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149: Lecture 15

• Chapter 6: Conservation of

– 6.3

– 6.4 Gravitational Energy

Lecture 15 Purdue University, Physics 149 1 ILQ 1

Mimas Saturn at a D. Enceladus orbits Saturn at a distance 4D. What is the ratio of the periods of their orbits?

A) Tm/Te = 1/8 T23∝ r 2 ⎛⎞ 3 B) T m/Te =1/4= 1/4 Tm ⎛⎞D ⎜⎟= ⎜⎟ ⎝⎠TDE ⎝⎠4 C) Tm/Te = 1/2 ⎛⎞T 11 D) T /T =2= 2 ⎜⎟m = = m e ⎝⎠TE 64 8

Lecture 15 Purdue University, Physics 149 2 ILQ 2

A bob swings back and forth along a circular path. Does the in the string do any on the bob? Does 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 • Forms – Kinetic Energy Stored – (E=mc2) • Units: = kg m2 /s/ s2

Lecture 15 Purdue University, Physics 149 4 Definition of “Work” in Physics

• Work is a quantity (not a vector quantity). • Units: J (), N⋅m, kg⋅m2/s2, etc. . – Unit conversion: 1 J = 1N1 N⋅m = 1kg1 kg⋅m2/s2 • Work is denoted by W (not to be confused by W).

Lecture 15 Purdue University, Physics 149 5 Total Work

• When several act on an object, the “total” work is the sum of the work done by each individually:

Lecture 15 Purdue University, Physics 149 6 ILQ 1 You are towing a car up a hill with constant . The work done on the car by the force is: A) positive B) negative

C) zero FN V

T

Normal force is perpendicular to direction of W disp lacemen t, so wor k is zero.

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 FN V

T

Gravity is pushing against the direction of W motion so it is nega tive.

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 FN V

T

The force of tension is in the same direction as W the motion of the car, making the work positi ve.

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 FN V

T

The total work done is positive because the car is moving up the hill. (Not quite!) W

2 2 W=KEf-KEi=(0.5mvf ) - (0.5mvi ). Because the final and initial are the same, there is no change in kinetic energy, and therefore no total work is done.

Lecture 15 Purdue University, Physics 149 10 Problem A box is pulled up a rough (μ > 0) incline by a rope- -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

Draw FBD of box: N v T z Consider direction of motion of the box

z Any force not perpendicular to the motion will do work: f N does no work (perp. to v) T does positive work mg f does negative work 3f3 forces do work mg does negative work

Lecture 15 Purdue University, Physics 149 12 Example: Block with

• A block is on a surface with an initial 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 N y-direction: FFma=ma f N-mg = 0 x N = mg mg Work W = Δ K 2 2 WN = 0 -μmg Δx = ½ m (vf –v0 ) 2 Wmg = 0 -μg Δx=½x = ½ (0 – v0 ) 2 Wf = f Δx cos(180) μg Δx = ½ v0 2 = -μmg Δx Δx = ½ v0 / μg 5 m/s = 313.1 meters

Lecture 15 Purdue University, Physics 149 13 Kinetic Energy: Motion

• Apply constant force along x-direction to a point m.

W = Fx Δx 1 = m ax Δx 2 2 recall:axΔx = (vx − vx0 ) 2 2 = ½ m (vf –v0 ) 2

• Work changes ½ m v2

• Define Kinetic Energy K=½mvK = ½ m v2 WkWork-Kine tic Energy W = Δ K For Point Theorem

Lecture 15 Purdue University, Physics 149 14 Translational Kinetic Energy

• When an object of mass m is moving with speed v (the maggy),jnitude 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 . Lecture 15 Purdue University, Physics 149 15 Work - Kinetic Energy Theorem

1 1 = K − K = mv2 − mv2 f i 2 f 2 i

• The work done on an object by the “net” force (whether the 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) K1 = 4K2

B) K 1 = K2

C) 2K1 = K2

D) K1 = 2K2

Lecture 15 Purdue University, Physics 149 17 Work Done by Gravity 1

•Examppple 1: Drop ball

Y = h Wg = (mg)(s)cosθ i s = h S 0 Wg = mghcos(0 ) = mgh mg y Δy = yf-yi = -h Yf = 0 x Wg = -mgΔy

Lecture 15 Purdue University, Physics 149 18 Work Done by Gravity 2

•Examppple 2: Toss ball up

Y = h Wg = (mg)(s)cosθ i s = h S W = mghcos(1800) = -mgh mg g y

Δy = yf-yi = +h Y = 0 f x

Wg = -mgΔy

Lecture 15 Purdue University, Physics 149 19 Work Done by Gravity 3

•Example 3: Slide block down incline

Wg = (mg)(s)cosθ s = h/cosθ h θ Wg = mg(h/cosθ)cosθ mg S Wg = mgh

Δy = yf-yi = -h

Wg = -mgΔy

Lecture 15 Purdue University, Physics 149 20 Work Done by Gravity

• Depends only on initial and final height!

• Wg = -mg(yf - yi)=) = -mgΔy – Independent of path

– If you end up where you began, Wg =0= 0 – Note: can do work “against” gravity, then get gravity to do work back. – Define: Potential Energy……

We call this a “” 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?

θ mg Δr

Left Case: Wgrav = F⋅Δr⋅cosθ = mg⋅|Δy|⋅cos0° = mg|Δy|

Middle Case: Wgrav = F⋅Δr⋅cosθ = mg|Δy| (because Δr⋅cosθ = |Δy|)

Right Case: Wgrav = 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 ƒ -m g Δy= ΔK

ƒ 0 = ΔK + m g Δy define Ug = mgy

ƒ 0 = ΔK + ΔUg

W = ΔK ΔU = -WC • Works for any CONSERVATIVE force

ƒ Gravity Ug = m g y 2 ƒ Us = 1/2 k x ƒ 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 ) is W =F= F⋅Δr⋅cosθ =mg= mg⋅Δy⋅cos180° = –mgΔy grav vf=0

In this problem, gravity is the only force. Thus,

Wttltotal = Wgrav = –mgΔy According to work-energy theorem, 2 2 Wtotal = –mgΔy = –½mvi (because ½mvf =0)

2 vi That is, the initial kinetic energy (Ki = ½mvi ) 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 of an object with something else (in this case, gravity) that can easily be recovered as kinetic energy is called potential energy.

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⋅m2/s2, 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 WFygg= (Δ=)cosθ mghcos(0) = mgh

Lecture 15 Purdue University, Physics 149 27 Work and Potential Energy • Work done by gravity independent of path

– Wg = -mgg(y (yf - yi)

• Define Ug = mgy

• Works for any CONSERVATIVE force

• Modify Work-Energy theorem

∑Wnc = ΔK + ΔU

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 ca lle d “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 W = ΔK ∑ i ΔU = -WC ΣWW=+=Δcons W nc K Move work by conservative forces to other side

∑Wnc = ΔK + ΔU WKWnc= Δ− cons If there are NO non-conservative forces 0 =ΔK +ΔU =ΔE mech EEi = f 0 = K f − Ki +U f −Ui Conservation of K +U = K +U i i f f

Lecture 15 Purdue University, Physics 149 30 Mechanical Energy

• The sum of the kinetic and potential is

called the (total) mechanical energy (Emech).

Emech ≡ K + U

• Mechanical energy is a scalar quantity. • Units: J, N⋅mkgm, kg⋅m2/s2, etc.

• Mechanical energy is denoted by Emech.

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.

Emech = Ki + Ui = Kf + Uf = const (if Wnc = 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?

Emech = Ki + Ui = Kf + Uf = const (b/c Wnc = 0) Only force/work done be gravity Ui = mgyi

Uf =0= 0 Ki = 0 2 2 2 Wg = m ½ (vf –vi ) Kf = ½mvf 2 mg Fg h½h = ½m vf 2 mgh = ½m vf

Vf = sqrt( 2 g h ) = 10 m/s

Lecture 15 Purdue University, Physics 149 33 Example: Pendulum

Vi = 0 mg • In this case, there are two forces acting on the object. • But, the direction of tension is perpendicular to the 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.

Emech = Ki + Ui = Kf + Uf = const (b/c Wnc = 0)

Ui = mgh Uf = 0 Ki = 0 2 Kf =½= ½mvf 2 Æ 0 + mgh = ½mvf + 0 Thus, vf = sqrt(2gh) Å 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 tim e to get to the bottom is height / y-compone nt of veloc ity

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))ppg Dropping B) Slide on ramp (no friction) C) Swinging down D) All the same correct A B C

Conservation of Energy (Wnc=0) Kinitial +U+ Uinitial = Kfinal +U+ Ufinal 2 0 + mgh = ½ m v final + 0 vfinal = 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 (Wnc=0)

ΣWnc = ΔK + Δ U

0 = Kfinal -Kinitial + Ufinal -Uinitial

Kinitial + Uinitial = Kfinal + Ufinal 0 + mgh = ½ m v2 + 0 final h vfinal =sqrt(2gh)= sqrt(2 g 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))y h < y B))y h = y C))y 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> h2 B) h 1 = h2 C) h1

Conservation of Energy (Wnc=0)

ΣWnc = ΔK + Δ U Kinitial + Uinitial = Kfinal + Ufinal 0 + mgh1 = 0 + mgh2 h1 =h= h2 m

h1 h2

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 ’s law mm FG= 12 r 2 • The gravitational potential energy is: mm UG=− 12 r if U==∞0 for r

Lecture 15 Purdue University, Physics 149 40