General Physics I: Day 17 Conservation of Energy & Energy Diagrams

Using Energy Diagrams Tool for visualizing how energy is transformed Example: Object oscillating on a spring

WarmUp: Adding Kinetic A truck initially at rest at the top of a hill is allowed to roll down. At the bottom, its speed is 14 m/s. Next, the truck is again rolled down the hill, but this time it does not start from rest. It has an initial speed of 14 m/s at the top before it starts rolling down the hill. How fast is it going when it gets to the bottom? ~24% → 14 m/s ~9% → 17 m/s ~18% → 20 m/s ~9% → 24 m/s ~40% → 28 m/s True if no energy were added (flat). 2x the energy means… 2x the speed?

An object hangs motionless from a spring An object hangs motionless from a spring. Think about the sum of the elastic potential energy of the spring and the gravitational potential energy of the object and Earth ( 𝑈 Spr +𝑈 g ). When the object is pulled down and held, this sum increases. stays the same. decreases.

Worked-Example: Spring Safety System Suppose the system does fail when the car is at an elevation of 25 meters. If the 600-kg car is carrying six 100-kg passengers, (a) how fast is the car moving when it connects with the spring and (b) how much does the spring compress when it finally stops the car? Mcar = 600 kg Mpassengers = 600 kg k = 36 kN/m h = 25 m vmax=? xspring = ?

Worked-Example: Spring Safety System

Worked-Example: Spring Safety System

Worked-Example: Spring Safety System

Worked-Example: Spring Safety System

Sample Problem (tricky, but cool) A small cap is placed on top of a smooth inverted spherical mixing bowl (as shown on the board). If the cap is nudged slightly, and we ignore friction, at what height will the cap leave the surface of the bowl? mg cos O = mv2/r → v2 = rg cos O mgr = ½ m v2 + mgr cos O → cos O = 2/3 Thus h = 2/3 r

Warm-Up: Coaster A real-world roller coaster is shown. A coaster car is released at point A and coasts without external power. Friction is not negligible in the real world. Does the roller coaster have the same mechanical energy at points B and C? ~33% → Yes ~67% → No Is the total energy conserved during the coaster ride? ~77% → Yes ~23% → No

Warm-Up: Coaster Give a qualitative statement about what forms the energy has when it is halfway down the hill after B. ~17% → Described 𝐾, 𝑈𝑔 and 𝑈 int ~42% → Left out 𝑈𝑔 ~17% → Left out 𝑈 int generated by friction ~17% → Left out 𝐾 ~17% → Described energy “lost” to friction ~8% → Described energies that don’t exist

Warm-Up: Coaster Conceptual miss-steps “C) force energy, potential energy” “c. The energy is still there. It turns into different energies: kinetic and potential.” “c. Some of the energy is lost due to friction and air resistance. The total energy it had at the beginning is not equal to the total after it reached point C.” Watch out for “potential energy” is too vague. Energy is never lost. It has to be there somewhere.

Warm-Up: Coaster “a) No, because Emech = U + K, we know that U is dependent upon the height per the equation U = ugh. So the coaster will have different mechanical energies at points B and C because they are different heights. b) The total energy is conserved but the mechanical energy is not conserved because the sum of KE and PE constantly decreases due to friction.”

Warm-Up: Coaster “a)For mechanical energy solely, yes. At the roller coaster hits points B and C, it has a relationship between potential energy and kinetic energy. 100% PE occurred at Point A, while at the other points the car converts some KE to PE as it climbs each hill. b)Yes, it is continuously converted between potential, kinetic, and internal/non-conservative energies (is friction and air resistance external or internal?) c)U[potential],U[internal], K”

Conservation of Energy! If we only allow conservative forces (quite limited): What if we allow other forces? They can appear in two ways:

Applying Conservation Energy Choose a system! Choose an initial and final situation. Are there non-conservative forces? Do they do work on your system? Alternatively, did the internal energy of the system change? Write down knowns and unknowns for initial and final potential and kinetic energies. Go:

Empire State Penny: If you drop a penny from the top of the Empire State Building (373 m), how fast will it be going at the bottom? First, lets ignore air resistance (a horrible approximation): 60 m/s 86 m/s 7300 m/s

Empire State Penny: A real penny (2 Empire State Penny: A real penny (2.5 g after 1982) has a terminal velocity. One empirical measurement gave 25 mph (11 m/s). How much thermal energy is created as the penny falls from the top of the Empire State Building (373 m) down to the pavement below? 9.0 J 9.1 J -9.1 J 9000 J

Friction & 𝐸 mech. Someone says: “friction causes a loss of mechanical energy”. True? Lots of counter-examples. Often this is about how we define our system. Static friction can do positive or negative work: Sit in the back of a truck, change speed (+ or –). If we choose our system poorly, even kinetic friction can increase Emech.: Pulling a table cloth out from under dishes. Motorcylce tablecloth video http://www.youtube.com/watch?v=KcSwxT_z6io

A child gets on a shallow playground slide A child gets on a shallow playground slide. After pushing himself to get started down the slide he slides at a constant speed all the way to the bottom. Compare the change in gravitational potential energy of the child to the work done on the child by non-conservative forces. 𝑊𝑓 is greater than Δ𝑈𝑔 𝑊𝑓 is less than Δ𝑈𝑔 𝑊𝑓 is equal to Δ𝑈𝑔

Coming up… Tuesday (10/21) → 8.1 – 8.2 WarmUp due Monday by 10:00 PM (w/Image) Ch. 7 Homework due Sunday by 11:59 PM