Exploring Work and Energy: Summary and Examples

Ch. 6, Work & Energy, Continued

Summary So Far • Work(constant force): • W = F||d = Fd cosθ • Work-Energy Theorem: • Wnet = (½)m(v2)2 - (½)m(v1)2 KE • Total work done by ALL forces! • Kinetic Energy: l • KE  (½)mv2

Potential Energy A mass can have a Potential Energydue to its environment Potential Energy (PE)  An energy associated with the position or configuration of a mass. Examples of Potential Energy: A wound-up spring A stretched elastic band An object at some height above the ground

Gravitational Potential Energy • When an object of mass m follows any path that moves through a vertical distance h, the work done by the gravitational force is always equal to W = mgh • So, we say that an object near the Earth’s surface has a Potential Energy (PE) that depends only on the object’s height, h • The PE is a property of the Earth-object system

Potential Energy (PE)  Energy associated with the position or configuration of a mass. • PotentialWork Done! • Example: • Gravitational Potential • Energy:PEgrav mgy • y = distance above Earth. • m has the potential to do workmgy when it falls • (W = Fy, F = mg)

Gravitational Potential Energy For constant speed: ΣFy = Fext – mg = 0 So, Wext = Fext hcosθ = mghcos(0) = mgh = mg(y2 – y1) Work-Energy Theorem Wnet = KE  (½)[m(v2)2 - m(v1)2] (1) In raising a mass m to a height h, the work done by the external force ismgh. So we define the gravitational potential energy at a height y above some reference point (y1) as (PE)grav = mgh

Consider a problem in which the height of a mass above the Earth changes from y1 to y2: Change in Gravitational PEis: (PE)grav= mg(y2 - y1) Work done on the mass: W = (PE)grav y = distance above Earth • Where we choose y = 0is arbitrary, since we take the difference in 2y’s in (PE)grav

Of course, this Potential energy can be converted to kinetic energy if the object is dropped. PE is a property of a system as a whole, not just of the object (it depends on external forces). If PEgrav = mgy, from where do we measure y? It turns out not to matter! As long as we are consistent about where we choose y = 0 that choice won’t matter because only changes in potential energy can be measured.

Example: PE Changes for a Roller Coaster A roller-coaster car, mass m = 1000 kg, moves from point 1 to point 2 & then to point 3. ∆PEdepends only on differences in height. a. Calculate the gravitational potential energy at points 2 & 3 relative to point 1. (That is, take y = 0 at point 1.) b. Calculate thechangein potential energy when the car goes from point 2 to point 3. c. Repeat parts a. & b., but take the reference point (y = 0) at point 3.

Many Other Types of Potential Energy Besides Gravitational Exist! Consider an IdealSpring An Ideal Spring, is characterized by a spring constant k, which is a measure of it’s “stiffness”. The restoring force of the spring on the hand is: L (Fs >0, x <0; Fs <0, x >0) This is known as Hooke’s “Law” (but, it isn’t really a law!) Fs = - kx It can be shown that the work done by the person is: W = (½)kx2 (PE)elastic We use this as the definition of Elastic Potential Energy

Elastic Potential Energy (PE)elastic≡(½)kx2 Relaxed Spring Work to compress spring distance x: W = (½)kx2 (PE)elastic The spring stores potential energy! When the spring is released, it transfers it’s potential energy PEe = (½)kx2 to mass in the form of kinetic energy KE = (½)mv2

The applied Force Fappis equal & opposite to the force Fs exerted by block on the spring: Fs = - Fapp = -kx

Force Exerted by a Spring on a Block x > 0, Fs < 0 x = 0, Fs = 0 x < 0, Fs > 0 The spring force Fs varies with the block position xrelative to equilibrium at x = 0. Fs = -kx. Spring constantk > 0 Fs(x) vs. x

Relaxed Spring Spring constant k x = 0  W In (a), the work to compress the spring a distance x: W = (½)kx2 So, the spring stores potential energy in this amount.  x  W W W W = (½)kx2 W In (b), the spring does work on the ball, convertingit’s stored potential energy into kinetic energy. W W W

Elastic PE PEelastic = (½)kx2 KE = 0 PEelastic = 0 KE = (½)mv2

Measuring k for a Spring Hang the spring vertically. Attach an object of mass m To the lower end. The spring stretches a distance d. At equilibrium, Newton’s 2nd Law says ∑Fy = 0. So, mg – kd = 0, mg = kd Knowing m & measuring d, k = (mg/d) Example: d = 2.0 cm, m = 0.55 kg  k = 270 N/m

In a problem in which compression or stretching distance of spring changes from x1 to x2, The change in PE is: (PE)elastic= (½)k(x2)2 - (½)k(x1)2 • The work done is: W = - (PE)elastic The PE belongs to the system, not to individual objects.

Conservative Forces

Conservative Forces • Conservative Force  The work done by that force depends only on initial & final conditions & not on path taken between the initial & final positions of the mass. A PE CAN be defined for conservative forces

Non-Conservative Force  The work done by that force depends on the path taken between the initial & final positions of the mass. A PE CAN’T be defined for non-conservative forces • The most common example of a non-conservative force is FRICTION

Definition:A force isconservativeif & only if the work done by that force on an object moving from one point to another depends ONLY on the initial & final positions of the object, & is independent of the particular path taken. Example: gravity.

Gravitational PE Again! • The work done by the gravitational forceas the object moves from its initial position to its final position is Independent of the path taken! • The potential energy is related to the work done by the force on the object as the object moves from one location to another. • Because of this property, the gravitational force is called a Conservative Force.

Conservative Force:Another definition: A force is conservative if the net work done by the force on an object moving around any closed path is zero.

Potential Energy • The relationship between work & PE: ΔPE = PEf– PEi = - W • W is a scalar, so PE is also a scalar • The Gravitational PE of an object when it is at a height y is PE = mgy Applies only to objects near the Earth’s surface • Potential Energy, PE is stored energy • The energy can be recovered by letting the object fall back down to its initial height, gaining kinetic energy

Potential Energy: Can only be defined for Conservative Forces! In other words, if a force is Conservative, a PECAN be defined. But, if a force is Non-Conservative, a PE CANNOTbe defined!!

If friction is present, the work done depends not only on the starting & ending points, but also on the path taken. Friction is a non-conservative force! Friction is non-conservative!!! The work done depends on the path!

If several forces act, (conservative & non-conservative), the total work done is: Wnet =WC + WNC WC ≡work done by conservative forces WNC ≡work done by non-conservative forces • The work energy theorem still holds: Wnet =WC + WNC = KE • For conservative forces (by the definition of PE): WC = -PE  KE = -PE + WNC or: WNC = KE + PE

 In general, WNC = KE + PE • The total work done by all non-conservative forces≡ The total change in KE + The total change in PE

Mechanical Energy & its Conservation GENERALLY:In any process, total energy is neither created nor destroyed. • Energy can be transformed from one form to another & from one object to another, but the Total Amount Remains Constant.  Law of Conservation of Total Energy

In general, for mechanical systems, we found: WNC = KE + PE • For the Very Special Case of Conservative Forces Only  WNC = 0 = KE + PE = 0  The Principle of Conservation of Mechanical Energy • Please Note!!This is NOT(quite) the same as the Law of Conservation of Total Energy! It is a very special case of this law (where all forces are conservative)

So, for conservative forces ONLY! In any process KE + PE = 0 Conservation of Mechanical Energy • It is convenient to define the Mechanical Energy: E  KE + PE  In any process (conservative forces!): E = 0 = KE + PE Or,E = KE + PE = Constant ≡ Conservation of Mechanical Energy

Conservation of Mechanical Energy • In any process with conservative forces ONLY!  E = 0 = KE + PE Or,E = KE + PE = Constant • In any process (conservative forces!), the sum of the KE & the PE is unchanged: That is, the mechanical energy may change from PE to KE or from KE to PE, but Their Sum Remains Constant.

Principle of Conservation of Mechanical Energy: If only conservative forces are doing work, the total mechanical energy of a system neither increases nor decreases in any process. It stays constant—it is conserved.

Conservation of Mechanical Energy:  KE + PE = 0 Or E = KE + PE = Constant • This is valid for conservative forces ONLY (gravity, spring, etc.) • Suppose that, initially: E = KE1 + PE1, & finally: E = KE2+ PE2. • But,E = Constant, so  KE1 + PE1= KE2+ PE2 A very powerful method of calculation!!

Conservation of Mechanical Energy  KE + PE = 0 or E = KE + PE = Constant • For gravitational PE: (PE)grav = mgy E = KE1 + PE1 = KE2+ PE2  (½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2 y1= Initial height, v1 = Initial velocity y2 =Final height, v2 = Final velocity

KE1+ PE1 = KE2+ PE2 = KE3+ PE3 all PE PE1 = mgh, KE1 = 0 but their sum remains constant! KE3+ PE3 = KE2+ PE2 = KE1+ PE1 half KE half PE KE1 + PE1 = KE2 + PE2 0 + mgh = (½)mv2+ 0 v2 = 2gh PE2 = 0 KE2 = (½)mv2 all KE

Example: Falling Rock • Energy “buckets” are not real!! • Speeds at y2 = 0.0, & y3 = 1.0 m? Mechanical EnergyConservation! • KE1 + PE1 = KE2 + PE2 • (½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2 =(½)m(v3)2 + mgy3 (Mass cancels!) • y1 = 3.0 m, v1 = 0, y2 = 1.0 m, v2 = ?, y3 = 0.0, v3 = ? • Results:v2 = 6.3 m/s, v3 = 7.7 m/s NOTE!! • Always use KE1+ PE1 = KE2+ PE2 = KE3+ PE3 • NEVERKE3= PE3!!!!In general, KE3≠ PE3!!! This is a very common error! WHY????

Speeds at y2 = 0.0, & y3 = 1.0 m? Mechanical EnergyConservation! • Cartoon Version! (½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2 =(½)m(v3)2 + mgy3 (Mass cancels!) y1 = 3.0 m, v1 = 0, y2 = 1.0 m, v2 = ? , y3 = 0.0 m, v3 = ? Results:v2 = 6.3 m/s, v3 = 7.7 m/s v3 = ? PE only part PE part KE v1 = 0 y2 = 1.0 m y1 = 3.0 m v2 = ? y3 = 0 KE only

Example: Roller Coaster Height of hill = 40 m. Car starts from rest at top. Calculate: a. Speed of the car at bottom of hill. b. Height at which it will have half this speed. Take y = 0 at bottom of hill. • Mechanical energy conservation!(Frictionless!)  (½)m(v1)2 + mgy1= (½)m(v2)2 + mgy2(Mass cancels!) Only height differences matter! Horizontal distance doesn’t matter! • Speed at the bottom? y1 = 40 m, v1 = 0 y2 = 0 m, v2 = ? Find: v2 = 28 m/s • What is y when v3 = 14 m/s? Use: (½)m(v2)2 + 0 = (½)m(v3)2 + mgy3 Find: y3 = 30 m In general, KE3≠ PE3!!! 1 3 2 NOTE!!Always use KE1+ PE1 = KE2+ PE2 = KE3+ PE3 NeverKE3= PE3 ! A very common error! WHY????

Conceptual Example :Speeds on 2 Water Slides • Who is traveling faster at the bottom? • Who reaches the bottom first? • Demonstration! v = 0, y = h Frictionless water slides! Both start here! y = 0 v = ? Both get to the bottom here!

Exploring Work and Energy: Summary and Examples