05 Mass Weight Gravity

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INERTIA, MASS, GRAVITY AND WEIGHT Introduction We have studied interactions and forces between objects. Now we will focus on the objects themselves. Objects are made of matter. Matter is the stuff that we touch, see, smell, lift. We have seen in a previous course that matter comes in bits called the molecules which are made of smaller structures, called atoms that are formed by three different sub atomic particles namely, the protons, neutrons and electrons. Matter has three general properties: extension (occupies a place in space that cannot be occupied by other mass at the same time), inertia (will never change its movement be it in speed or direction), and gravitation (pieces of matter attract themselves to form lumps, planets and finally stars) Inertia and Mass Inertia is the resistance an object has to a change in its state of motion. At the time, Newton's concept of inertia was in direct opposition to the more popular conceptions about motion. The dominant thought prior to Newton's day was that it was the natural tendency of objects to come to rest. Moving objects, or so it was believed, would eventually stop moving since a force was necessary to keep an object moving. If left to itself, a moving object would eventually come to rest and an object at rest would stay at rest; thus, the idea which dominated the thinking for nearly 2000 years prior to Newton was that it was the natural tendency of all objects to assume a rest position. Galileo, the premier scientist of the seventeenth century, developed the concept of inertia. Galileo reasoned that moving objects eventually stop because of a force called friction. In experiments using a pair of inclined planes facing each other, Galileo observed that a ball will roll down one plane and up the opposite plane to approximately the same height. If smoother planes were used, the ball would roll up the opposite plane even closer to the original height. Galileo reasoned that any difference between initial and final heights was due to the presence of friction and he postulated that if friction could be eliminated entirely, then the ball would reach exactly the same height up the opposite plane. Galileo further observed that regardless of the angle at which the planes were oriented, the final height was almost always equal to the initial height. If the slope of the opposite incline was reduced, then the ball would roll a further distance in order to reach that original height. Galileo's reasoning continued – if the opposite incline was elevated at close to a 0-degree angle, then the ball would roll almost forever in an effort to reach the original height. And if the opposing incline was not inclined at all (that is, if it were oriented along the

2 horizontal), then ... an object in motion would continue in motion in an effort to reach the original height. The more mass an object has - the more inertia it has - the stronger the tendency it has to resist changes in its state of motion. Suppose that there are two seemingly identical bricks at rest on a table. However, one brick consists of mortar and the other brick consists of Styrofoam. Without lifting the bricks, how could you tell which brick was the Styrofoam brick? You could give the bricks an identical push in an effort to change their state of motion. The brick which offers less resistance is the brick with less inertia – and therefore the brick with less mass (i.e., the Styrofoam brick). The behaviour of all objects can be described by saying that objects tend to "keep on doing what they're doing" (unless acted upon by an unbalanced force). If at rest, they will continue in this same state of rest. If in motion with an eastward velocity of 5 m/s, they will continue in this same state of motion (5 m/s, East). The state of motion of an object is maintained as long as the object is not acted upon by an unbalanced force. All objects resist changes in their state of motion – they tend to "keep on doing what they're doing." Have you ever experienced inertia (resisting changes in your state of motion) in an automobile while it is braking to a stop? The force of the road on the locked wheels provides the unbalanced force to change the car's state of motion, yet there is no unbalanced force to change your own state of motion. Thus, you continue in motion, sliding forward along the seat. A person in motion tends to stay in motion with the same speed and in the same direction ... unless acted upon by the unbalanced force of a seat belt. Yes, seat belts are used to provide safety for passengers whose motion is governed by Newton's laws. The seat belt provides the unbalanced force which brings you from a state of motion to a state of rest. Perhaps you could speculate what would occur when no seat belt is used. The tendency of an object to resist changes in its state of motion is dependent upon its mass. The mass of an object is the measure of its inertia. To measure inertia (or mass) we should find how an object does move compared to a standard that is called the kilogram (kg). We should give both the object and the standard kilogram the same push and compare how their movement (speed) changed. As it is obvious this procedure is absolutely unpractical but we will see later how we cope with this problem.

3 The mass of an object is just a property of its own: it depends on how many particles it has, the mass of each of the particles and how tightly these particles are packed Newton's First Law of Motion Isaac Newton built on Galileo's thoughts about motion. Newton's first law of motion declares that a force is not needed to keep an object in motion. Slide a book across a table and watch it slide to a stop. The book in motion on the table top does not come to rest because of the absence of a force; rather it is the presence of a force – the force of friction – which brings the book to a halt. In the absence of a frictional force, the book would continue in motion with the same speed and in the same direction – forever! (Or at least to the end of the table’s top.) A force is not required to keep a moving book in motion; in actuality, it is a force which brings the book to rest. Newton’s First Law of Movement is stated as follows “An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force and vice versa.” If we consider that to be at rest is just moving with v = 0 m / s and that velocity is a vector, so that unchanged direction is implicit in the term “constant velocity” we can shorten the statement to: An object will show rectilinear uniform movement unless acted by an unbalanced force and vice versa A sophisticated mathematical expression for this law is n Σ Fi = 0 ↔ v =0 (where v is the velocity vector not just speed) i There are many applications of Newton's first law of motion. Consider some of your experiences in an automobile. Have you ever observed the behaviour of coffee in a coffee cup filled to the rim while starting a car from rest or while bringing a car to rest from a state of motion? Coffee tends to "keep on doing what it is doing." When you accelerate a car from rest, the road provides an unbalanced force on the spinning wheels to push the car forward; yet the coffee (which is at rest) wants to stay at rest. While the car accelerates forward, the coffee remains in the same position; subsequently, the car accelerates out from under the coffee and the coffee spills in your lap. On the other hand, when stopping from a state of motion, the coffee continues to move forward with the same speed and in the same direction, ultimately hitting the windshield or the dashboard. Coffee in motion tends to stay in motion.

4 Universal Gravitation The Apple, the Moon, and the Inverse Square Law In the early 1600's, German mathematician and astronomer Johannes Kepler mathematically analyzed known astronomical data in order to develop three laws to describe the motion of planets about the sun. Kepler's three laws emerged from the analysis of data carefully collected over a span of several years by his Danish predecessor and teacher, Tycho Brahe. Kepler's three laws of planetary motion can be briefly described as follows: • • •

The paths of the planets about the sun are elliptical in shape, with the centre of the sun being located at one focus. (The Law of Ellipses) An imaginary line drawn from the centre of the sun to the centre of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas) The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

While Kepler's laws provided a suitable framework for understanding the motion and paths of planets about the sun, there was no accepted explanation for why such paths existed. The cause for how the planets moved as they did was never stated. Kepler could only suggest that there was some sort of interaction between the sun and the planets which provided the driving force for the planet's motion. To Kepler, the planets were somehow "magnetically" driven by the sun to orbit in their elliptical trajectories. There was however no interaction between the planets themselves. Newton was troubled by the lack of explanation for the planet's orbits. To Newton, there must be some cause for such elliptical motion. Even more troubling was the circular motion of the moon about the earth. Newton knew that there must be some sort of force which governed the heavens; for the motion of the moon in a circular path and of the planets in an elliptical path required that there be an inward component of force. Circular and elliptical motion was clearly departures from the inertial paths (straight-line) of objects; and as such, these celestial motions required a cause in the form of an unbalanced force. The nature of such a force - its cause and its origin - bothered Newton for some time and was the fuel for much mental pondering. And according to legend, a breakthrough came at age 24 in an apple orchard in England. Newton never wrote of such an event, yet it is often claimed that the notion of gravity as the cause of all heavenly motion was instigated when he was struck in the head by an apple while lying under a tree in an orchard in England. Whether it is a myth or a reality, the fact is certain that it was Newton's ability to relate the cause for heavenly motion (the orbit of the moon about the earth) to the cause for Earthly motion (the fall of an apple to the Earth) which led him to his notion of universal gravitation.

5 Newton's Law of Universal Gravitation Isaac Newton compared the force the moon to the force on objects on earth. Believing that the same kind of forces was responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's centre from the object's centre. But distance is not the only variable affecting the magnitude of a gravitational force. Newton knew that the force which pulls the apple down (gravity) must be dependent upon the mass of the apple. Big apples are heavier, more difficult to lift. And since the force acting on the apple downward is equal and opposite to the force acting on the earth upward (Newton's third law), that force must also depend upon the mass of the earth. So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance which separates the centres of the earth and the object. But Newton's law of universal gravitation extends gravity beyond earth. Newton's law of universal gravitation is about the universality of gravity. All objects attract each other with a force of gravitational attraction. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance which separates their centres. Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as

Since the gravitational force is directly proportional to the mass of both interacting objects, more massive objects will attract each other with a greater gravitational force. Hence, as the mass of either object increases, the force of gravitational attraction between them also increases. If the mass of one of the objects is doubled, then the force of gravity between them is doubled; if the mass of one of the objects is tripled, then the force of gravity between them is tripled; if the mass of both of the objects is doubled, then the force of gravity between them is quadrupled; and so on. Since gravitational force is inversely proportional to the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces. So as two objects are separated from each other, the force of gravitational attraction between them also decreases. If the separation distance between two objects is doubled (increased by a factor of 2), then the force of gravitational attraction is decreased by a factor of 4 (2 raised to the second power). If the separation distance between any two objects is tripled (increased by a factor of 3), then the force of gravitational attraction is decreased by a factor of 9 (3 raised to the second power).

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Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below.

The constant of proportionality (G) in the above equation is known as the universal gravitation constant. The precise value of G was determined experimentally by Henry Cavendish in the century after Newton's death. G = 6.67 x 10-11 N m2/kg2 The units on G may seem rather odd; nonetheless they are sensible. When the units on G are substituted into the equation above and multiplied by m1.m2 units and divided by d2 units, the result will be in newtons - the unit of force. Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance. Gravitational interactions do not simply exist between the earth and other objects; and not simply between the sun and other planets; gravitational interactions exist between all objects with an intensity which is directly proportional to the product of their masses. So as you sit in your seat in the physics classroom, you are gravitationally attracted to your lab partner, to the desk you are working at, and even to your physics book. Newton's revolutionary idea was that gravity is universal - ALL objects attract in proportion to the product of their masses. Of course, most gravitational forces are too minimal to be noticed. Gravitational forces only are recognizable as the masses of the objects become large. Today, Newton's law of universal gravitation is a widely accepted theory. It guides the efforts of scientists in their study of planetary orbits. Knowing that all objects exert gravitational influences on each other, the small perturbations in a planet's elliptical motion can be easily explained. Newton's comparison of the acceleration of the apple to that of the moon led to a surprisingly simple conclusion about the nature of gravity which is woven into the entire universe. Cavendish and the Value of G The constant of proportionality in Newton’s equation is G - the universal gravitation constant. The value of G was not experimentally determined until nearly a century later by Lord Henry Cavendish using a torsion balance. Cavendish's apparatus for experimentally determining the value of G involved a light, rigid rod which was 6-feet long. Two small metal spheres were attached to the ends of the rod and the rod was suspended by a wire. When the long rod becomes twisted, the torsion of the wire begins to exert a torsional force which is proportional to the angle of rotation of the rod. Cavendish had calibrated his instrument to determine the

7 relationship between the angle of rotation and the amount of torsional force. A diagram of the apparatus is shown below. Cavendish then brought two large lead spheres near the smaller spheres attached to the rod. Since all masses attract, the large spheres exerted a gravitational force upon the smaller spheres and twisted the rod a measurable amount. Once the torsional force balanced the gravitational force, the rod and spheres came to rest and Cavendish was able to determine the gravitational force of attraction between the masses. By measuring m1, m2, d and Fgrav, the value of G could be determined. Cavendish's measurements resulted in an experimentally determined value of 6.75 x 10-11 N m2/kg2. Today, the currently accepted value is 6.67259 x 10-11 N m2/kg2. The value of G is an extremely small numerical value. Its smallness accounts for the fact that the force of gravitational attraction is only appreciable for objects with large mass. While two students will indeed exert gravitational forces upon each other, these forces are too small to be noticeable. Yet if one of the students is replaced with a planet, then the gravitational force between the other student and the planet becomes noticeable. The Value of “g” Let us look at the law of gravitation from a different point of view

If object 1 is the Earth and object 2 is any object at its surface, d will be the radius of the Earth. We can rearrange the previous equation slightly.

The first factor on the right side depends on the mass of the earth and its radius and not on any property of the second object. This factor measures the gravitational field of the Earth at that specific point: it is called gravity and is represented by g. It can be calculated (do it as an exercise) and is 9,8 N / kg. For practical purposes it is many times rounded up to 10 N / kg. Thus the equation above will get simplified to Fgrav = m . g Mass and Weight

8 The force of gravity is the force with which the earth, moon, or other massive body attracts an object towards itself. By definition, this is the weight of the object. All objects upon earth experience a force of gravity which is directed "downward" towards the centre of the earth. The force of gravity on an object on earth is always equal to the weight of the object. The weight (Fgrav) of an object is calculated ass its mass times the local gravity Fgrav = m . g

where:

g = gravitational field of the Earth = 9.8 N / kg (on Earth at sea level) m = mass of the object (in kg) Thus, the Earth pulls an object that has a 1kg mass with a force of 9,8 (10) N The table below shows some values of g at various distances from Earth's centre

Location Earth's surface

Distance from

Value of g

Earth's centre (m)

m/s2

6.38 x 106 m

9.8

1000 km above surface 7.38 x 106 m

7.33

6000 km above surface 1.24 x 107 m

2.60

10000 surface 50000 surface

km

above

km

above

1.64 x 107 m

1.49

5.64 x 107 m

0.13

As is evident from both the equation and the table above, the value of g varies inversely with the distance from the centre of the earth. In fact, the variation in g with distance follows an inverse square law where g is inversely proportional to the distance from earth's centre. This inverse square relationship means that as the distance is doubled, the value of g decreases by a factor of 4; as the distance is tripled, the value of g decreases by a factor of 9; and so on. This inverse square relationship is depicted in the graphic at the right. The same equation used to determine the value of g on earth can also be used to determine gravity on the surface of other planets. The value of g on any other planet can be calculated from the mass of the planet and the radius of the planet. Using this equation, the following acceleration of gravity values can be calculated for the various planets.

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Planet

Radius (m)

Mass (kg)

g (m/s2)

Mercury

2.43 x 106

3.2 x 1023

3.61

Venus

6.073 x 106

4.88 x1024

8.83

Mars

3.38 x 106

6.42 x 1023

3.75

Jupiter

6.98 x 107

1.901 x 1027

26.0

Saturn

5.82 x 107

5.68 x 1026

11.2

Uranus

2.35 x 107

8.68 x 1025

10.5

Neptune

2.27 x 107

1.03 x 1026

13.3

Pluto

1.15 x 106

1.2 x 1022

0.61

A most important consideration The force of gravity is a source of much confusion to many students of physics. The mass of an object refers to the amount of matter that is contained by the object; the weight of an object is the force of gravity acting upon that object. Mass is related to "how much stuff is there" and weight is related to the pull of the Earth (or any other planet) upon that stuff. The mass of an object (measured in kg) will be the same no matter where in the universe this object is located. Mass is never altered by location, the pull of gravity, speed or even the existence of other forces. For example, a 2-kg object will have a mass of 2 kg whether it is located on Earth, on the moon, or on Jupiter; its mass will be 2 kg whether it is moving or not (at least for purposes of this study); and its mass will be 2 kg whether it is being pushed or not. On the other hand, the weight of an object (measured in newtons) will vary according to where in the universe the object is. Weight depends upon which planet is exerting the force and the distance the object is from the planet. Weight, being equivalent to the force of gravity, is dependent upon the value of g. On Earth's surface, g is 9.8 N / kg (often approximated to 10 N / kg). On the moon's surface, g is 1.7 N / kg. Go to another planet, and there will be another g value. In addition, the g value is inversely proportional to the distance from the centre of the planet. Always be cautious of the distinction between mass and weight. It is the source of much confusion for many students of physics.

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PROBLEMS ON MASS, WEIGHT AND GRAVITY 1- Dr Krazykoff focuses his telescope night after night on a very narrow region in the sky. He observes a star that circles slowly but he doesn’t see any other star or celestial object nearby. Proudly, he communicates the scientific community that he has very probably discovered a “black hole”. Why is he assuming this? 2- Gravity on the Moon is about 1/6 of the Earth’s gravity. Find the weight of a block on the Moon if its mass is 144 Kg. Suppose a force meter has an elastic limit of 250 N: could you use it to weigh it on the Earth? And what about using it on the Moon? 3- Tracy’s car weighs 9000 N. It is travelling along a straight path at 120 km / hr. The motor makes a 300 N force on the car. Find the friction force that the pavement and the air are making on Tracy’s car. 4- Peter is standing on the Earth. He holds with his finger a spring balance with a stone hanging from it. The spring balance reads 50 N. a- Calculate the stone’s mass. b- If Peter were standing in the Moon: what would have been your answer? c- If Peter goes (with the force meter and the hanging stone) to Planet X and the force meter reads 42 N. What is the value of gravity at X? 5- James is Peter’s friend and he has carried out the three experiments of problem 4 (on Earth, Moon and Melmac) but he has used a two beam balance and a set of brass weights instead of a force meter. What were James’ results? 6- Which is bigger, the gravitational pull of the Earth on the Moon or of the Moon on the Earth? Write the equation showing how these pulls can be calculated knowing the masses of both bodies and the distance between their centres. 7- Calculate the gravitational pull on a 70 kg man standing on the ground of a planet that has a mass twice the mass of the Earth and a radius twice as big. Remember g is 10 N / kg. Compare with his weight here on Earth. 8- Write the expression of Newton’s law of gravitation between the Earth and man standing on the ground. Draw a circle around the parts of the equation that correspond to gravity. 9- Mars’ radius is ½ of the Earth’s radius and gravity at its surface is just 4 N /kg. Estimate Mars’ mass and density compared to the Earth’s 10- If the Earth’s radius is about 6.380 km (6,4 x 106 m), γ is 6,67 x 10-11 and gravity is considered to be 9,8 N / kg estimate: a- Our planet’s mass b- Its density

11 c- The average density of rocks in the earth’s crust is about 3.000 kg / m 3. What does it suggest about the Earth’s “unreachable” mantle and core? 11- The two-block system in the diagram is falling at a constant speed (10 cm/s). Calculate the friction force (Ff). (Suppose there are no extra friction forces acting on the pulley) Ff

v = 10 cm / s pulley X

m =2 kg 12- After point X in the previous diagram the table is lubricated. What do you think will happen to the blocks? 13- Calculate the force of gravity between the following familiar objects.

Mass of Object Mass of Object 2 Separation Distance Force of Gravity 1 (kg) (m) (N) (kg) a.

Football Player 100 kg

Earth 5.98 x1024 kg

6.37 x 106 m (on surface)

b.

Ballerina 40 kg

Earth 5.98 x1024 kg

6.37 x 106 m (on surface)

c.

Physics Student 70 kg

Physics Student 70 kg

1m

d.

Physics Student 70 kg

Physics Student 70 kg

0.2 m

e.

Physics Student 70 kg

Physics Book 1 kg

1m