Ch. 6, Work & Energy, Continued

Summary So Far Work-Energy Theorem: W net = (½)m(v 2 ) 2 - (½)m(v 1 ) 2   KE Total work done by ALL forces! Kinetic Energy: l KE  (½)mv 2 Work (constant force): W = F || d = Fd cosθ

Potential Energy A mass can have a Potential Energy due 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. Potential Work Done! Example: Gravitational Potential Energy: PE grav  mgy y = distance above Earth. m has the potential to do work mgy when it falls (W = Fy, F = mg)

Gravitational Potential Energy In raising a mass m to a height h, the work done by the external force is mgh. So we define the gravitational potential energy at a height y above some reference point (y 1 ) as (PE) grav = mgh For constant speed: ΣF y = F ext – mg = 0 So, W ext = F ext hcosθ = mghcos(0  ) = mgh = mg(y 2 – y 1 ) Work-Energy Theorem W net =  KE  (½)[m(v 2 ) 2 - m(v 1 ) 2 ] (1)

Consider a problem in which the height of a mass above the Earth changes from y 1 to y 2 : Change in Gravitational PE is:  (PE) grav = mg(y 2 - y 1 ) Work done on the mass: W =  (PE) grav y = distance above Earth Where we choose y = 0 is arbitrary, since we take the difference in 2 y’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 PE grav = 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. a. Calculate the gravitational potential energy at points 2 & 3 relative to point 1. (That is, take y = 0 at point 1.) b. Calculate the change in 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. ∆PE depends only on differences in height.

Many Other Types of Potential Energy Besides Gravitational Exist! It can be shown that the work done by the person is: W = (½)kx 2  (PE) elastic We use this as the definition of Elastic Potential Energy Consider an Ideal Spring 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 (F s >0, x 0) This is known as Hooke’s “Law” (but, it isn’t really a law!) F s = - kx

Elastic Potential Energy (PE) elastic ≡ (½)kx 2 Relaxed Spring Work to compress spring distance x: W = (½)kx 2  (PE) elastic The spring stores potential energy! When the spring is released, it transfers it’s potential energy PE e = (½)kx 2 to mass in the form of kinetic energy KE = (½)mv 2

The applied Force F app is equal & opposite to the force F s exerted by block on the spring: F s = - F app = -kx

Force Exerted by a Spring on a Block The spring force F s varies with the block position x relative to equilibrium at x = 0. F s = -kx. Spring constant k > 0 x > 0, F s < 0 x = 0, F s = 0 x 0 F s (x) vs. x

W = (½)kx 2  Relaxed Spring Spring constant k x = 0   x  W W In (a), the work to compress the spring a distance x: W = (½)kx 2 So, the spring stores potential energy in this amount. W W W W W In (b), the spring does work on the ball, converting it’s stored potential energy into kinetic energy. W

Elastic PE PE elastic = (½)kx 2 KE = 0 PE elastic = 0 KE = (½)mv 2

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 2 nd Law says ∑F y = 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 x 1 to x 2, The change in PE is:  (PE) elastic = (½)k(x 2 ) 2 - (½)k(x 1 ) 2 The work done is: W = -  (PE) elastic The PE belongs to the system, not to individual objects.

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 is conservative if & 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 force as 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 = PE f – PE i = - 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

In other words, if a force is Conservative, a PE CAN be defined. But, if a force is Non-Conservative, a PE CANNOT be defined !! Potential Energy: Can only be defined for Conservative Forces!

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: W net = W C + W NC W C ≡ work done by conservative forces W NC ≡ work done by non-conservative forces The work energy theorem still holds: W net = W C + W NC =  KE For conservative forces (by the definition of PE): W C = -  PE   KE = -  PE + W NC or: W NC =  KE +  PE

 In general, W NC =  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: W NC =  KE +  PE For the Very Special Case of Conservative Forces Only  W NC = 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 = KE 1 + PE 1, & finally: E = KE 2 + PE 2. But, E = Constant, so  KE 1 + PE 1 = KE 2 + PE 2 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 = KE 1 + PE 1 = KE 2 + PE 2  (½)m(v 1 ) 2 + mgy 1 = (½)m(v 2 ) 2 + mgy 2 y 1 = Initial height, v 1 = Initial velocity y 2 = Final height, v 2 = Final velocity

PE 1 = mgh, KE 1 = 0 PE 2 = 0 KE 2 = (½)mv 2 KE 3 + PE 3 = KE 2 + PE 2 = KE 1 + PE 1 but their sum remains constant! KE 1 + PE 1 = KE 2 + PE mgh = (½)mv v 2 = 2gh all PE half KE half PE all KE KE 1 + PE 1 = KE 2 + PE 2 = KE 3 + PE 3

Example: Falling Rock This is a very common error! WHY???? Energy “buckets” are not real!! Speeds at y 2 = 0.0, & y 3 = 1.0 m? Mechanical Energy Conservation! KE 1 + PE 1 = KE 2 + PE 2 (½)m(v 1 ) 2 + mgy 1 = (½)m(v 2 ) 2 + mgy 2 = (½)m(v 3 ) 2 + mgy 3 (Mass cancels!) y 1 = 3.0 m, v 1 = 0, y 2 = 1.0 m, v 2 = ?, y 3 = 0.0, v 3 = ? Results: v 2 = 6.3 m/s, v 3 = 7.7 m/s NOTE!! Always use KE 1 + PE 1 = KE 2 + PE 2 = KE 3 + PE 3 NEVER KE 3 = PE 3 !!!! In general, KE 3 ≠ PE 3 !!!

PE only part PE part KE KE only v 1 = 0 y 1 = 3.0 m v 2 = ? y 2 = 1.0 m v 3 = ? y 3 = 0 (½)m(v 1 ) 2 + mgy 1 = (½)m(v 2 ) 2 + mgy 2 = (½)m(v 3 ) 2 + mgy 3 (Mass cancels!) y 1 = 3.0 m, v 1 = 0, y 2 = 1.0 m, v 2 = ?, y 3 = 0.0 m, v 3 = ? Results: v 2 = 6.3 m/s, v 3 = 7.7 m/s Speeds at y 2 = 0.0, & y 3 = 1.0 m? Mechanical Energy Conservation! Cartoon Version!

Example: Roller Coaster Mechanical energy conservation! (Frictionless!)  (½)m(v 1 ) 2 + mgy 1 = (½)m(v 2 ) 2 + mgy 2 (Mass cancels!) Only height differences matter! Horizontal distance doesn’t matter! Speed at the bottom? y 1 = 40 m, v 1 = 0 y 2 = 0 m, v 2 = ? Find: v 2 = 28 m/s What is y when v 3 = 14 m/s? Use: (½)m(v 2 ) = (½)m(v 3 ) 2 + mgy 3 Find: y 3 = 30 m 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 NOTE!! Always use KE 1 + PE 1 = KE 2 + PE 2 = KE 3 + PE 3 Never KE 3 = PE 3 ! A very common error! WHY???? In general, KE 3 ≠ PE 3 !!!

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