Mechanics Work and Energy Chapter 6

Work  What is “work”?  Work is done when a force moves an object some distance  The force (or a component of the force) must be parallel to the object’s motion W = F ║ d W = Fdcosθ Work is measured in Joules (J); 1 J = 1 N · m  Work is the bridge between force (a vector) and energy (a scalar)

Work  SI unit for Work & Energy:  Joule (N · m)  1 Joule of work is done when 1 N acts on a body, moving it a distance of 1 meter  Other units for Work & Energy:  British: foot-pound  Atomic Level: electron-Volt (eV) ← we’ll use this later!

Work  A 5-N force pushes a box 1-m. How much work was done?  A 5-N force pushes a box, but the box doesn’t budge. How much work was done?  A 5-N force pushes upward on a box, and the box moves 1-m to the right. How much work was done?

Work  There is NO WORK done by a force if it causes NO DISPLACEMENT!  Forces perpendicular to displacement can do no work. The normal force and gravity do no work when an object is slid on a flat floor, for instance.  Forces can do positive, negative, or zero work

Work  A person pulls a rolling suitcase at an angle of 30 ° with the horizontal, with a force of 200 N. How much work does she do to pull it 160 m along a flat surface?

More Work Practice  Jane uses a vine wrapped around a pulley to lift a 70-kg Tarzan to a tree house 9.0 meters above the ground  How much work does Jane do when she lifts Tarzan?  How much work does gravity do when Jane lifts Tarzan?

Work & Energy  Work transfers energy to an object or a system  If a force does positive work on a system, the mechanical energy of the system increases  If a force does negative work on a system, the energy of the system decreases  The two forms of mechanical energy are Potential Energy and Kinetic Energy

Kinetic Energy  Moving objects have Kinetic Energy. K = ½ mv 2 K is measured in Joules (J)

Constant Force and Work  If force is constant over the distance traveled:  W = F Δ r can be used to calculate the work done by the force when it moves an object some distance r  For a Force vs. distance graph, the area under the curve can be used to calculate the work done by the force  This is true even if force is not constant!

Work and graphs The area under the curve of a graph of force vs displacement gives the work done by the force in performing the displacement. F(x) x xaxa xbxb

The Work-Energy Theorem  W net =  KE  When net work due to all forces acting on an object is positive, the kinetic energy of the object will increase (positive acceleration).  When net work due to all forces is negative, the kinetic energy of the object will decrease (deceleration).  When there is no net work due to all forces acting on an object, the kinetic energy is unchanged (constant speed).

Kinetic Energy  A 10.0 g bullet has a speed of 1.2 km/s.  What is the kinetic energy of the bullet?  What is the bullet’s kinetic energy if the speed is halved?  What is the bullet’s kinetic energy if the speed is doubled?

Work & Energy  A 0.25-kg ball falls for 5 seconds.  What force does work on the ball?  Find the work done on the ball after  1.0 second, 3.0 seconds, and 5.0 seconds  Find the kinetic energy of the ball after  1.0 second, 3.0 seconds, and 5.0 seconds

Work & Energy  Where did the ball get the energy to speed up?  Potential Energy (PE or U) is energy stored in an object from its position  The ball had stored energy due to its height

Potential energy  Energy an object possesses by virtue of its position or configuration.  Represented by the letter U.  Examples:  Gravitational Potential Energy  Spring Potential Energy

Energy  Gravitational Potential Energy (PE g – measured in Joules): energy stored in any object that has the ability to fall PE g = mgh h is the height of the object Find the gravitational potential energy of the falling ball if it was originally 10.0 m above the ground.

Energy  Elastic Potential Energy: stored in objects that can stretch or compress  It takes force to stretch or compress a spring: F P = kx, where k is the spring constant, or resistance to stretching, and x is the distance stretched/compressed  Remember Hooke’s Law! The force of the spring is opposite the direction of displacement F = -kx

Springs  A spring does NEGATIVE WORK on an object, since it pushes or pulls opposite the direction of stretch/compression  The force doing the stretching/compressing does positive work, equal but opposite the work done by the spring

Springs: stretching mm x 0 F app = kx F(N) x (m) W app = ½ kx 2 F

Springs:compressing mm x 0 F app = kx F(N) x (m) W app = ½ kx 2 F

Spring Practice  It takes 180 J of work to compress a certain spring 0.10 m  What is the force constant of the spring?  To compress the spring an additional 0.20 m, does it take 180 J, more than 180 J, or less than 180 J? Verify your answer with a calculation.

More Spring Practice/Review  A physics student hangs various masses on a spring using a kg hanger. He determines the spring constant to be 18.2 N/m. He then hangs a kg mass on the spring, and a few seconds later, the mass falls off and the hanger is propelled upward by the restoring force of the spring.  Find the energy stored in the spring when it is stretched.  When it is stretched, what force does it exert on the mass and hanger?  When the hanger is launched upward, it has kinetic energy. Where did that energy come from?

Energy Review  Moving objects have kinetic energy K = ½ mv 2  Objects at some height have gravitational potential energy PE g = mgy  Compressed/Stretched objects have elastic potential energy Elastic PE = ½ kx 2

Power  Power is the rate at which work is done  Remember: Work is a transfer of energy!  P = W/ Δ t  W: work in Joules  Δ t: elapsed time in seconds  P = F V  (force )(velocity)  The SI unit for Power is the Watt (W)  1 Watt = 1 Joule/second  The British unit is horsepower (hp)  1 hp = 746 W

Power  The rate of which work is done.  When we run upstairs, t is small so P is big.  When we walk upstairs, t is large so P is small.

Power Practice Problem  A record was set for stair climbing when a man ran up the 1600 steps of the Empire State Building in 10 minutes and 59 seconds. If the height gain of each step was 0.20 m, and the man’s mass was 70.0 kg, what was his average power output during the climb? Give your answer in both watts and horsepower!

Work & Energy Force Types

 Forces acting on a system can be divided into two types according to how they affect potential energy:  Conservative forces can be related to potential energy changes  Non-conservative forces cannot be related to potential energy changes

Conservative and Nonconservative Forces  Forces like friction “use up” energy. It cannot be recovered later as kinetic energy. It is converted to other forms of energy (like heat)  Work done by a nonconservative force cannot be recovered later as kinetic energy.  Nonconservative forces are “path dependent” - knowing starting and ending points is not sufficient – you have to know the total distance traveled

Conservative and Nonconservative Forces  Other forces CAN be recovered as kinetic energy later, and are Conservative Forces.  Gravity is also a conservative force. Gravitational potential energy is stored in objects and can be released at a later time.  Conservative forces are “path independent”  Work can be calculated from the starting and ending points – the actual path can be ignored

Law of Conservation of Energy  In any isolated system, the total energy remains constant  Energy can neither be created nor destroyed, but can only be transferred to other objects or transformed from one type of energy to another

Law of Conservation of Mechanical Energy

Pendulums and Energy Conservation  Energy goes back and forth between K and U  At the highest point, all energy is U  As it drops, U transforms into K  At the bottom, energy is all K

Pendulums:  A 5.0-kg swinging pendulum encounters a frictional force from air resistance. The pendulum is released from rest at a height of 0.50 m above its lowest point. After making one complete swing forward and back, the pendulum only reaches a height of 0.49 m. What amount of mechanical energy was lost to air resistance?

Springs and Energy Conservation  Energy is transformed back and forth between K and U  When fully stretched, all energy is U  When passing through equilibrium, all energy is K  At other points, energy is a mixture of U and K

Mechanical Energy  Along an ideal rollercoaster (with no friction) the mechanical energy of the car will always remain constant.  Realistically, frictional forces transform kinetic energy into thermal energy.  Mechanical energy is not conserved, but friction does work to transform KE into heat

Nonconservative Forces  The work done by a nonconservative force is equal to the change in mechanical energy: W NC = ΔKE + Δ PE  The work done by the frictional force is: W NC = -F fr d  So, Δ KE + Δ PE = -F fr d

Energy Conservation in Oscillators (general)  K + U = constant  K 1 + U 1 = K 2 + U 2  ΔK = -ΔU

Energy Conservation in Springs  K 1 + U 1 = K 2 + U 2  K = 1 / 2 mv 2  U = 1 / 2 kx 2 x

 K 1 + U 1 = K 2 + U 2  K = 1 / 2 mv 2  U = mgh Energy Conservation in Pendulums h