Unimportable 8279 AE 55 4 00 Lecture 10

Unimportable #8279 AE 55 4. 00 Lecture 10 Purdue University, Physics 220 1

PHYSICS 220 Lecture 10 Potential Energy and Energy Conservation Lecture 10 Purdue University, Physics 220 2

Work Done by Gravity Ball is tossed up with velocity v. A. What is the work done by gravitational force? How far up it will go? B Only force/work done be gravity Wg = - mgh = KEB – KEA = ½m v. B 2 - ½m v. A 2 -mgh = -½m v. A 2 h h = v. A 2 / 2 g A Lecture 9 Purdue University, Physics 220 3

Work Done by Gravity Slide block down incline Wg = (mg)(s)cos s = h/cos Wg = mg(h/cos )cos h Wg = mgh Lecture 9 Purdue University, Physics 220 mg S 4

Work Done by Gravity • Depends only on initial and final height! • Wg = -mg(yf - yi) – Independent of path • If you end up where you began, Wg = 0 Whenever the work that is done by a force is independent of its path and it's only determined by the starting point and the end point that force is called a "conservative force“. Lecture 9 Purdue University, Physics 220 5

Potential Energy • Work done by gravity is independent of path • Wg = -mg (yf - yi) = - PEg • Define PEg = mgy • Only the difference in potential energy is physically meaningful, i. e. , you have the freedom to choose the reference (or zero potential energy) point. • Works for any CONSERVATIVE force Lecture 10 Purdue University, Physics 220 6

Work-Energy with Conservative Forces Work-Energy Theorem Move work by conservative forces to other side If there are NO non-conservative forces Conservation of mechanical energy Lecture 10 Purdue University, Physics 220 7

Skiing Example (no Friction) A skier goes down a 78 meter high hill with a variety of slopes. What is the maximum speed the skier can obtain starting from rest at the top? Conservation of energy: KEi + PEi = KEf + PEf ½ m v i 2 + m g y i = ½ m v f 2 + m g y f 0 + g y i = ½ v f 2 + g y f vf 2 = 2 g (yi-yf) vf = sqrt( 2 g (yi-yf)) Lecture 10 vf = sqrt( 2 x 9. 8 x 78) = 39 m/s Purdue University, Physics 220 8

Skiing with Friction A 50 kg skier goes down a 78 meter high hill with a variety of slopes. She is observed to be going 30 m/s at the bottom of the hill. How much work was done by friction? Work Energy Theorem: Wnc = (KEf + PEf) - (KEi + PEi) = (½ m vf 2 + m g yf) - (½ m vi 2 + m g yi) = ½ (vf 2 - g yi )m = (½ (30)2 – 9. 8 x 78) 50 = (450 – 764) 50 Joules = -15700 Joules Lecture 10 Purdue University, Physics 220 9

i. Clicker 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) B) C) D) Dropping Slide on ramp (no friction) Swinging down correct All the same A B C Conservation of Energy (Wnc=0) KEinitial + PEinitial = KEfinal + PEfinal 0 + mgh = ½ m v 2 final + 0 vfinal = sqrt(2 g h) Lecture 10 Purdue University, Physics 220 10

Pendulum Exercise • As the pendulum falls, the work done by the string is A) Positive B) Zero C) Negative W = F d cos . But = 90 degrees so Work is zero. • How fast is the ball moving at the bottom of the path? Conservation of Energy (Wnc=0) SWnc = KE + PE 0 = KEfinal - KEinitial + PEfinal - PEinitial KEinitial + PEinitial = KEfinal + PEfinal 0 + mgh = ½ m v 2 final + 0 vfinal = sqrt(2 g h) Lecture 10 Purdue University, Physics 220 h 11

Pendulum Exercise With no regard for his own personal safety your physics professor will risk being smashed by a bowling ball pendulum! If released from a height h, how far will the bowling ball reach when it returns? Conservation of Energy (Wnc=0) SWnc = KE + PE 0 = KEfinal - KEinitial + PEfinal- PEinitial KEinitial + PEinitial = KEfinal + PEfinal 0 + mghinitial = 0 + mghfinal hinitial = hfinal Lecture 10 Purdue University, Physics 220 h 12

Power (Rate of Work) • Pav = W / t Units: Joules/Second = Watt • W = F r cos = F (v t) cos • P = F v cos • How much power does it take for a (70 kg) student to run up the stairs (5 meters) in 7 seconds? Pav = W / t = mgh /t = (70 kg) (9. 8 m/s 2) (5 m) / 7 s = 490 J/s Lecture 10 or 490 Watts Purdue University, Physics 220 13

Gravitational Potential Energy • If the gravitational force is not constant or nearly constant, we have to start from Newton’s gravitational force law • The gravitational potential energy is: Lecture 10 Purdue University, Physics 220 14

Problem: How High? • A projectile of mass m is launched straight up from the surface of the earth with initial speed v 0. What is the maximum distance from the center of the earth RMAX it reaches before falling back down. RMAX RE m v 0 M Lecture 10 Purdue University, Physics 220 15

Problem: How High. . . • All forces are conservative: • And we know: WNC = 0 KE = - PE RMAX RE m v 0 h. MAX M Lecture 10 Purdue University, Physics 220 16

Problem: How High. . . RMAX RE m v 0 h. MAX M Lecture 10 Purdue University, Physics 220 17

Escape Velocity • If we want the projectile to escape to infinity we need to make the denominator in the above equation zero: We call this value of v 0 the escape velocity, vesc Lecture 10 Purdue University, Physics 220 18

i. Clicker 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 (Wnc=0) SWnc = KE + PE KEinitial + PEinitial = KEfinal + PEfinal 0 + mgh 1 = 0 + mgh 2 h 1 = h 2 m h 1 Lecture 10 h 2 Purdue University, Physics 220 19