conservative Forces  A force is conservative if the work done by it in moving an object from one location to another is independent of the path taken. Ex. Gravity is conservative  A force is non-conservative if the work done by it in moving an object from one location to another is dependent on the path taken. Ex. Friction is non-conservative

Conservation of Mechanical Energy  In a conservative system (only conservative forces acting), the Total Energy of the System remains unchanged!  KE f + U f = KE 0 + U 0

Examples  5.8 – Calculate Energy  5.9 – Energy Transformations  5.11 – Conservation of Energy  5.12 – Energy of a Spring  5.13 / Non Conservative Forces

Ex 5.9  A 0.50 kg ball is thrown vertically upward with an initial velocity of 10m/s. What is the change in the balls kinetic energy between the starting point and the maximum height? What is the change in the balls potential energy between the starting point and the maximum height?  LOOK AT pg. 155 – reference points

Example 5.11  A painter on a scaffold drops a 1.5 kg can pf paint from a height of 6.00m. What is the kinetic energy of the can when the can is at a height of 4.00m? With what speed will the can hit the ground?

Example 5.12  Three balls of equal mass, m, are projected with the same speed, in different directions, from a cliff as shown. Neglecting air resistance, which ball would you expect to strike the ground with the greatest speed?

Example 5.14  A skier with a mass of 80kg starts from rest at the top of a slope and skis down from an elevation of 110m. The speed of the skier at the bottom is 20m/s. Show that the system is non-conservative. How much work is done by the force of friction?