Work and Energy

Potential Energy Work is the measure of energy transfer .

Objectives: After completing this unit, you should be able to: Define kinetic energy and potential energy, along with the appropriate units. Describe the relationship between work and kinetic energy, and apply the WORK-ENERGY THEOREM. Define and apply the concept of POWER.

Energy is the capability for doing work. Energy is anything that can be converted into work; i.e., anything that can exert a force through a distance. The SI unit for energy is the joule (J) Energy is the capability for doing work.

Potential Energy Potential Energy: Ability to do work by virtue of position or condition. The SI unit for energy is the joule (J)

Potential Energy Potential energy is measured as a difference from an arbitrarily chosen reference point

Gravitational Potential Energy Example Problem: What is the potential energy of a 50-kg person in a skyscraper if he is 480 m above the street below? Gravitational Potential Energy What is the P.E. of a 50-kg person at a height of 480 m? U = mgh = (50 kg)(9.8 m/s2)(480 m) U = 235 kJ

Kinetic Energy Kinetic Energy: Ability to do work by virtue of motion. (Mass with velocity) Examples – A speeding car A bullet

Examples of Kinetic Energy What is the kinetic energy of a 5-g bullet traveling at 200 m/s? 5 g 200 m/s What is the kinetic energy of a 1000-kg car traveling at 14.1 m/s?

Work and Kinetic Energy A force acting on an object changes its velocity, and does work on that object. m vi vf d F

The Work-Energy Theorem Work is equal to the change in ½mv2 If we define kinetic energy as ½mv2 then we can state a very important physical principle: The Work-Energy Theorem: The work done by a force is equal to the change in kinetic energy that it produces.

Work to stop bullet = change in K.E. for bullet Example 1: A 20 g projectile strikes a mud bank, penetrating a distance of 6 cm before stopping. Find the stopping force F if the entrance velocity is 80 m/s. d F = ? 80 m/s 6 cm Work = ½ mvf2 - ½ mvi2 F d= - ½ mvi2 F (0.06 m) cos 1800 = - ½ (0.02 kg)(80 m/s)2 F = 1067 N F (0.06 m)(-1) = -64 J Work to stop bullet = change in K.E. for bullet

Work = DK Work = F(cos q) d d=25 m f F= mk.FN= mk mg Example 2: A bus slams on brakes to avoid an accident. The tread marks of the tires are 25m long. If mk = 0.7, what was the speed before applying brakes? Work = DK Work = F(cos q) d d=25 m f F= mk.FN= mk mg DK = ½ mvf 2 - ½ mvi 2 Work = - mk mg d vi= 2mkgd -½ mvi 2 = -mk mgd vi = 18.52 m/s vi = 2(0.7)(9.8 m/s2)(25 m)

Summary Potential Energy: Ability to do work by virtue of position or condition. Kinetic Energy: Ability to do work by virtue of motion. (Mass with velocity) The Work-Energy Theorem: The work done by a force is equal to the change in kinetic energy that it produces. Work = ½ mvf2 - ½ mvi2

Conservation of Energy Law of conservation of energy – the total energy of a system remains unchanged unless an outside force does work on the system, or the system does work on something outside the system.

Conservation of Energy The energy of a system can interconvert between potential and kinetic energy, but the sum of the two will always remain constant.

Conservation of Energy

Conservation of Energy

Conservation of Energy Sample problem1 : A 5.0 kg ball is dropped from a height of 10.0 meters. a. What is its velocity when it is 8.0 meters high? b. What is its velocity immediately prior to hitting the ground?

Conservation of Energy Solution: a. Before it is dropped the ball only has gravitational potential energy since there is no motion. Therefore: E=U+K=U+0=mgh=(10)(5)(9.81)= 490.5J=U

Conservation of Energy Solution (continued): (continued). At 8.0 meters the ball’s potential energy is U=(8)(5)(9.81)=392.4J Since its total energy is conserved, E=U+K and K=E-U K=490.5-392.4=98.1J

Conservation of Energy Solution: b. Right before the ball hits, h=0, U=0, and E=K+0=490.5J

Conservation of Energy Pendulum – the energy of a pendulum is also a result of work done against gravity. The pendulum is at equilibrium when the bob hangs straight down. As the bob is moved from equilibrium it rises and acquires potential energy.

Conservation of Energy

Conservation of Energy Pendulum – the potential energy of a pendulum displaced at an angle q from equilibrium is U=mgh = mgl(1-cos q) Where l is the length of the pendulum The energy of the system E is E=mgl(1-cos qmax) Where qmax is the angle at the maximum displacement from equilibrium.

Conservation of Energy As the pendulum swings it loses potential energy and gains kinetic energy. At the bottom of the swing U=0 and K=E

Conservation of Energy Example: A pendulum is 2.0m long, and has a bob that has a mass of 2.0 kg. The bob is pulled from the equilibrium position, so that the angle with the vertical axis θ=10o, and released. What is the potential energy of the system before the bob is released? b. What is the speed of the bob as it goes through the equilibrium position? c. What is the speed of the bob when the angle with the vertical is 5.0o?

Conservation of Energy Example: What is the potential energy of the system before the bob is released? U=mgl(1-cos qmax)=E=(2)(9.81)(2)(1-cos10o)=0.59J

Conservation of Energy Example: b. What is the speed of the bob as it goes through the equilibrium position? At equilibrium, U=0, and E=K

Conservation of Energy Example: c. What is the speed of the bob when the angle with the vertical is 5.0o? K=E-U U=mgl(1-cos q)=(2)(9.81)(2)(1-cos5.0o)=0.149J K=E-U=0.59J-0.149J=.441J

Conservation of Energy

Conservation of Energy After the mass is released, the spring stretches or compresses the velocity of the mass changes and the energy of the system interconverts between potential and kinetic energy. The maximum displacement from the equilibrium position is called the amplitude A.

Conservation of energy

Conservation of Energy

Conservation of Energy

Conservation of Energy

Conservation of Energy Example: A system consists of a spring attached to a 1.0kg mass that slides on a frictionless horizontal surface. The spring has a force constant of 200N/m. a. If the spring is stretched to a distance of 0.20m from the equilibrium position, what is the potential energy of the system? b. What is the speed of the mass when it is going through the equilibrium position? c. What is the speed of the mass when it is at a distance of 0.1 m from the equilibrium position?

Conservation of Energy Example: a. If the spring is stretched to a distance of 0.20m from the equilibrium position, what is the potential energy of the system? Potential energy = the work done on the system

Conservation of Energy Example: b. What is the speed of the mass when it is going through the equilibrium position? At the equilibrium position, U=0 and K=E

Conservation of Energy Example: c. What is the speed of the mass when it is at a distance of 0.1 m from the equilibrium position? At 0.1m from equilibrium E=U+K

Conservation of Energy During an elastic collision – when two moving objects collide, the kinetic energy of the system is the same before and after the collision. During an inelastic collision – when two moving objects collide, the kinetic energy of the system is not the same before and after the collision.