Ch 8 Potential energ and Conservation of Energ Question: 2, 3, 8, 9 Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63
Potential energ Kinetic energ energ due to motion Potential energ energ due to the arrangement of the objects in a sstem (U) Potential energ is due to an objects location. gravitational potential energ due to the separation between two objects elastic potential energ energ associated with the compression or stretching of an elastic object Potential energ is a form of energ so the units are Joules.
Potential energ is energ that can be stored and later released to produce motion. Examples: Shove a ball into a spring loaded gun. The elastic potential energ is stored and when ou shoot the ball the potential energ is converted to kinetic energ. After climbing a tree ou have gravitational potential energ stored up. Fall out of the tree that potential energ is converted to kinetic energ as ou fall down.
Work and potential energ Throw ball verticall upward. As the ball rises, gravit does negative work on the ball. The ball slowed down (lost kinetic energ). The kinetic energ is transferred to the gravitational potential energ of the ball-earth sstem. As the ball is falling, gravit does positive work on the ball. The ball speeds up (gains kinetic energ). The kinetic energ comes from the gravitational potential energ of the ball-earth sstem.
As ball rises: K < 0 When the gravitational force did negative work, the GPE increased, U > 0. As ball falls: K > 0 When the gravitational force did positive work, the GPE decreased, U < 0. From chapter 7: W = K W = - U
Block hitting spring Replace throwing a ball upward with a block sliding into a spring that is fixed to a wall. When the block hits the spring the block has kinetic energ. As the block compresses the spring, it loses kinetic energ and spring gets elastic potential energ. When block comes to rest, all the kinetic energ has been turned into potential energ. As the spring expands the transfer of energ is reversed from potential energ to kinetic energ.
Conservative forces Assume a force acts between a sstem of 2 objects. As the force changes the configuration of the sstem, the force does work, W 1, energ is transferred from kinetic energ to some other form of energ. When the change in configuration is reversed, the force does work, W 2, and the transfer of energ is reversed. If W 1 = -W 2 is true the other tpe of energ was potential energ. The force is a conservative force. examples: of conservative forces: gravit, springs, electric force
If W 1 = -W 2 is not alwas true, the force is nonconservative. Examples of nonconservative forces: friction, drag Nonconservative forces convert kinetic energ to thermal energ. Transfers to thermal energ cannot be reversed. Example: Kinetic energ can be lost to friction, but friction cannot be used to produce kinetic energ.
Conservative forces are path independent. The work done b a conservative force onl depends on the endpoint. How ou get from the initial configuration to the final configuration does not matter. The net work done b a conservative force around a closed loop is zero. Example: Throw a ball up and let it fall back down. The total work done b gravit is zero. Since conservative forces are path independent, we usuall onl care about the endpoints. This can make a complicated problem easier to solve.
Nonconservative forces are path dependent. When dealing with nonconservative forces, such as friction, we need to consider the path the object takes. W f = F f d where d is the path length. A 2 1 B Drag a box over a rough surface from point 1 to point 2. Friction will do more work when using path A than path B. If ou dragged the box along A from 1 to 2 then dragged it back using path B the total work done b friction is not zero.
How to calculate values of potential energ. In general, the work done b a force is: W x x F( x) dx Since U = -W U i f x x i f F( x) dx Gravitational Potential Energ (picking up to be positive) U U mg i f ( mg) d mg i f d mg( f i )
Gravitational Potential Energ Change in GPE from one point to another is: U i f ( mg) d mg mg( U mg Onl changes in GPE are phsicall meaningful. U U i = mg( i ) i f We will often compare GPE to the GPE at a reference point. Usuall we use: U i = 0 at i = 0. So: U() = mg d You can set an height to be the zero of GPE. This will be a big convenience in solving problems. f i )
Elastic potential energ (EPE) x f x f 1 2 1 U ( kx) dx k xdx kx f kx i xi xi 2 2 When spring is relaxed, x i = 0, the U i = 0 U 0 = ½ kx 2 0 U(x) = ½ kx 2 2 Remember from last chapter W s = - ½ kx 2 So this fits U = -W
Conservation of mechanical energ Mechanical energ: E mech = K + U If ou onl have conservative forces, the total mechanical energ never changes. K 2 + U 2 = K 1 + U 1 and E mech = K + U = 0
E mech = K + U = 0 This is definitel one of the most important and most useful rules in phsics. If there are onl conservative forces involved, and there are multiple steps, ou can ignore the intermediate configurations of the sstem. Will see a good example of this later.
Potential Energ Curves Graph of potential energ vs. position See fig. 8-9 For a particle moving in the x-direction U(x) = -W = -F(x) x F U( x) x du( x) dx The derivative (slope) of the potential energ curve is related to the force.
The force doing the work, is the negative of the slope in the potential energ curve. Example: spring U(x) ½ kx 2 du( x) d 1 2 F ( x) ( kx ) dx dx 2 kx U(x) x
U(x) + K(x) = E mech K(x) = E mech U(x) Where U(x) = E mech the kinetic energ will be zero. This is a turning point. Think of the potential energ curve as a rollercoaster track. The height of the track is U(x). When the cart reach a point where its total energ is U(x), the cart stops and turns around. In Newtonian (classical phsics), particles are confined between turning points.
Equilibrium points. Points on the potential energ curve are said to be either unstable or stable. A point is unstable is a small displacement in either direction will lead to a greater displacement. (relative maxima) A point is stable is after a small displacement in either direction, the particle will return to its original position. (relative minima)
Work done b an external force. If positive work is done b an external force on a sstem, energ is transferred to the sstem. Negative work done on a sstem results in energ transferred from the sstem. W = U + E = E mech
Friction Look at case where a force, F, dragged a box across a floor. Newton s 2 nd law gives: F f k = ma F a is constant so: v a 2 Fd Fd ma v 2 o f 2 ( v v 2d 1 mv 2 K k 2ad 2 0 2 ) f k 1 mv 2 d 2 o f k d
The friction between the two surfaces causes their temperatures to rise. The sliding increases the thermal energ b an amount equal in magnitude to the work done b friction: E th = f k d Fd = E mech + f k d = E mech + E th Work done on a sstem when there is friction: W = E mech + E th
Conservation of Energ The total energ of a sstem can change onl b amounts of energ that are transferred to or from the sstem. Energies to be considered are mechanic, thermal, and internal When work is done, the work is equal to the sum of the changes in these tpes of energies. W = E = E mech + E th + E int
Conservation of energ for an isolated sstem: E mech + E th + e int = 0 For an isolated sstem, the total energ cannot change.
Power: Average power is the average rate that a force transfers energ from, one tpe to another. P ave E t Instantaneous power: P de dt Problems: 18, 26, 34, 44, 46, 58