Welcome back to Physics 215 Today s agenda: rolling friction & review Newtonian gravity Planetary orbits Gravitational Potential Energy Physics 215 Spring 2017 Lecture 13-1 1 Midterm 3 Thursday April 13th Material covered: Ch 9 Ch 12 (and conservation of momentum) Lectures through last Thursday Also draws on material from Ch 1-8 HW, lectures, demos, recitations No office hours after exam on Thursday April 13 th or Friday April 14 th. Practice Exam available online Midterm review with Society of Physics students TODAY (Tuesday) 6-8pm Physics 233 Pizza and soft drinks and mentoring help review for the midterm, too. Physics 215 Spring 2017 Lecture 13-1 2 1
Rolling without slipping translation rotation v cm = a cm = Physics 215 Spring 2017 Lecture 13-1 3 Kinetic energy of rolling The total kinetic energy of a rolling object is the sum of its rotational and translational kinetic energies: Physics 215 Spring 2017 Lecture 13-1 4 2
SG. Two cylinders with the same radius and same total mass roll down a ramp. In cylinder A, a set of 8 point masses are equally spaced in a circle with radius r1 around the cylinder s axis of rotation, while in cylinder B, the 8 point masses are a distance r2 > r1 from the center. Which cylinder reaches the bottom of the ramp first? A. Cylinder A B. Cylinder B C. They both reach the bottom at the same time D. Not enough information to tell Physics 215 Spring 2017 Lecture 13-1 5 A block of mass m=2.0kg is set into motion down a rough inclined plane with speed 4.0 m/s as shown in the picture. The coefficient of kinetic friction between the block and the plane is 0.4 (take g=9.8 m/s2). What is the speed of the block after it has moved 1.0 m down the incline? v=2.0m/s 30 0 Physics 215 Spring 2017 Lecture 13-1 6 3
A toy race car of mass 20 g is traveling on a frictionless loopthe-loop track. The diagram shows the track from the side. The radius of the loop track is R=0.1 m. The initial speed of the car at the bottom of the loop is 2.5 m/s. What is the speed of the car at the top of the loop? What is the normal force on the car at the top? al) R Physics 215 Spring 2017 Lecture 13-1 7 A sled slides along a horizontal surface with a coefficient of kinetic friction of 0.2. Its velocity at point A is 8.0 m/s and its velocity at point B is 5.0 m/s. How long does it take the sled to travel from point A to point B? How far did it slide? Use the work-kinetic energy and the impulse-momentum theorems. Physics 215 Spring 2017 Lecture 13-1 8 4
A child sits on a seesaw of length L = 2m with his father and his pet dog. The mass of the seesaw is 20 kg, the mass of the father is 50 kg, the mass of the dog is 5 kg and the mass of the child is 20 kg. The father sits on the right edge of the seesaw, while the child sits on the left edge. The dog is 0.5 m from the father. If the seesaw is in equilibrium, what is the distance from the pivot to the child? Confirm that the net torque about this pivot point is zero. Physics 215 Spring 2017 Lecture 13-1 9 Gravity Before 1687, large amount of data collected on motion of planets and Moon (Copernicus, Galileo, Brahe, Kepler) Newton showed that this could all be understood with a new Law of Universal Gravitation Physics 215 Spring 2017 Lecture 13-1 10 5
Universal Gravity Mathematical Principles of Natural Philosophy: Every particle in the Universe attracts every other with a force that is directly proportional to their masses and inversely proportional to the square of the distance between them. Physics 215 Spring 2017 Lecture 13-1 11 Inverse square law m 1 F 12 r m 2 Physics 215 Spring 2017 Lecture 13-1 12 6
Interpretation F acts along line between bodies F 12 = -F 21 Law in accord with Newton s Third Acts at a distance (even through a vacuum) G is a universal constant = 6.7 x 10-11 N. m 2 /kg 2 Physics 215 Spring 2017 Lecture 13-1 13 SG: The Earth exerts a gravitational force of 800 N on a physics professor. What is the magnitude of the gravitational force (in Newtons) exerted by the professor on the Earth? 1. 800 divided by mass of Earth 2. 800 3. zero 4. depends on how fast the Earth is spinning Physics 215 Spring 2017 Lecture 13-1 14 7
Motivations for law of gravity Newton reasoned that Moon was accelerating so a force must act Assumed that force was same as that which caused apple to fall Assume this varies like r -p Compare acceleration with known acceleration of Moon à find p Physics 215 Spring 2017 Lecture 13-1 15 Apple and Moon calculation a M = a apple = a M /a apple = But: a rad = v = a M = a M /a apple = 2.7 x 10-4 r M /r E = à p = Physics 215 Spring 2017 Lecture 13-1 16 8
QuickCheck 13.2 SG The gravitational force between two asteroids is 1,000,000 N. What will the force be if the distance between the asteroids is doubled? 1. 250,000 N. 2. 500,000 N. 3. 1,000,000 N. 4. 2,000,000 N. 5. 4,000,000 N. Physics 215 Spring 2017 Lecture 13-1 17 Slide 13-31 What is g? Force on body close to r E = GM E m/r E 2 = mg à g = GM E /r E 2 = 9.81 m/s 2 Constant for bodies near surface Assumed gravitational effect of Earth can be thought of as acting at center (ultimately justified for p = 2) Physics 215 Spring 2017 Lecture 13-1 18 9
SG Planet X has free-fall acceleration 8 m/s 2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y 1. g = 2 m/s 2. 2. g = 4 m/s 2. 3. g = 8 m/s 2. 4. g = 16 m/s 2. 5. g = 32 m/s 2. Physics 215 Spring 2017 Lecture 13-1 19 Slide 13-37 Kepler s Laws experimental observations 1. Planets move on ellipses with the sun at one focus of the ellipse (actually, CM of sun + planet at focus). Physics 215 Spring 2017 Lecture 13-1 20 10
Kepler s Laws experimental observations 2. A line from the sun to a given planet sweeps out equal areas in equal times. *Conservation of angular momentum Physics 215 Spring 2017 Lecture 13-1 21 Clicker 14-1.4. The following is a log-log plot of the orbital period (T) compared to the distance to the sun(r). What is the relationship between T and r? 1. Log T = C r 2. T 2 = C r 3 3. T = C r 4. T 3 = C r 2 For some constant C Physics 215 Spring 2017 Lecture 13-1 22 11
Kepler s Laws experimental observations 3. Square of orbital period is proportional to cube of semimajor axis. We can actually deduce this (for circular orbit) from gravitational law assume gravity responsible for acceleration in orbit à GM S M/r 2 = M(2p r/t) 2 /r T 2 = (4p 2 /GM S )r 3!! Physics 215 Spring 2017 Lecture 13-1 23 Orbits of Satellites Following similar reasoning to Kepler s 3rd law à GM E M sat /r 2 = M sat v 2 /r v = (GM E /r) 1/2 Physics 215 Spring 2017 Lecture 13-1 24 12
The moon is in free fall around the earth. Physics 215 Spring 2017 Lecture 13-1 25 Slide 13-25 Gravitational Field Newton never believed in action at a distance Physicists circumvented this problem by using new approach imagine that every mass creates a gravitational field G at every point in space around it Field tells the magnitude (and direction) of the gravitational force on some test mass placed at that position à F = m test G Physics 215 Spring 2017 Lecture 13-1 26 13
Gravitational Potential Energy Work done moving small mass along path P W = S F. D x But F acts along the radial direction M m D x P Therefore, only component of F to do work is along r W = - S F(r) D r Independent of P! Physics 215 Spring 2017 Lecture 13-1 27 Gravitational Potential Energy Define the gravitational potential energy U(r) of some mass m in the field of another M as the work done moving the mass m in from infinity to r è U = S F(r) D r = -GMm/r Physics 215 Spring 2017 Lecture 13-1 28 14
SG A football is dropped from a height of 2 m. Does the football s gravitational potential energy increase or decrease? 1. decreases 2. increases 3. stays the same 4. depends on the mass of football Physics 215 Spring 2017 Lecture 13-1 29 Gravitational Potential Energy near Earth s surface U h = For small h/r E : D U =(GM E m/r E2 )h = mgh!! as we expect Called the flat earth approximation Physics 215 Spring 2017 Lecture 13-1 30 15
Recall: Energy conservation Consider mass moving in gravitational field of much larger mass M Since W = -D U = D K we have: D E = 0 where E = K+U = 1/2mv 2 - GmM/r Notice E < 0 if object bound Physics 215 Spring 2017 Lecture 13-1 31 Escape speed Object can just escape to infinite r if E=0 à (1/2)mv esc 2 = GM E m/r E à v esc 2 = 2GM E /R E Magnitude? 1.1x10 4 m/s on Earth What about on the moon? Sun? Physics 215 Spring 2017 Lecture 13-1 32 16
Consequences for planets Planets with large escape velocities can retain light gas molecules, e.g. Earth has an atmosphere of oxygen, nitrogen Moon does not Conversely Jupiter, Sun manage to retain hydrogen Physics 215 Spring 2017 Lecture 13-1 33 Sample problem: A less-than-successful inventor wants to launch small satellites into orbit by launching them straight up from the surface of the earth at a very high speed. A) with what sped should he launch the satellite if it is to have a speed of 500 m/s at a height of 400 km? Ignore air resistance. B) By what percentage would you answer be in error if you used a flat earth approximation (i.e. U = mgh)? Physics 215 Spring 2017 Lecture 13-1 34 17
Black Holes Suppose light travels at speed c Turn argument about is there a value of M/R for some star which will not allow light photons to escape? Need M/R = c 2 /2G è density = 10 27 kg/m 3 for object with R = 1m approx Need very high densities possible for collapsed stars Physics 215 Spring 2017 Lecture 13-1 35 18