PHYSICS 149: Lecture 15

PHYSICS 149: Lecture 15 Chapter 6: Conservation of Energy 6.3 Kinetic Energy 6.4 Gravitational Potential Energy Lecture 15 Purdue University, Physics 149 1

ILQ 1 Mimas orbits Saturn at a distance D. Enceladus orbits Saturn at a distance 4D. What is the ratio of the periods of their orbits? A) T m /T e = 1/8 B) T m /T e =1/4 C) T m /T e = 1/2 T r 2 3 2 3 T m D = TE 4D T 1 1 = = m D) T =2 m /T e 64 8 T E Lecture 15 Purdue University, Physics 149 2

ILQ 2 A pendulum bob swings back and forth along a circular path. Does the tension in the string do any work on the bob? Does gravity do work on the bob? A) only tension does work B) both do work C) neither do work D) only gravity does work Lecture 15 Purdue University, Physics 149 3

Energy Energy is conserved meaning it can not be created nor destroyed Can change form Can be transferred Total Energy of an isolated system does not change with time Forms Kinetic Energy Motion Potential Energy Stored Heat Mass (E=mc 2 ) Units: Joules = kg m 2 /s 2 Lecture 15 Purdue University, Physics 149 4

Definition of Work in Physics Work is a scalar quantity (not a vector quantity). Units: J (Joule), N m, kg m 2 /s 2, etc.. Unit conversion: 1 J = 1N m = 1kg m 2 /s 2 Work is denoted by W (not to be confused by weight W). Lecture 15 Purdue University, Physics 149 5

Total Work When several forces act on an object, the total work is the sum of the work done by each force individually: Lecture 15 Purdue University, Physics 149 6

ILQ 1 You are towing a car up a hill with constant velocity. The work done on the car by the normal force is: A) positive B) negative C) zero F N V T Normal force is perpendicular to direction of displacement, so work is zero. W Lecture 15 Purdue University, Physics 149 7

ILQ 2 You are towing a car up a hill with constant velocity. The work done on the car by the gravitational force is: A) positive B) negative C) zero F N V T Gravity is pushing against the direction of motion so it is negative. W Lecture 15 Purdue University, Physics 149 8

ILQ 3 You are towing a car up a hill with constant velocity. The work done on the car by the tension force is: A) positive B) negative C) zero F N V T The force of tension is in the same direction as the motion of the car, making the work positive. W Lecture 15 Purdue University, Physics 149 9

ILQ 4 You are towing a car up a hill with constant velocity. The total work done on the car by all forces is: A) positive B) negative C) zero F N V T The total work done is positive because the car is moving up the hill. (Not quite!) W=KE f -KE i =(0.5mv f2 ) - (0.5mv i2 ). Because the final and initial velocities are the same, there is no change in kinetic energy, and therefore no total work is done. W Lecture 15 Purdue University, Physics 149 10

Problem A box is pulled up a rough (μ > 0) incline by a ropepulley-weight arrangement as shown below. How many forces are doing (non-zero) work on the box? A) 0 B) 1 C) 2 D) 3 E) 4 Lecture 15 Purdue University, Physics 149 11

Solution Draw FBD of box: T v N Consider direction of motion of the box Any force not perpendicular to the motion will do work: N does no work (perp. to v) f T does positive work f does negative work mg does negative work 3f forces do work mg Lecture 15 Purdue University, Physics 149 12

Example: Block with Friction A block is sliding on a surface with an initial speed of 5 m/s. If the coefficient of kinetic friction between the block and table is 0.4, how far does the block travel before stopping? y-direction: F=ma N-mg = 0 N = mg Work W = Δ K W N = 0 -μmg Δx = ½ m (v f2 v 02 ) W = Δx=½ 2 mg 0 -μg = ½ (0 v 0 ) W f = f Δx cos(180) μg Δx = ½ v 2 0 = -μmg Δx Δx = ½ v 02 / μg 5 m/s = 31 3.1 meters f N mg y x Lecture 15 Purdue University, Physics 149 13

Kinetic Energy: Motion Apply constant force along x-direction to a point particle m. W = F x Δx = m a x Δx = ½ m (v f2 v 02 ) 1 2 2 recall: a x Δ x = ( vx vx0 ) 2 Work changes ½ m v 2 Define Kinetic Energy K=½mv 2 Work-Kinetic Energy W = Δ K For Point Particles Theorem Lecture 15 Purdue University, Physics 149 14

Translational Kinetic Energy When an object of mass m is moving with speed v (the magnitude of instantaneous velocity), the object s translational kinetic energy is defined as follows: Kinetic energy is a scalar quantity. Units: J, N m, kg m2/s2, etc. Kinetic energy is denoted by K. Translational kinetic energy means the total work done on the object to accelerate it to that speed starting from rest. Translational kinetic energy is often called the kinetic energy if it is clearly distinguished from the rotational energy or internal energy. Lecture 15 Purdue University, Physics 149 15

Work - Kinetic Energy Theorem = K f K i = 1 2 1 2 mv f mvi 2 2 The work done on an object by the net force (whether the net force is constant or variable) is equal to the change in the kinetic energy. Or, the total work done on the object is equal to the change in the kinetic energy. Lecture 15 Purdue University, Physics 149 16

ILQ Compare the kinetic energy of two balls: Ball 1: mass m thrown with speed 2v Ball 2: mass 2m thrown with speed v A) K 1 = 4K 2 B) K 1 = K 2 C) 2K 1 = K 2 D) K 1 = 2K 2 Lecture 15 Purdue University, Physics 149 17

Work Done by Gravity 1 Example 1: Drop ball W = (mg)(s)cosθ Y i = h W g s = h W g = mghcos(0 0 ) = mgh Δy = y f -y i = -h W g = -mgδy mg S Y f = 0 y x Lecture 15 Purdue University, Physics 149 18

Work Done by Gravity 2 Example 2: Toss ball up W = (mg)(s)cosθ i W g s = h W g = mghcos(180 0 ) = -mgh mg S Y i = h Δy = y f -y i = +h Y f = 0 W g = -mgδy y x Lecture 15 Purdue University, Physics 149 19

Work Done by Gravity 3 Example 3: Slide block down incline W g = (mg)(s)cosθ s = h/cosθ W h g = mg(h/cosθ)cosθ θ W g = mgh h mg S Δy = y f -y i = -h W g = -mgδy Lecture 15 Purdue University, Physics 149 20

Work Done by Gravity Depends only on initial and final height! W g = -mg(y f - y i )= -mgδy Independent of path If you end up where you began, W g =0 Note: can do work against gravity, then get gravity to do work back. Define: Potential Energy We call this a Conservative Force because we can define a Potential Energy to go with it. Lecture 15 Purdue University, Physics 149 21

Work Done by Gravity Question: Does the work done by gravity depend on the path taken? mg θ Δr Left Case: W grav = F Δr cosθ = mg Δy cos0 = mg Δy Middle Case: W grav = F Δr cosθ = mg Δy (because Δr cosθ = Δy ) Right Case: W grav = mg Δy (because each segment can be treated like the middle case) Answer: The work done by gravity is independent of path that is, the work depends only on the initial and final positions ( Δy ). This kind of force is called conservative force. Lecture 15 Purdue University, Physics 149 22

Potential (stored) Energy Stored gravitational energy can be converted to kinetic energy -m g Δy= ΔK 0 = ΔK + m g Δy y define U g = mgy 0 = ΔK + ΔU g W = ΔK ΔU = -W C Works for any CONSERVATIVE force Gravity U g = m g y g Spring U s = 1/2 k x 2 NOT friction Lecture 15 Purdue University, Physics 149 23

Potential Only change in potential energy is important Lecture 15 Purdue University, Physics 149 24

What is Potential Energy? An object is thrown up vertically, and it reaches top. Assume no air resistance. The work done by gravity (near the surface of Earth) is W grav =FΔr F Δr cosθ =mgδy mg Δy cos180 cos180 = mgδy v f =0 In this problem, gravity is the only force. Thus, W total t = W grav = mgδy According to work-energy theorem, W total = mgδy = ½mv i2 (because ½mv f2 =0) That is, the initial kinetic energy (K i = ½mv i2 ) has been stored in (or transformed into) the form of mgδy at the top. And, it has the potential to do work (or to become kinetic energy). Stored energy due to the interaction of an object with something else (in this case, gravity) that can easily be recovered as kinetic energy is called potential energy. v i Lecture 15 Purdue University, Physics 149 25

Definition of Potential Energy The change in potential energy is equal to the negative of the work done by the conservative forces. Potential energy can be defined only for the conservative forces. For the non-conservative forces, potential energy can not be defined in the first place. There is no way to calculate the absolute value of the potential energy. Only the change in potential energy is important. The choice of the zero point of potential energy is arbitrary. Potential energy is a scalar quantity. Units: J, N m, kg m 2 /s 2, etc. Potential energy is denoted by U (so ΔU means the change in potential energy). Lecture 15 Purdue University, Physics 149 26

ILQ A hiker descends from the South Rim of the Grand Canyon to the Colorado River. During this hike, the work done by gravity on the hiker is A) positive and depends on the path taken B) positive and independent of the path taken C) negative and depends on the path taken D) negative and independent of the path taken E) zero W = F ( Δ y)cosθ = g g mghcos(0) = mgh Lecture 15 Purdue University, Physics 149 27

Work and Potential Energy Work done by gravity independent of path W g = -mg (y f - y i ) g g(y f y i ) Define U g = mgy Works for any CONSERVATIVE force Modify Work-Energy theorem W nc = Δ K + Δ U Lecture 15 Purdue University, Physics 149 28

Conservative Force If the work done by a force is independent of path (that is, depends only on the initial and final positions), the force is called conservative. Example: gravitational force, spring force, and electrical force Note that the work done by conservative forces for any closed loop is zero. If the work done by a force depends on the path taken, the force is called non-conservative. Example: frictional force and air resistance Lecture 15 Purdue University, Physics 149 29

Work - Energy with Conservative Forces Work Energy Theorem Σ W = W + W =ΔK W i = ΔK ΔU = -W C cons nc Move work by conservative forces to other side W nc = ΔK + ΔU W = ΔK W If there are NO non-conservative forces nc cons 0 =ΔK +ΔU =ΔE mech E = E 0 = K f K i + U K + U = K + U i i f f U f i i Conservation of mechanical energy Lecture 15 Purdue University, Physics 149 30 f

Mechanical Energy The sum of the kinetic and potential energies is called the (total) mechanical energy (E mech ). E mech K + U Mechanical energy is a scalar quantity. Units: J, N m, kg m 2 /s 2, etc. Mechanical energy is denoted by E mech. Lecture 15 Purdue University, Physics 149 31

Conservation of Mechanical Energy If the work is done by only conservative forces (this is, the work done by non-conservative forces is zero), the mechanical energy is conserved. E mech = K i + U i = K f + U f = const (if W nc = 0) If the work is done by also non-conservative forces, the mechanical energy is not conserved and the following relations are satisfied. or Lecture 15 Purdue University, Physics 149 32

Falling Ball Example Ball falls a distance of 5 meters. What is its final speed? E mech = K i + U i = K f + U f = const (b/c W nc = 0) mech i i f f nc U i = mgy i U f =0 K i = 0 K f = ½mv 2 f Only force/work done be gravity W g = m ½ (v f2 v i2 ) F g h = ½m v 2 mg f mgh = ½m v 2 f V f = sqrt( 2 g h ) = 10 m/s Lecture 15 Purdue University, Physics 149 33

Example: Pendulum V i = 0 mg In this case, there are two forces acting on the object. But, the direction of tension is perpendicular to the displacement of the object, so the work done by tension is zero. Gravity (conservative force) is the only force which does work, so the mechanical energy is conserved. E mech = K i + U i = K f + U f = const (b/c W nc = 0) U i = mgh U f = 0 K i = 0 K =½mv 2 f f 0 + mgh = ½mv f2 + 0 Thus, v f = sqrt(2gh) the same result as two previous results Lecture 15 Purdue University, Physics 149 34

ILQ Imagine that you are comparing three different ways of having a ball move down through the same height. In which case does the ball get to the bottom first? A) Dropping B) Slide on ramp (no friction) C) Swinging down D) All the same correct A B C The time to get to the bottom is height / y-component nt of velocity Lecture 15 Purdue University, Physics 149 35

ILQ Imagine that you are comparing three different ways of having a ball move down through the same height. In which case does the ball reach the bottom with the highest speed? A) Dropping B) Slide on ramp (no friction) C) Swinging down D) All the same correct A B C Conservation of Energy (W nc =0) K initial +U initial = K final +U final 0 + mgh = ½ m v 2 final + 0 v final = sqrt(2 g h) Lecture 15 Purdue University, Physics 149 36

Pendulum ILQ As the pendulum falls, the work done by the string is A) Positive B) Zero C) Negative W = F d cos θ. But θ = 90 degrees so Work is zero. How fast is the ball moving at the bottom of the path? Conservation of Energy (W nc =0) ΣW nc = ΔK + Δ U 0 = K final -K initial + U final -U initial K initial + U initial = K final + U final 0 + mgh = ½ m v 2 final + 0 v final =sqrt(2gh) h Lecture 15 Purdue University, Physics 149 37

Pendulum Demo A pendulum is released from a height h above the minimum. At the bottom of its swing, the string hits a peg, reducing the length. What is the final height y the ball reaches? A) h < y B) h = y C) h > y Lecture 15 Purdue University, Physics 149 38

Galileo s Pendulum How high will the pendulum swing on the other side now? A) h 1 >h 2 B) h 1 = h 2 C) h 1 <h 2 Conservation of Energy (W nc =0) ΣW nc = ΔK + Δ U K initial + U initial = K final + U final m 0 + mgh 1 = 0 + mgh 2 h 1 =h 2 h 1 h 2 Lecture 15 Purdue University, Physics 149 39

Gravitational Potential Energy If the gravitational force is not constant or nearly constant, we have to start from Newton s law mm F = G r 1 2 2 The gravitational potential energy is: mm U = G 1 2 r if U = 0 for r = Lecture 15 Purdue University, Physics 149 40