Chapter 5: Energy Energy Energy is present in a variety of forms: mechanical, chemical, electromagnetic, nuclear, mass, etc. Energy can be transformed from one from to another. The total amount of energy in the Universe never changes. If a collection of objects can exchange energy with each other but not with the rest of the Universe (an isolated system), the total energy of the system is constant. If one form of energy in an isolated system decreases, another form of energy must increase. In this chapter, we focus on mechanical energy: kinetic energy and potential energy.

Work when the object is displaced by Dx by the force: The work W done on an object by a constant force F when the object is displaced by Dx by the force: SI unit: joule (J) = newton-meter (N m) = kg m2/s2 Work is a scalar quantity. If the force exerted on an object is not in the same direction as the displacement: component of the force along the direction of the displacement dot product or inner product

Work a force exerted, then no work is done. If an object is displaced vertical to the direction of a force exerted, then no work is done. If an object is displaced in opposite direction to that of an exerted force, the work done by the force is negative (if F<Fg).

Work The friction force between two objects in contact and in relative Work and dissipative forces The friction force between two objects in contact and in relative to each other always dissipate energy in complex ways. Friction is a complex process caused by numerous microscopic interactions over the entire area of the surfaces in contact. The dissipated energy above is converted to heat and other forms of energy. Frictional work is extremely important: without it Eskimos can’t pull sled, cars can’t move, etc.

Work Example 5.1: Sledding through the Yukon (a) How much work is done if q=0? m=50.0 kg F= 1.20x102 N Dx=5.00 m (b) How much work is done if q=30o?

Work Example 5.2: Sledding through the Yukon (with friction) (a) How much work is done if q=0? m=50.0 kg F= 1.20x102 N Dx=5.00 m fk=0.200 (b) How much work is done if q=30o?

Kinetic Energy Kinetic energy (energy associated with motion) Consider an object of mass m moving to the right under action of a constant net force Fnet directed to the right. (constant acceleration) Define the kinetic energy KE as: SI unit: J work-energy theorem

Kinetic Energy Example 5.3: Collision analysis m=1.00x103 kg An example Example 5.3: Collision analysis m=1.00x103 kg vi = 35.0 m/s -> 0 =8.00x103 N (a) The minimum necessary stopping distance? (b) If Dx=30.0 m what is the speed at impact?

Kinetic Energy Conservative and non-conservative forces Two kinds of forces: conservative and non-conservative forces Conservative forces : gravity, electric force, spring force, etc. A force is conservative if the work it does moving an object between two points is the same no matter what path is taken. It can be derived from “potential energy”. Non-conservative forces : friction, air drag, propulsive force, etc. In general dissipative – it tends to randomly disperse the energy of bodies on which it acts. The dispersal of energy often takes the form of heat or sound. The work done by a non-conservative force depends on what path of an object that it acts on is taken. It cannot be derived from “potential energy”. Work-energy theorem in terms of works by conservative and non- conservative force

Gravitational Potential Energy Gravitational work and potential energy Gravity is a conservative force and can be derived from a potential energy. Work done by gravity on the book:

Gravitational Potential Energy Gravitational work and potential energy Gravity is a conservative force and can be derived from a potential energy. Let’s define the gravitational potential energy of a system consisting of an object of mass m located near the surface of Earth and Earth as: y : the vertical position of the mass to a reference point ( often at y=0 ) g : the acceleration of gravity SI unit: J where

Gravitational Potential Energy Reference levels for gravitational potential energy As far as the gravitational potential is concerned, the important quantity is not y (vertical coordinate) but the difference Dy between two positions. You are free to choose a reference point at any level (but usually at y=0). yi yf

Gravitational Potential Energy Gravity and the conservation of mechanical energy When a physical quantity is conserved the numeric value of the quantity remains the same throughout the physical process. When there is no non-conservative force involved, Define the total mechanical energy as: The total mechanical energy is conserved. In general, in any isolated system of objects interacting only conservative forces, the total mechanical energy of the system remains the same at all times.

Gravitational Potential Energy Examples Example 5.5: Platform diver (a) Find the diver’s speed at y=5.00 m. (b) Find the diver’s speed at y=0.0 m.

Gravitational Potential Energy Examples Example 5.8: Hit the ski slopes (a) Find the skier’s speed at the bottom (B). (b) Find the distance traveled on the horizontal rough surface.

Spring Potential Energy Spring and Hooke’s law Force exerted by a spring Fs Hooke’s law If x > 0, Fs <0 If x < 0, Fs >0 Fs to the left x>0 Fs to the right k : a constant of proportionality called spring constant. SI unit : N/m Fs The spring always exerts its force in a direction opposite the displacement of its end and tries to restore the attached object to its original position. Restoring force

Spring Potential Energy Potential due to a spring The spring Fs is associated with elastic potential energy. Between xi -1/2Dx and xi+1/2Dx the work exerted by the spring is approximately: Between x=0 and x, the total work exerted by the spring is approximately: -Fs width = Dx xi-1/2Dx xi+1/2Dx -Fi In general when the spring is stretched from xi to xf, the work done by the spring is: x -Ws,i= areai xi xi+1

Spring Potential Energy Potential due to a spring (cont’d) The energy-work theorem including a spring and gravity elastic potential energy Extended form of conservation of mechanical energy

Spring Potential Energy Examples Example 5.9: A horizontal spring (a) Find the speed at x=0 without friction. m=5.00 kg k=4.00x102 N/m xi=0.0500 m mk=0 (b) Find the speed at x=xi/2.

Spring Potential Energy Examples Example 5.9: A horizontal spring (cont’d) (c) Find the speed at x=0 with friction m=5.00 kg k=4.00x102 N/m xi=0.0500 m mk= 0.150

Spring Potential Energy Examples Example 5.10 : Circus acrobat What is the max. compression of the spring d? m=50.0 kg h =2.00 m k = 8.00 x 103 N/m

Spring Potential Energy Examples Example 5.11 : A block projected up a frictionless incline m=0.500 kg xi=10.0 cm k=625 N/m q=30.0o Find the max. distance d the block travels up the incline. (b) Find the velocity at hafl height h/2.

Spring Potential Energy Systems and energy conservation Work-energy theorem Consider changes in potential The work done on a system by all non-conservative forces is equal to the change in mechanical energy of the system. If the mechanical energy is changing, it has to be going somewhere. The energy either leaves the system and goes into the surrounding environment, or stays in the system and is converted into non- mechanical form(s).

Systems and Energy Conservation Forms of energy Forms of energy stored kinetic, potential, internal energy Forms of energy transfer between a non-isolated system and its environment Mechanical work : transfers energy to a system by displacing it with a force. Heat : transfers energy through microscopic collisions between atoms or molecules. Mechanical waves : transfers energy by creating a disturbance that propagates through a medium (air etc.). Electrical transmission : transfers energy through electric currents. Electromagnetic radiation : transfers energy in the form of electromagnetic waves such as light, microwaves, and radio waves.

Systems and Energy Conservation Principle of energy conservation: Energy cannot be created or destroyed, only transferred from one form to another. The principle of conservation of energy is not only true in physics but also in other fields such as biology, chemistry, etc.

Power Power The rate at which energy is transferred is important in the design and use of practical devices such as electrical appliances and engines. If an external force is applied to an object and if the work done by this force is W in time interval Dt, then the average power delivered to the object during this interval is the work done divided by the time interval: SI unit : watt (W) = J/s = 1 kg m2/s3 W=FDt More general definition U.S. customary unit : 1 hp = 550 ft lb/s = 746 W 1 kWh = (103 W)(3600 s) = 3.60 x 105 J

Power Example 5.12 : Power delivered by an elevator Examples Example 5.12 : Power delivered by an elevator What is the min. power to lift the elevator with the max. load? M=1.00x103 kg m=8.00x102 kg f =4.00x103 N v = 3.00 m/s

Power Example 5.14 : Speedboat power How much power would a 1.00x103 kg speed boat need to go from rest to 20.0 m/s in 5.00 s, assuming the water exerts a constant drag force of magnitude fd=5.00x102 N and the acceleration is constant?

Power Center of mass (CM) Energy and power in a vertical jump Center of mass (CM) The point in an object at which all the may be considered to be concentrated. h=0.40 m depth of crouch Dt=0.25 s time for extension m=68 kg Stationary jump Two phases: (1) Extension, (2) free flight