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1 l LON-CAPA #3 and Mastering Physics due next Tuesday help room hours (Strosacker Help Room, 1248 BPS): M: 5-8 PM W: 5-8 PM F: 2-6 PM l Register for Mastering Physics >95% of you have l Register your iclicker on LON-CAPA >98% of you have l First iclicker question from Tuesday will not be graded because of iclicker problems l First exam: Feb 6 in Life Sciences A133

2 Resolving vectors l Just as we can add any two vectors to get a resultant vector, we can also resolve any vector into two components perpendicular to each other as for example, the vertical and horizontal components of a thrown ball Note that there is a change in the velocity of the ball. It is accelerating in the downward direction.

3 Forces l A girl hangs from a rope l What are the forces involved? l Which end of the rope is more likely to break? l The force of gravity is acting downward l so there must be an equal and opposite reaction force l That force can be divided into vector components along the two ropes l The larger force is on the rope segment to the right so that is where the rope is more likely to break

4 Friction l Consider the forces acting on a sled initially moving with a velocity v l Gravity is pulling downward with a force F l There s an equal and opposite normal force N applied from the sled to the ground l There s a frictional force proportional to the normal force f k =µ k N where µ k is called the coefficient of kinetic friction l So there s no net force in the vertical direction l There is a net force in the horizontal direction l So what happens to the sled N N F

5 Static friction l What about when there s no motion l Is there still friction? l Yes l There s a gravitational force acting downward l It can be resolved into two components one perpendicular to the surface, balanced by the normal force one along the surface l There s a frictional force also along the surface, proportional to the normal force f s =µ s N l If the horizontal forces are balanced, the frog stays put l What happens if I start tilting the frog?

6 l Consider all of the forces acting in this situation Motion of a car air resistance normal force l As the tire moves over the road, the tire flattens. The force exerted by the road on the tire acts to slow the tire s rotation. l Rolling resistance l Less pronounced for steel wheels gravity

7 Gravitation l Remember the story about Isaac Newton and the apple l Maybe it didn t happen in reality, but the reasoning is roughly correct l Newton had the insight to realize that the force between the Earth and the apple is the same as the force between the Earth and the Moon

8 Gravitation l The apple falls to the ground with an acceleration of 9.8 m/ s 2 l The Moon falls away from the straight line path it would follow if there were no forces acting on it remember Newton s first law l Isaac Newton compared the measurements for the Moon s acceleration to his calculations and found they didn t agree so he put his theory away for 20 years, and worked on optics and alchemy

9 Gravitation l Inspired by the appearance of Halley s comet in 1682, he returned to the problem and found errors in the experimental data l Now the data agreed with his predictions l He then published the universal law of gravitation

10 What did he find? l He found the acceleration of the moon to be ~2.7X10-3 m/s 2 l Radius of Earth =6.37X10 6 m l Average Earth-Moon distance is 3.84X10 8 m l So the Moon is about 60 times as far away from the center of the Earth as is someone standing on the surface of the Earth 9.83m /s 2 2.7X = 602 m /s l The acceleration due to gravity decreases as the square of the distance from the center of the Earth video

11 Universal law of gravitation l Incredibly simple and beautiful relation Force mass 1Xmass 2 distance 2 Force m 1m 2 l The gravitational force between two masses is proportional to the product of the two masses and is inversely proportional to the square of the distance between them l Before we dealt with mass in the equation below, technically known as the inertial mass F = ma l Here we are dealing with the gravitational mass l It is experimentally established that the two are equal and in fact the equivalence of the two is an integral part of the general theory of relativity

12 Universal law of gravitation l We need a real equation, which means we need a constant of proportionality F = G m 1m 2 l Newton didn t know the value of G by itself, only the product of G times the mass of the Earth l The measurement of G was big news in 1798 l Knowing G meant that the mass of the Earth could be calculated 6X10 24 kg G was determined by measuring the attraction of two masses; a difficult measurement since the force is very small G=6.67X10-11 Nm 2 /kg 2

13 How small of a force? F = G m 1m 2 F = 6.67X10 11 Nm 2 /kg2 (1kg)(5000kg) (1m) 2 F = 3.34X10 7 N m 1 =1kg m 2 =5000kg d=1m The small value of G is an indication of how weak the gravitational force is. G was determined by measuring the attraction of two masses; a difficult measurement since the force is very small G=6.67X10-11 Nm 2 /kg 2

14 Weight l Let s consider another force, your weight, i.e. the force the Earth exerts on you l Suppose you weigh 60 kg l m 1 =m Earth =6X10 24 kg l m 2 =m you =60 kg l d=r earth =6.37X10 6 m F = G m 1 m 2 F = 6.67X10 11 Nm 2 /kg 2 (6X10 24 kg)(60kg) (6.37X10 6 m) 2 F = 591N

15 Clicker question l What is the gravitational force that you exert on the Earth? A) 0 N B) 591 N C) 6.67X10-11 N D) 6X10 24 N E) cannot be determined with the information given F = G m 1 m 2 F = 6.67X10 11 Nm 2 /kg 2 (6X10 24 kg)(60kg) (6.37X10 6 m) 2 F = 591N

16 Clicker question l What is the gravitational force that you exert on the Earth? A) 0 N B) 591 N C) 6.67X10-11 N D) 6X10 24 N E) cannot be determined with the information given F = G m 1 m 2 F = 6.67X10 11 Nm 2 /kg 2 (6X10 24 kg)(60kg) (6.37X10 6 m) 2 F = 591N

17 Acceleration l What is your acceleration due to the gravitational force of the Earth? a = F m you = G m Earthm you m you a = G m Earth = 9.8N /kg = 9.8m /s 2 = g l That is, your acceleration is the same as your friend who has twice the mass that you do (and weighs twice as much)

18 F = G m 1m 2 The gravitational force grows weaker as the square of the distance. Inverse square law.

19 Inverse square law l The further away from the center of the Earth that you travel, the smaller your weight will be l If you weigh 600 N at sea level, you ll weigh 598 N on top of Mt Everest l The gravitational force falls off to zero as the distance from the center of the Earth goes to infinity

20 Weightlessness l Suppose that you re in an elevator that s stationary (or moving with a constant velocity) you stand on a scale and it gives your normal weight l Suppose the elevator accelerates upward the scale registers a larger weight; you feel heavier l The elevator accelerates downward the scale registers a lower weight; you feel lighter l The elevator cable snaps the scale registers zero weight; you feel weightless

21 Weightlessness l If you have enough money, you book a flight on a vomit comet that lets you experience weightlessness for a brief period of time l Or you can spend some time on the International Space Station

22 Space elevator l One of the problems refers to a space elevator l This would be a convenient way of transporting material into a geosynchronous orbit without the use of any rockets l One end of the elevator is attached to the surface of the Earth, the other to a counterweight l It s stable l But we don t currently have the technology to build a strong enough tower l Carbon nanotubes are close; they would work for an elevator on the Moon or on Mars

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24 Central force l l l l Suppose I have a mass m 1 and I want to calculate the gravitational force on this mass from the Earth What is the direction of the force on the mass? towards the center of the Earth What is the direction of the force on the Earth? along the line joining m 1 and the center of the Earth, pointing towards m 1 What is the magnitude of the force? F = G m 1m 2 m 1 d l The Earth has a radius of 6.37X10 6 m, but from a gravitational point of view, it acts like all of the mass is concentrated in a point at the center of the Earth How do we know that? Isaac Newton had to develop calculus to prove it.

25 Roundness l Why is the Earth round? l Because every part of the Earth attracts every other part and so the Earth is pulled together as tightly as possible a sphere l The Moon is round for a similar reason l The asteroids, except for the largest ones, are not round, because they don t have sufficient mass

26 Oblateness of the Earth l We discussed that gravity is responsible for the planets being round l But they re not exactly round due to the rotational motion of the Earth l The Earth tends to be somewhat thicker at the equator than at the poles l Not by much km at equator km at the poles l So a round sphere is still a very good representation of the shape of the Earth How different is the force of gravity at the North Pole and at the Equator? Take a mass of 50 kg. F = G m 1 m 2

27 Oblateness of the Earth l The Earth tends to be somewhat thicker at the equator than at the poles l Not by much km at equator km at the poles F equator = W equator = G mm Earth F equator = (6.67X10 11 Nm 2 /kg 2 ) (50kg)(6X1024 kg) (6.378X10 6 m) 2 F equator = 491.9N F pole = (6.67X10 11 Nm 2 /kg 2 ) (50kg)(6X1024 kg) (6.357X10 6 m) 2 F pole = 495.2N How much did the mass change? How different is the force of gravity at the North Pole and at the Equator? Take a mass of 50 kg. F = G m 1 m 2

28 Jupiter l Jupiter is even more oblate since it s much larger than the Earth it rotates much faster (one day = 10 hours) it s composed mostly of fluid l How much would this 50 kg person weigh on the equator of the surface of Jupiter? l Jupiter is 300 times as massive as the Earth l Why isn t the weight 300 times as much? F = W = G mm Jupiter F = (6.67X10 11 Nm 2 /kg 2 ) (50kg)(1.90X1027 kg) (7.13X10 7 m) 2 F = 1246N

29 Neutron star l A neutron star has a mass of 4 X kg (about twice the sun s mass) and a radius of 10 km (about 1/70000 th that of the sun) l Remember that most of an atom is empty space; in a neutron star all of the atoms collapse down to the size of their nuclei l What would be the weight of a 50 kg person on the surface of this neutron star? F = G mm neutronstar F = (6.67X10 11 Nm 2 /kg 2 ) (50kg)(4 X1030 kg) (1X10 4 m) 2 F = 1.33X10 14 N l Or about 300 billion times as much as on the surface of the Earth l A neutron star has the same density as the nucleus of an atom