c Swinburne University of Technology, 2010 35 OF 1 AGE P Mass HET625-M04A01: The EquivalenceMass Principle: A Question of The Equivalence Principle: A Question of c Swinburne University of Technology, 2010 35 OF 2 AGE P HET625-M04A01: The EquivalenceMass Principle: A Question of the differences between inertial mass and gravitational mass;the and Weak Equivalence Principle. gravitational acceleration; Galileo’s experiments with pendulums and inclined planes; • • • • In this Activity, we will investigate: Summary c Swinburne University of Technology, 2010 35 OF 3 AGE P to “forced” experienced and exactly the same rate created of the object which is being mass states that gravity is a property depends on the . mass law of motion . second showed that objects with different masses and compositions fell at gravity , whatever it is, seems to be a very important part of gravitation. HET625-M04A01: The EquivalenceMass Principle: A Question of Newton’s law of universal gravitation Newton’s Galileo by all objects which have accelerate. under mass • • • So Let’s recap a couple of important results that we will need for this Activity. The story so far... c Swinburne University of Technology, 2010 35 OF 4 AGE P will increase: velocity 2 − 8 ms . = 9 g causes an acceleration. HET625-M04A01: The EquivalenceMass Principle: A Question of gravitational force If an object is dropped in a gravitational field, its Near the surface ofclose the to 9.8 Earth, metres per theof second acceleration the per Earth). due second (directed to towards the gravity centre isThis is very often written as: the Gravitational acceleration c Swinburne University of Technology, 2010 35 OF 5 AGE P traveled each of: distance speed after two seconds after three seconds after four seconds after five seconds. . . after one second 1 1 1 1 1 − − − − − HET625-M04A01: The EquivalenceMass Principle: A Question of 9.8 m s 19.6 m s 29.4 m s 39.2 m s 49.0 m s second also increases. After letting go, an object will have a ...unless it hits the ground first! As the velocity increases, the vertical c Swinburne University of Technology, 2010 35 OF 6 AGE P ! 1 − HET625-M04A01: The EquivalenceMass Principle: A Question of Could Galileo really havedropping performed them an from the experiment Leaning todropped Tower of show from Pisa? that this The height objects Tower is120 would fell around km take at hour 56 3.4 the metres seconds high, same to so rate reach thatIt by a the ball would ground with have a beenespecially final without quite the speed benefit an of of photography nearly effort orof a air to stopwatch. resistance... Plus determine there was which the added object disadvantage So hit if Galileo the didn’t perform his ground experimentwas at first independent Pisa, of how at did mass? he this show that speed, the gravitational acceleration Galileo’s experiment? c Swinburne University of Technology, 2010 35 OF 7 AGE P of the length ) depends on the period g L r . ∝ g Period , and the acceleration due to gravity, L HET625-M04A01: The EquivalenceMass Principle: A Question of pendulum, He used a pendulum! The time it takes a pendulum to complete one swing (the Swinging into action c Swinburne University of Technology, 2010 35 OF 8 AGE P /100 /10 total total total T ), the more accurately you can T T = = = T total T T T 1 swing gives 1 measurement, so 10 swings gives 10 measurements, so 100 swings gives 100 measurements, so HET625-M04A01: The EquivalenceMass Principle: A Question of ...in theory, anyway. If it wasn’t forpendulum would air keep resistance swinging and foreveras with friction the the height at same of period. each the upswing In join decreases. reality,Galileo between the was able period the to slowly show pendulum changes thatthe and the gravitational “bob” the acceleration on was ceiling, the the same“bob”, the pendulum for the all with period objects remained different by the replacing masses. same! Regardless of the mass or composition of the The more times themeasure the pendulum period, swings (total elapsed time, c Swinburne University of Technology, 2010 35 OF 9 AGE P  1 2 2 R M GM 2 1 R  acceleration × GM × come from? And how do we actually measure it? 2 = 2 M − F = = mass F F HET625-M04A01: The EquivalenceMass Principle: A Question of We can write this as: This looks a lot like Newton’s second law: Where does this mysterious value of 9.8 ms 9.8 metres per second per second Let’s go back to Newton’s universal law of gravitation: c Swinburne University of Technology, 2010 35 OF 10 AGE P 1 M g 1 × 2 2 R GM M = = g F . 1 M will experience a force: from 2 M R is called the gravitational mass of the body, as it determines . 2 1 M M HET625-M04A01: The EquivalenceMass Principle: A Question of how that object willwith respond mass to the gravitational field produced by the body The mass when it is a distance as: then a body of mass If we define the gravitational acceleration produced by a body with mass Gravitational mass c Swinburne University of Technology, 2010 35 OF 11 AGE P or somewhere in between as the Earth equator 2 − kg 2 Nm 2 kg m 11 − 1 24 6 − 2 R 10 10 10 GM 8 ms . × × × = 1 98 38 67 = 9 g . . . M R g = 5 = 6 = 6 G at the surface of the Earth changes a little Mass = Radius = g HET625-M04A01: The EquivalenceMass Principle: A Question of This is the gravitational acceleration at theand surface is of directed the towards Earth, the Earth’s centre. The value of depending on whether you are at the poles, the we get: is not a perfect sphere. If we take the values for the Earth: and substitute them into: c Swinburne University of Technology, 2010 35 OF 12 AGE P y coordinate (i.e. it written with a minus y g pointing away from the with the ground level at of the axis 2 - − y axes 8 ms . 9 − = g negative values = 0). We have to include the minus sign in order y . HET625-M04A01: The EquivalenceMass Principle: A Question of to get the directionvector of the acceleration correct, as acceleration is a sign as: Suppose we draw a set of= coordinate 0 (for convenience), and theground. positive If we haveacceleration due an to the objectto Earth’s move gravitational at towards field more causes somemoves the towards height object above the ground, the Just to sidetrack briefly, sometimes you will see Why the minus sign? c Swinburne University of Technology, 2010 35 OF 13 AGE P -axis = 0 to y y -axis pointing towards the ground, y ! g to mean the magnitude of the gravitational acceleration. If we want 2 − = 9.8 ms g HET625-M04A01: The EquivalenceMass Principle: A Question of to be explicit about the directionwill (which include starts the minus to sign matter or when draw you an are arrow doing to actual indicate calculations), the we direction. However. . . sometimes it can be a nuisance to incorporate the minus sign, as the choice of a Mathematical notation pointing away from the ground isbe not the always location the best of one thewe to object do use. we not For are need example, about if the we to minus had sign drop, set in andFor convenience had (and the to savewill us use having the to symbol include a whole lot of extra mathematical notation!), we c Swinburne University of Technology, 2010 35 OF 14 AGE P HET625-M04A01: The EquivalenceMass Principle: A Question of The gravitational force causes theslower, ball and to therefore accelerate easier down to the observe slope, than but objects at in speeds free-fall. which are much Another experiment which Galileo performed involved rolling balls down inclined planes. Experimenting with inclined planes c Swinburne University of Technology, 2010 35 OF 15 AGE P HET625-M04A01: The EquivalenceMass Principle: A Question of The shallower the slope, theheight). greater the horizontal motion (if both balls start moving from the same Galileo observed that by using twothat slopes, the one ball for would the reach ball almost to the roll sameIt down height doesn’t and that quite one it for get the had there ball started because to of at. roll friction, up, which takes away some of the ball’s energy. c Swinburne University of Technology, 2010 35 OF 16 AGE P of Motion, also Newton’s First Law which causes it to keep moving in the absence of inertia . Law of Inertia HET625-M04A01: The EquivalenceMass Principle: A Question of The ball has aany special force. property This is called incausing contrast the to motion. Aristotle’s belief that objectsGalileo’s did experiments not with move inclined unless planes there providedcalled was the the a basis force for Inertia Galileo reasoned that if there wascontinue no to friction, travel when without the requiring ball a reached force the to bottom propel of it the further. slope it would c Swinburne University of Technology, 2010 35 OF 17 AGE P acceleration × = inertial mass F of an object is its resistance to a force. Any time a force is applied to a body in its motion (that is, produce an acceleration), the body opposes that change through change HET625-M04A01: The EquivalenceMass Principle: A Question of inertial mass The greater the inertial mass, the harder it is to accelerate for a given strength of force. its inertial mass. We should really write Newton’s Second Law as: The order to Inertial mass c Swinburne University of Technology, 2010 35 OF 18 AGE P . inertial mass . Are these two masses the same? What are the arguments for and against this idea? HET625-M04A01: The EquivalenceMass Principle: A Question of Discuss! If I push an object, it responds to that push with its So, if Igravitational put mass an object in a gravitational field, it responds with its There is absolutely no reasonphysicists why have they been have able to to be perform, the they same, are. but according to the best experiments Gravitational or inertial mass? c Swinburne University of Technology, 2010 35 OF 19 AGE P ¨ os and his colleagues ¨ otv ¨ os ¨ otv HET625-M04A01: The EquivalenceMass Principle: A Question of ´ and Baron von E ´ or L At the turnperformed of a the series 20th ofmasses century, experiments of to the different substances determine Hungarian were whether the physicist same. theConsider Baron ratio a of von ball tied the E to inertial a string and with gravitational the other end attached to a pole in the ground. c Swinburne University of Technology, 2010 35 OF 20 AGE P HET625-M04A01: The EquivalenceMass Principle: A Question of If the ball had no gravitationalthe mass, Earth. its inertial mass would cause it to be flung out by the rotation of c Swinburne University of Technology, 2010 35 OF 21 AGE P HET625-M04A01: The EquivalenceMass Principle: A Question of If the ball had no inertial mass, its gravitational mass would pull it towards the centre of the Earth. c Swinburne University of Technology, 2010 35 OF 22 AGE P . angle HET625-M04A01: The EquivalenceMass Principle: A Question of Since a real ball has both inertial and gravitational mass, it will hang at a slight c Swinburne University of Technology, 2010 35 OF 23 AGE P ¨ os was used by three physicists, P. G. Roll, R. Krotkov and ¨ otv HET625-M04A01: The EquivalenceMass Principle: A Question of on various materials. Sun In their experiment, they made use of a torsion balance. A torsion balance allowsmasses a (in this very case accurate gold and measurement aluminium) of are attached the via rotation a of beam. a thread, to which two A similar idea to that of Baron von E Experimenting with torsion balances R. H. Dicke, who performed an experiment in 1964 which compared the force of gravity due to the c Swinburne University of Technology, 2010 35 OF 24 AGE P of the two materials, there would inertial m / = 1 inertial gravitational M gravitational m M HET625-M04A01: The EquivalenceMass Principle: A Question of be a detectable rotation of the torsion balanceNo over detectable a rotation 24 was hour measured: period. Twelve hours later, the Earth has rotated, andIf the there position was of a the apparatus difference changes between with the it. ratio Consider the view ofexperimental arrangement the is shown Earth at 0600 looking hours. down from the North Pole at the torsion balance. The c Swinburne University of Technology, 2010 35 OF 25 AGE P (WEP) and it is one of the most important By neutral, we just meangravity, and that not the by body electrostatic is or only magnetic being forces. acted on by Weak Equivalence Principle the motion of a neutral test body released at a . He called it the HET625-M04A01: The EquivalenceMass Principle: A Question of Universe The WEP says that: given point in space-time is independent of its composition. pieces of General Relativity. Einstein looked on this equality ofthe inertial and gravitational masses as something fundamental about It’s fundamental c Swinburne University of Technology, 2010 35 OF 26 AGE P all materials fall at . 12 HET625-M04A01: The EquivalenceMass Principle: A Question of This is exceptional experimental evidence that theis Weak Equivalence valid, Principle and as Galileothe suspected same over rate regardless 300 of years their earlier: composition. Some of the mostperformed accurate by experiments R. H. performed Dicke toat and exactly test V. the B. the same Braginsky. rate Weak They to showed Equivalence one that part Principle gold, in were platinum 10 and aluminium fall Experimental evidence for the WEP c Swinburne University of Technology, 2010 35 OF 27 AGE P determines the strength of a gravitational field. is the response of an object to a gravitational field. HET625-M04A01: The EquivalenceMass Principle: A Question of Active gravitational mass Passive gravitational mass • • We have now shownthese that are there identical according are to two the types Weak EquivalenceWe of Principle. can mass, go further gravitational and mass distinguish and between two inertial different mass, types of but gravitational mass: Active versus passive mass c Swinburne University of Technology, 2010 35 OF 28 AGE P HET625-M04A01: The EquivalenceMass Principle: A Question of The gravitational field generated by the Earth is due to its passive gravitational mass. c Swinburne University of Technology, 2010 35 OF 29 AGE P HET625-M04A01: The EquivalenceMass Principle: A Question of The forces we experiencegravitational masses. as we walk aroundBut, on as the with Earth’s inertialidentical. surface and are gravitational a mass, result the of active our and active passive masses of a body are also c Swinburne University of Technology, 2010 35 OF 30 AGE P is fundamental to Einstein’s . are the same thing. This may seem weightlessness inertial mass and Weak Equivalence Principle gravitational HET625-M04A01: The EquivalenceMass Principle: A Question of development of his General Theory. Before we leave mass toweight. delve further Or more into importantly General for Relativity, the we next need Activity, to consider one more thing: obvious, but it is actually quite profound. The Newtonian model of universal gravitationmasses is of not an seriously object affected are if different. the But gravitational the and inertial So, according to Einstein, More weighty matters c Swinburne University of Technology, 2010 35 OF 31 AGE P apparent with your bathroom scales. Miraculously enough, you Moon : NASA Credit about 1/6 of what you did on Earth. Celebrate by having some chocolate or ice-cream! is 70 kilograms. Now travel to the HET625-M04A01: The EquivalenceMass Principle: A Question of weigh now How much do youanswered truthfully. weigh? It’s aSuppose common you enough were to question, stand and on a one pair which of bathroom doesn’t scales always on the get Earth, and find that your Weight or mass? weight c Swinburne University of Technology, 2010 35 OF 32 AGE P in kilograms or pounds and depends on the strength of the force apparent weight HET625-M04A01: The EquivalenceMass Principle: A Question of Next, take your scales withagain. you Oh to no! Jupiter, You now and weigh nearlyMaybe stand 3 on you times your them shouldn’t Earth-weight. haveMoon. . celebrated . quite so muchIf on the you give your or stones,because then weight is you a aren’tgravitational field acting quite on you. telling the whole truth, c Swinburne University of Technology, 2010 35 OF 33 AGE P : NASA, Johnson Credit Space Center, 1972 g × M . 2 − = 9.8 m s g gravitational acceleration = × Mass HET625-M04A01: The EquivalenceMass Principle: A Question of Your bathroom scales measure howforce the much scales of must a push downwardthe back force force at on you you you (Newton’s are due Third to applying, Law). gravity and is: According how to much Newton’s Second Law, Never leave your mass behind Your scales were built on Earth, andacceleration so at are the calibrated surface against of the the gravitational Earth, Whether you arethe on same the amount Earth,gravitational of the acceleration matter. at Moon the or Yourthe surface same apparent Jupiter, of as weight for the your the changes Moon Earth. body and becauseAnd contains at the strictly Jupiter speaking, are the not Newtons! correct units of weight (which is simply a force) is c Swinburne University of Technology, 2010 35 OF 34 AGE P is not g = 0? g = 0 g × Mass ! But you would still have the same mass as before. In the weightless HET625-M04A01: The EquivalenceMass Principle: A Question of zero... In other words, you would be next Activity, we will look at some things you can do to lose all of your weight, even when What if there was no gravitational acceleration, so that, locally, Guaranteed weight loss Then the force due to gravity is: c Swinburne University of Technology, 2010 35 OF 35 AGE P HET625-M04A01: The EquivalenceMass Principle: A Question of In this activity we looked(inertial at and early gravitational) and experiments the of formulation gravity, of two theIn different Weak the kinds Equivalence next Principal. of activity we’re experiential goingstart masses to by investigate jumping gravity in and an acceleration elevator in and quite going a for bit a more ride... detail. We’ll Summary