Chapter 5 Work, Energy and Power What You Need to Know

Problem Types Work Work at an angle Kinetic Energy Gravitational Potential Elastic Potential Conservation Power Alta Physics

Work The Physics definition of work requires a displacement, i.e. an object must be moved in order for work to be done! The Applied force which causes the displacement contributes to the work, i.e. in order to contribute to the work, the applied force must be parallel to the displacement.

Work: A Mathematical Definition Work = (Force)(Displacement) Units of Work = (Newton)(Meter) 1 Newton•Meter = 1 Joule A Joule is a unit of Energy and it takes energy to do work and work done on an object either causes it to move (kinetic energy) or is stored (potential energy)

Sample Problem What work is done sliding a 200 Newton box across the room if the frictional force is 160 Newtons and the room is 5 meters wide? W = Fk • d = (160 N)(5 m) 800 Joules

Sample Problem 5A How much work is done on a vacuum cleaner pulled 3.0 m by a force of 50.0 N at an angle of 30.0o above the horizontal?

Sign of Work is Important Work is a scalar quantity and can be positive or negative 0o – 90o = (+) work 90o – 180o = (-) work 180o – 270o = (-) work 270o – 360o = (+) work Net work (+) the object speeds up Net force does work on the object Net work (-) the object slows down Work is done by the object on the other object

Energy Facts There are different types of energy Energy of all types is measured in Joules Law of Conservation of Energy – Energy can be neither created nor destroyed, merely changed from one form to another

Types of Energy Mechanical Potential Energy Kinetic Energy Heat Energy Energy of Position Gravitational Elastic Kinetic Energy Energy of Motion If it moves it has kinetic energy Heat Energy Heat is a form of Energy Transfer Other Forms of Stored Energy Chemical Fuels - usually release energy by combustion Food – energy released by digestion Electrical Generated from other forms of energy

Kinetic vs. Potential Energy

Energy of a Rollercoaster

Kinetic Energy Kinetic Energy is energy of Motion Any moving object has kinetic energy Dependent on the mass of the object and its velocity. Mathematically expressed as: KE = ½ mv2

Sample Problem What is the kinetic energy of a car with a mass of 2000 kg moving at 30 m/s? KE = ½ mv2 = (½)(2000 kg)(30 m/s)2 = 900,000 Joules

Sample Problem 5B A 7.00 kg bowling ball moves at 3.00 m/s. How much kinetic energy does the bowling ball have? How fast must a 2.45 g table-tennis ball move in order to have the same kinetic energy as the bowling ball?

Work-Kinetic Energy Theorem Work done by a net force acting on an object is equal to the change in the kinetic energy of the object. Wnet = ∆KE Wnet = KEf - KEi Must include all the forces that do work on the object when calculating the net work done.

Sample Problem 5C On a frozen pond, a person kicks a 10.0 kg sled, giving it an initial speed of 2.2 m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10?

Energy of Position: Gravitational Potential Energy Potential energy is stored energy Occurs due to the accelerating force of gravity Is determined by the position of the object in the gravitational field Mathematically determined by: PE = mgh where m is mass, g is the acceleration due to gravity and h is the height above a determined baseline.

Sample Problem What is the potential energy of a 10.0 kg rock sitting on a cliff 30.0 meters high? PE = mgh = (10 kg)(9.8 m/s2)(30 m) 2940 Joules

Elastic Potential Energy Bungee cords, rubber bands, springs any object that has elasticity can store potential energy. Each of these objects has a rest or “zero potential” position When work is done to stretch or compress the object to a different position elastic potential energy is stored

Elastic Potential Energy Top picture is “rest position”; x = 0 This is a point where the elastic potential energy = 0 Bottom picture is “stretched position” Here elastic potential energy is stored in the spring PE = ½ kx2 where k is the “spring constant” in N/m

Spring Constant Flexible spring = small spring constant Stiff spring = large spring constant

Where Does “K” Come From? K is measured in Newtons/meter. It is defined as the force required to displace a spring 1 meter. So: K = F/x Often K is determined by hanging a known weight from the spring and measuring how much it is stretched from its rest position.

Sample Problem What is the Elastic potential energy of a car spring that has been stretched 0.50 meters? The spring constant for the car spring is 90. N/m. PE = ½ kx2 = (½)(90 N/m)(0.5 m)2 =11 Joules

Sample Problem A spring is hung from a hook and a 10. Newton weight is hung from the spring. The spring stretches 0.25 meters. What is the spring constant? If this spring were compressed 0.50 meters, how much energy would be stored? If this spring were used to power a projectile launcher, which fires a 0.20 kg projectile, with what velocity would the projectile leave the launcher? Assume 0.50 m compression. (PE = KE)

Solution K = F/x K =10. N/0.25 m = 40 N/m PE = ½ Kx2 PE = ½ (40. N/m)(0.50 m)2 = 5.0 Joules PE = KE = ½ mv2 5.0 Joules = ½ (0.20 kg)(v2) V = 7.1 m/s

Sample Problem 5D A 70.0 kg stuntman is attached to a bungee cord with an unstretched length of 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m. When he finally stops, the cord has a stretched length of 44.0 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord. Assuming the spring constant of the bungee cord is 71.8 N/m, what is the total potential energy relative to the water when the man stops falling?

Work is Exchange of Energy 4/23/2008 Work is Exchange of Energy Energy is the capacity to do work Two main categories of energy Kinetic Energy: Energy of motion A moving baseball can do work A falling anvil can do work Potential Energy: Stored (latent) capacity to do work Gravitational potential energy (perched on cliff) Mechanical potential energy (like in compressed spring) Chemical potential energy (stored in bonds) Nuclear potential energy (in nuclear bonds) Energy can be converted between types Lecture 9 27

4/23/2008 Conversion of Energy Falling object converts gravitational potential energy into kinetic energy Friction converts kinetic energy into vibrational (thermal) energy makes things hot (rub your hands together) irretrievable energy Doing work on something changes that object’s energy by amount of work done, transferring energy from the agent doing the work Lecture 9 28

4/23/2008 Energy is Conserved! The total energy (in all forms) in a “closed” system remains constant This is one of nature’s “conservation laws” Conservation applies to: Energy (includes mass via E = mc2) Momentum Angular Momentum Electric Charge Conservation laws are fundamental in physics, and stem from symmetries in our space and time Aside: Earth is just another big ball: Obeys Newton’s laws * Force on superball by earth countered by force on earth * Earth accelerates towards dropped ball (F = ma) - tiny, tiny acceleration * Tries to continue in straight line, but deflected by Sun - acceleration changes direction of velocity vector Included in conservation laws * Dropped ball appears to get momentum out of nowhere - but earth’s motion towards ball counters with same momentum * Energy conservation must include huge input from sun - the activity around us gets its energy from the sun Lecture 9 29

Conservation of Energy Formula KEi + PEi = KEf + PEf Your book calls this the change in Mechanical Enegy (MEi = MEf) Kinetic/Potential Energy Before = Kinetic/Potential Energy After In a wild shot, Bo flings a pool ball of mass m off a 0.68 m high pool table, and the ball hits the floor with a speed of 6.0 m/s. How fast was the ball moving when it left the pool table?

Frank, a San Francisco hot dog vendor, has fallen asleep on the job. When an earthquake strikes, his 300 kg hotdog cart rolls up and down a street. It passes the top of hill A, which is 50.0 m high, at a speed of 8.00 m/s. How fast is the hot dog cart going to pass the top of the next hill ﴾B﴿, which is 30.0 m high when Frank finally wakes up and starts to run after it? A B 50.0 m 30.0 m

Sample Problem 5E Starting from rest, a child zooms down a frictionless slide from an initial height of 3.00 m. What is her speed at the bottom of the slide? Assume she has a mass of 25.0 kg.

Energy Conservation Demonstrated 4/23/2008 Energy Conservation Demonstrated Roller coaster car lifted to initial height (energy in) Converts gravitational potential energy to motion Fastest at bottom of track Re-converts kinetic energy back into potential as it climbs the next hill Lecture 9 33

Kinetic Energy The kinetic energy for a mass in motion is KE = ½mv2 4/23/2008 Kinetic Energy The kinetic energy for a mass in motion is KE = ½mv2 Example: 1 kg at 10 m/s has 50 J of kinetic energy Ball dropped from rest at a height h (PE = mgh) hits the ground with speed v. Expect ½mv2 = mgh h = ½gt2 v = gt  v2 = g2t2 mgh = mg(½gt2) = ½mg2t2 = ½mv2 sure enough Ball has converted its available gravitational potential energy into kinetic energy: the energy of motion Lecture 9 34

Kinetic Energy, cont. Kinetic energy is proportional to v2… 4/23/2008 Kinetic Energy, cont. Kinetic energy is proportional to v2… Watch out for fast things! Damage to car in collision is proportional to v2 Trauma to head from falling anvil is proportional to v2, or to mgh (how high it started from) Hurricane with 120 m.p.h. packs four times the punch of gale with 60 m.p.h. winds Lecture 9 35

Energy Conversion/Conservation Example 4/23/2008 Energy Conversion/Conservation Example 10 m P.E. = 98 J K.E. = 0 J Drop 1 kg ball from 10 m starts out with mgh = (1 kg)(9.8 m/s2)(10 m) = 98 J of gravitational potential energy halfway down (5 m from floor), has given up half its potential energy (49 J) to kinetic energy ½mv2 = 49 J  v2 = 98 m2/s2  v  10 m/s at floor (0 m), all potential energy is given up to kinetic energy ½mv2 = 98 J  v2 = 196 m2/s2  v = 14 m/s 8 m P.E. = 73.5 J K.E. = 24.5 J 6 m P.E. = 49 J K.E. = 49 J 4 m P.E. = 24.5 J K.E. = 73.5 J 2 m P.E. = 0 J K.E. = 98 J 0 m Lecture 9 36

Power Power = Work/time = Joules/Second (Watt) Mathematically there are two formulas for Power: or since then

Sample Problem What power is developed by a 55 kg person who does 20 chin ups, h = 3 m, in 45 seconds? P= w/t = FΔd/t = mgh/t (20(55 kg)(9.8 m/s2)(3 m))/45 sec = 718.6 Watts = 700 Watts

Sample Problem 5F A 193 kg curtain needs to be raised 7.5 m, at a constant speed, in as close to 5.0 s as possible. The power ratings for three motors are listed as 1.0 kW, 3.5 kW, and 5.5 kW. Which motor is best for the job?

QOD Mike pulls a sled across level snow with a force of 225 N along a rope that is 35.0o above the horizontal. If the sled moves a distance of 65.3 m, how much work did Mike do? A student librarian picks up a 22 N book from the floor to a height of 1.25 m. He carries the book 8.0 m to the stacks and places the book on a shelf that is 0.35 m high. How much work was done on the book? A 188 W motor can lift a load at the rate of 6.50 cm/s. What is the maximum load (kg) this motor can lift at this rate?