Chapter 6: Conservation of Energy

Transcription

1 Chapter 6: Conservation o Energy Introduction Energy ability to perorm work. Unit o energy (and o work) Joules (J) 1 J = 1 kg- m 2 / s 2 Forms o energy: Light, chemical, nuclear, mechanical, electrical, sound, heat, kinetic, elastic, magnetic etc. Each type o energy is calculated dierently

2 Law o Conservation o Energy In any natural process, total energy is always conserved, i.e. energy can not be created nor destroyed. Can be transormed rom one orm to another. Can be transerred rom one system to another. In science, any law o conservation is a very powerul tool in understanding the physical universe. Work by Constant Force Work transer o energy as a result o orce. Constant orce its magnitude and direction unchanged. The orce acts on the object throughout the process. Only component o orce parallel to direction o motion perorms work!

3 Work by Constant Force Work done by orce F in moving the object at constant speed through displacement x : W = (F cos θ) x Fcosθ = component o F parallel to x. Work is a scalar quantity. Unit = Joule (J). N θ 1 J = 1 N-m F x W Work by Constant Force W = (F cos θ) x 1. I x = 0, then W = 0 I you held a 50 kg bag on your head without moving or 3 hours, would you have done work?

4 W = (F cos θ) x 2. I θ = 0, cos 0 = 1 ie F and x are parallel and W = F x F x W = (F cos θ) x 3. I θ = 90 o, cos 90 ο = 0, ie F and x are perpendicular to each other and W = 0 F N θ x W Work done by the normal orce N is zero

5 W = (F cos θ) x 4. I θ = 180 o, cos 180 ο = 1, ie F and x are antiperpendicular to each other and W = negative. N θ F x W Work done by the rictional orce is - F x Example: A 30 N chest was pulled 5 m across the loor at a constant speed by applying a orce o 50 N at an angle o 30 o. How much work was done by tension T? W = F x cos θ = (50 Ν) (5 m) cos (30) = 217 Joules T 50 N N 30 o

6 Example: A 30 N chest was pulled 5 m across the loor at a constant speed by applying a orce o 50 N at an angle o 30 o. How much work was done by gravity? 50 N N T 30 o W = F x cos θ = 50 x 5 cos(90) = 0 x 90 o Example: A 30 N chest was pulled 5 m across the loor at a constant speed by applying a orce o 50 N at an angle o 30 o. How much work was done by riction? 50 N To ind the magnitude o : Consider x-component o orces acting. Since constant velocity: F net = 0 So Tcos30 o = 0 Thus = 43.3 N. W = 43.3 x (cos180 o ) x 5 = -217 J N x 180 o T

7 W = F x x Work by Variable Force Work = area under F vs x plot Force F x x Force F = kx Spring F = k x Area = ½ k (x) 2 =W spring x Example A dart gun with a spring constant k = N/m is compressed 8.0 cm. How much energy will it transer to a dart when the spring is released? W = ½ k( x) 2

8 Kinetic Energy Suppose a constant orce F is applied horizontally on a small rigid object o mass m. Its velocity will change, say rom v o to v as it moves through a distance x (without rotation). Work done on the object W = F x x [F parallel to x] = m a x x [Recall: v 2 = v o2 + 2a(x-x o )] = ½ m (v 2 v 02 ) = ½ mv 2 -½ mv 2 0 The quantity ½. mass. (velocity) 2 is called kinetic energy o the object. An object o mass m, moving with velocity v, has kinetic energy K = ½ mv 2 K m: I m is doubled, K will double. K v 2 I v is doubled, K will quadrupled. Can K ever be zero? Can K ever be negative? W = K Can W ever be negative?

9 Kinetic Energy F v o v x I more than one orce acts on the object, the change in kinetic energy is the total work done by all the orces acting on the object. Total work W total = K This is called the Work- Energy Principle Total Work Total work W total = K W total = First ind work done by each orce, then add them. = irst add all the orces and then then calculate work done by the net orce Q: What is the total work done on an object moving with constant velocity? (A) Positive (B) Negative (C) Zero

10 Example 1: A 30 N chest was pulled 5 m across the loor at a constant speed by applying a orce o 50 N at an angle o 30 o. How much work was done by all the orces acting on it? W T = F x cos θ = (50 Ν)(5 m) cos (30) = 217 J W N = 0, W = 0 W = -217J W total = (-217) = 0 T = 50 N N T 30 o Example 2: A 30 N chest was pulled 5 m across the loor by applying a orce o 50 N at an angle o 30 o. How much work was done by all the orces acting on it i its speed changed rom 2 m/s to 5 m/s in the process?

11 Example You are towing a car up a hill with constant velocity. The work done on the car by the normal orce is: 1. positive 2. negative 3. zero correct F N W T V Work done by gravity? Work done by tension? Since the direction o the orce is positive the value o work will be positive. it's negative because it's trying to slow down the car. The normal orce does no work because it acts in a direction perpendicular to the displacement o the car. Example: Block w/ riction A block is sliding on a surace with an initial speed o 5 m/s. I the coeicient o kinetic riction between the block and table is 0.4, how ar does the block travel beore stopping? y Y direction: F=ma N- = 0 N = Work W N = 0 W = 0 W = x cos(180) = -µ x 5 m/s N W = K -µ x = ½ m (v 2 v 02 ) -µg x = ½ (0 v 02 ) µg x = ½ v 0 2 x = ½ v 02 / µg = 3.1 meters x

12 POTENTIAL ENERGY Potential Energy (U) = stored energy. It is the energy an object possess due to its position or its coniguration. There are dierent types o potential energy: Gravitational potential energy. Elastic potential energy. Electrical potential energy. Gravitational Potential Energy To move mass m rom initial point to inal point at constant velocity, an external orce F EXT, equal but opposite to the orce o gravity must be applied. W EXT W g = F cosθ y = h = ()(cos0)h = h = cos(180 o )(h) = -h F EXT Y = h Y o = 0

13 Gravitational Potential Energy The quantity h is called gravitational potential energy near the earth s surace: U = h where U = 0 wherever we have chosen h = 0 More general way o writing gravitational potential energy is Gm1m ( r) = r U 2 U = 0 at r = F EXT U = h U = 0 Work done by orce o gravity = W g = - h The quantity h is called gravitational potential energy. The gravitational potential energy o an object o mass m at a height h is U g = h 1. U g = h and W g = -h ie U g = - W g. 2. U g m 3. U g h 4. A reerence level where h = 0 can be chosen to be at any convenient location. 5. Only change in U g is important. U g = h = (h h i )

14 Gravitational Potential Energy Example 3: A block slid down a plane I a block is slid down a plane. Distance moved down the plane = s Vertical height moved = h How much work will the orce o gravity do? W = F cosθ x W g =. cosθ. S But S = y /cosθ h θ S W g = ()(cosθ)(h/cosθ) = h Elastic Potential Energy Elastic (Spring) orce F = -kx Work done by spring orce W = ½ kx 2 Energy stored in the spring when compressed or stretched by distance x Elastic Potential Energy U = ½ kx 2

15 Example Emil throws an orange straight up and catches it at the same point it was thrown. (a) How much work is done during the orange s ree all? (b) I the orange was thrown upward rom a 1.2 m height above the ground and alls to the ground, how much work is done by gravity? Conservative and Non Conservative Forces Sliding down a plane h Dropped θ Thrown Work done by orce o gravity, W g = - h Independent o path

16 Conservative Forces Forces which perorm same amount o work independent o the path taken are called conservative orces. Eg: gravity, elastic orce, electric orce. 1. Work done by a conservative orce depends only on the initial and inal position and not on the path taken to get there. 2. Potential energy only goes with conservative orces. There is potential energy associated with gravity, elastic orce, electric orce. 3. W C = - U [ U g = h and U spring = ½ kx 2 ] Non Conservative Forces 1. Amount o work done by them depend on the path taken. Eg: riction, tension, push/pull. 2. There is no potential energy associated with non conservative orces.

17 Work - Energy w/ Conservative Forces Total work = Work done by all types o orces. = Work by conservative orces + work by non-conservative orces W Total = W C + W NC But W Total = K and W C = - U So, K = - U + W NC ie, W NC = K + U W NC = K + U Work - Energy w/ Conservative Forces W NC = K + U When only conservative orces are acting; W NC = 0 0 = K + U OR 0 = (K K i ) + (U U i ) OR K + U = K i + U i The quantity K + U is called mechanical energy E Thus E = E i (Conservation o mechanical energy)

18 Skiing Example (no riction) A skier goes down a 78 meter high hill with a variety o slopes. What is the maximum speed she can obtain i she starts rom rest at the top? [Assume riction is negligible] Conservation o energy: K i + U i = K + U ½ m v i2 + m g y i = ½ m v 2 + m g y 0 + g y i = ½ v 2 + g y v 2 = 2 g (y -y i ) v = sqrt( 2 g (y -y i )) v = sqrt( 2 x 9.8 x 78) = 39 m/s Example Suppose the initial kinetic and potential energies o a system are 75J and 250J respectively, and that the inal kinetic and potential energies o the same system are 300J and -25J respectively. How much work was done on the system by non-conservative orces? 1. 0J 2. 50J J J J Initially = 75 J J = 325 J. Finally = 300 J - 25 J = 275 J. Final -Initial = = -50 J. No energy goes into or comes out o the system overall, so some nonconservative orce has to be accounted or.

19 Example A dart gun with a spring constant k = N/m is compressed 8.0 cm. (a) How much energy will it transer to a dart o mass 100 g when the spring is released? (b) What would be the escape speed o the dart? (c) I the dart was aimed upward, how high would it travel? U spring = ½ k( x) 2 K = ½ mv 2, U g = h Power Average Power P = rate at which work is being done. = rate at which energy is being transormed. P = W / t Units: J/s = Watt W/ t = [F x cos(θ)]/ t = F (v t) cos(θ) i.e., P = F v cos(θ)