Acceleration due to Gravity

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Acceleration due to gravity

Concept flow

Some of the key ideas we will cover in this section are:

• Gravitational force due to the Earth produces an acceleration in the objects. This is the force acting on a freely falling object.

• The value of acceleration is not dependent on the mass.

• All freely falling bodies gain same acceleration.

Free fall and acceleration due to gravity

A freely falling body undergoes acceleration. This acceleration is caused by the gravitational force exerted by the larger mass of the Earth. This is referred to as acceleration due to gravity. The Earth also undergoes an acceleration due to the gravitational force exerted by the object. We do not notice it because of the mass of the Earth.

The acceleration that an object experiences because of gravity when it falls freely close to the surface of a massive body, such as a planet is also known as the acceleration of free fall, its value can be derived as follows.

Let M be the mass of the Earth and m be the mass of the object that is subjected to free fall. Let R be the radius of the Earth and h be the height above the surface of the Earth.

The gravitational force on the object due to the Earth is given by

Fg = G Mm (R + h) 2 = m a (by Newton's second law of motion)

If R >>h, then this can be written as

a = GM/ R2

We call this acceleration due to gravity and is indicated by “g”. Please note that this is very different from the Universal Gravitational Constant “G”.

Notice the important result here that the acceleration due to gravity is independent of the mass of the object.

In the case of the Earth, g comes out to be approximately 9.8 m/s2, though the exact value depends on location because of two main factors: the Earth's rotation and the Earth's equatorial bulge.

An object that is allowed to fall freely will, if the effects of air resistance are ignored, gather speed (accelerate) at a rate of about 9.8 m/s2. If dropped from rest, it will have fallen 4.9 m and be traveling at a speed of 9.8 m/s after 1 second. After 2 seconds, it will have fallen a further 14.7 m and be traveling at 19.6 m/s. After 3 seconds, it will have fallen a further 24.5 m and be traveling at 29.4 m/s.

Variation of 'g' at various places on Earth

The value of “g” varies according to the effect of the Earth's rotation. If we have a mass hanging in equilibrium from a spring balance at the North Pole, there are two forces acting on the mass, Fg (= mg) and “w” which is the force with which the spring will pull on the mass. An equal and opposite force “w” acts on the spring downwards and this “w” will be read as the weight of the object.

Since the body is in equilibrim, there is no net force and there is no acceleration.

Therefore, mg – w = 0; w = mg.

However at the equator, the gravitational force is still Fg (= mg) and an equal and opppsite force w1 causes the spring to pull on the mass. An equal and opposite force “w1 ” acts on the spring downwards and this “w1 ” will be read as the weight of the object.

The object at the equator, has an acceleration, which is given by the centripetal acceleration.

mg - w1 = m v2/ R

where v is the rotational speed of the Earth and R is the radius.

W1 = m g' = m (g - v2/ R)

Difference in acceleration, g - g' = 0.0337 m/s2

Please see section on Additional Information for details of difference in “g”.

Variation due to the shape of the Earth

When the Earth was formed it was still molten. Due to the rotation, more mass moved towards the centre. This has resulted in the Earth being flatter at the poles and fatter at the equator. There is a difference of about 20 km in the distance from the centre at the equator and the poles. Therefore, an object closer to the equator will have a higher velocity and therefore, higher centripetal acceleration. This will result in a difference in the acceleration due to gravity. Nonfree fall

In the discussion above, we have assumed that an object is falling through vacuum – there are no other forces other than gravitational forces acting on it Although a feather and a coin will fall equally fast in a vacuum, they will fall differently in air. This is due to the presence of resistive forces.

Newton's laws apply both in vacuum and in the presence of air resistance. The important thing to remember is that acceleration is dependent on the net force.

If the Earth had no atmosphere, an object dropped from a great height would keep accelerating at a rate of 9.8 m/s2 until it hit the ground. For example, if a person fell from an aircraft at an altitude of 10,000 m, they would be travelling at about 442 m/s (1500 km/ hr) by the time they landed. In practice, this doesn't happen because of air resistance. The faster an object falls, the greater is the air resistance (called air drag) acting on it. Air drag depends on the surface area of the falling object and the speed.

At a certain velocity, known as the terminal velocity, the downward force of gravity is balanced out by the upward force of air resistance and there is no further acceleration. And it continues to move at the same terminal velocity till it reaches the ground.

The effect of air resistance

If there were no atmosphere, all objects would fall at the same rate. This happens, for example, on the Moon. In one of the most memorable moments of the space program, David Scott, commander of the Apollo 15 mission, standing on the Moon's surface, dropped two objects – a geological hammer and a falcon's feather (the Apollo 15 lunar module was called Falcon) – at the same time from the same height. The feather didn't drift down, meanderingly, as it would have done on Earth. Instead, in the airless vacuum of space, it fell straight, without a flutter, keeping pace with the hammer and reaching the lunar surface at the same instant.

Variance of “g” on Earth

In the case of the Earth, g comes out to be approximately 9.8 m/s2 (32 ft/s2), though the exact value depends on location because of two main factors: the Earth's rotation and the Earth's equatorial bulge. We saw that the value of “g” depends on the mass of the Earth and the distance from the centre. At a distance of twice the radius of the Earth, the value of “g” drops to 2.45 m/s2.

The shape of the curve suggests that the value of “g” changes according to the inverse square law

Value of “g” at different places

Weight

Concept flow

• Every particle has mass; weight is a force acting on a mass due to the gravitational pull.

• This force experienced by an object due to the gravitational pull of the Earth is what we call the weight. Weight is nothing but the force exerted on a mass due to the gravitational pull of the Earth.

How do we perceive this weight?

When you stand on a surface, the force of the Earth's gravity is acting upon you downwards and there is a normal force exerted by the surface on which you stand. Since you stand on a firm surface and there is no acceleration, the normal force is equal to the gravitational force and this is equal to mg. If an object is suspended from a spring, the gravitational force will be balanced by the tension force in the string.

Weight is that supporting force felt by an object in equilibrium; this opposes and balances the gravitational pull of the Earth. Thus, humans experience their own body weight as a result of this supporting force, which results in a normal force applied to a person by the surface of a supporting object, on which the person is standing or sitting. In the absence of this force, a person would be in free-fall, and would experience weightlessness. It is the transmission of this reaction force through the human body, and the resultant compression and tension of the body's tissues, that results in the sensation of weight.

When an object is in equilibrium, it only experiences the gravitational and restoring force/ Weight is mass multiplied by the acceleration due t gravity

Measured weight can change:

• when acceleration due to gravity changes

• when the object is accelerating (non-inertial frame)

When used to mean force, magnitude of weight (a scalar quantity), often denoted by an italic letter W, is the product of the mass, m, of the object and the magnitude of the local gravitational acceleration g;. thus: W = mg. When considered a vector, weight is often denoted by a bold letter W. The unit of measurement for weight is that of force, which in the International System of Units (SI) is the newton.

For example, an object with a mass of one kilogram has a weight of about 9.8 newtons on the surface of the Earth, about one-sixth as much on the Moon, and very nearly zero when in deep space far away from all bodies imparting gravitational influence.

Earlier concepts of weight

Concepts of heaviness (weight) and lightness (levity) date back to the ancient Greek philosophers. These were typically viewed as inherent properties of objects. Plato described weight as the natural tendency of objects to seek their kin. To Aristotle weight and levity represented the tendency to restore the natural order of the basic elements: air, earth, fire and water. He ascribed absolute weight to earth and absolute levity to fire. Archimedes saw weight as a quality opposed to buoyancy, with the conflict between the two determining if an object sinks or floats. The first operational definition of weight was given by Euclid, who defined weight as: "weight is the heaviness or lightness of one thing, compared to another, as measured by a balance.". Satellites and weightlessness

It is a very common misconception that when astronauts are in orbit they are weightless because they are somehow far enough from the earth that the force of earth's gravity does not pull on them. This is totally incorrect. If they were that far away, earth's gravity would not pull on the shuttle either and it would be impossible for it to be in orbit around the earth.

Gravity (a force we call weight) is actually responsible for keeping the space craft and the astronaut in orbit around the earth. Gravity is still pulling on the astronaut. The feeling of weight;ess is no differenet than when in ree fall. What they are not experiencing is the normal force, which is the opposing force. When that force is gone, we feel say we feel "weightless." In fact, whenever a person is in freefall they feel weightless even though gravity is still causing them to have weight. While in orbit, the space shuttle does not have to push on the astronaut (or anything else in the cabin) to keep him up. The space shuttle and the astronaut are in a constant state of freefall around the earth.

Significance of the gravitational force

Discovery of planets

Accurate measurements on the orbits of the plantes indicated that they did not precisely follow Kepler's laws. Slight deviations from perfectly elliptical orbits were observed. Newton was aware that this was to be expected from the Law of Universal Gravitation. The derivation of perfectly elliptical ignores the forces due to the other planets. These deviations called perturbations are observed and led to the discovery of Neptune and Pluto. Planets around distant stars were also inferred from the regular wobble of each star due to the gravitational attraction of the revolving plant.

Ocean tides

Ocean tides are caused by differences in the gravitational pull between the Moon and the Earth on the opposite sides of the Earth. Gravitational force is stronger on the side of the Earth nearer to the Moon and is weaker on the side of the Earth farther from the Moon. The bulge that is caused in the Earth's oceans due to this gravitational pull results in two sets of tides on the Earth.

Activity 2 – Observe a freely falling body

Objective: To observe the behaviour of a freely falling object

Procedure:

Ask a child to drop a piece of chalk from terrace, Start the stop clock as soon as the child drops it. Put off the clock as soon as the chalk touches the ground, note down the time taken , Repeat the same expt with a stone,& calculate the time, Time taken will be same in both the cases. Inference : All bodies accelerate equally irrespective of their mass.

Activity 3 : Thought Experiment

Objective: To understand the nature of gravitational force

Procedure:

Ask the children to think about what will happen if we had a hollow tunnel running through the centre of the Earth and we dropped a ball into it. What will happen to the ball?

Projectile and Satellite Motion

Concept flow

• A projectile motion of a body thrown is due to the gravitational force.

• Satellites are projectiles that are continuously falling in the orbit around planets

Let us study this picture below and analyze what happens in each of the cases.

In the first case, the ball is just dropped from the cliff and it falls down in a straight line, subject to the force of gravity. In the second and third instances, the ball is thrown upwards, reaches a certain height and still falls down. In the third case, the ball covers a horizontal range as well.

In all these cases, gravity is the only force acting. Without gravity, we could throw a rock upwards at an angle and it follow a straight line path. Because of gravity, however, the path curves.

Such an object, when thrown/ projected and continues its motion on its own inertia is called a projectile. A projectile will have two components to ots velocity – the horizontal and the vertical. The horizontal component is similar to an object rolling on a plane/ along a straight line. The vertical component of the velocity is subject to the acceleration due to gravity. A projectile moves horizontally as it moves downwards or upwards.

Imagine throwing a ball straight up. It will fall down to the same place. In this case the ball has a velocity in the vertical direction which changes with time as the force due to gravity causes an acceleration in the downward direction all the time. But suppose on were to throw a ball with only a horizontal velocity. The ball will now move with a velocity that has two components to it - one the horizontal velocity which remains unchanged as long as there is no force such as air resistance acting on it and a vertical velocity that is continually changing. This vertical velocity starts at zero - we threw the ball horizontally - and keeps increasing with an acceleration g.

The resultant velocity is a combination of the two. This is what causes objects to follow a parabolic path when they are thrown with a combination of horizontal and vertical velocities. The greater the horizontal component the farther the ball will travel. For short distances, and small velocities, the curvature of the Earth will make no difference. But suppose that we throw it so hard that the horizontal distance is very large and we can no longer ignore the curvature of the Earth?

Suppose we throw it so hard that that ball will continue to fall but will never reach the ground - the curvature of the fall of the ball is greater than the curvature of the Earth? Then the ball will become a satellite! It will move round and round the Earth - constantly falling but never reaching the ground!

The satellite and everything in it are constantly falling towards the Earth but will never reach it. Since they are all falling with the same velocity, the satellite does not exert any force on the objects or people inside. The people inside, therefore feel weightless. remember we have a sense of weight because of the Normal force. Here the Normal force is zero and so we feel weightless.

This is very similar to the sense of loss of weight in a lift that is accelerating downwards - except that here the acceleration is the acceleration due to gravity.

Satellite

An Earth satellite is simply a projectile that falls around the Earth rather than into it. That means the horizontal falling distance matches the Earths curvature. Geometrically, the curvature of the surface is that its surface drops a vertical distance of 5 metres for every 8000 metres tangent to the surface.

Therefore, if we throw a rock or a ball at a high enough speed (about 29000 km/s), it would follow the curvature of the Earth. But at this speed, atmospheric friction (due to air drag) would burn up everything. This is why satellites are launched at an altitude high enough for the air drag to be negligible.

Satellite motion was understood by Newton who reasoned that the Moon was simply a projectile that was circling the Earth.

Activity 4 - Demonstration of satellite motion using simulation

The following simulation can illustrate how satellite motion takes place.

Click to run PhET simulation

Kepler's Laws

Concept flow

• The key concept to understand here is that gravitational forces play an important role in planetary motion.

• Three laws of planetary motion that describe the motion of the planets have been postulated based on detailed astronomical observations

Laws of Planetary Motion

We now know that satellites are continually falling towards the Earth following a curved path whose curvature is greater than that of the curvature of the Earth. The Moon is just such a satellite that moves around the Earth. In a similar way, all the planets that move around the Sun are satellites of the Sun. The motion described in such a situation is not strictly circular - it is elliptical.

Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky.

1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.

2. The Law of Areas: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.

3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well.

The Law of Orbits

All planets move in elliptical orbits, with the sun at one focus. An ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (called the foci, F1 and F2) remains constant.

Orbit eccentricity

The semi major axis of the ellipse is a and represents the planet's average distance from the Sun. The eccentricity, “e” is defined so that “ea” is the distance from the centre to either focus. A circle is a special case of an ellipse where the two foci coincide. The Earth and most of the other planets have nearly circular orbits. For Earth, “e” = 0.017.

The Law of Equal Areas

Kepler's second law states that each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.

This can be shown to be true using the law of conservation of angular momentum.

If “v” is the velocity of the planet, in time “dt” the planet moves a distance vdt and sweeps out an area equal to the area of a triangle of base “r” and altitude vdt sinα.

Hence dA = ½ (r) (“v” x “dt” x sinα)

dA/ dt = ½ rv sinα

The magnitude of the angular momentum of the planet about the Sun is L = mvr sinα.

dA/ dt = (½)L/m

Because the angular momentum is conserved, the rate of change of area covered is constant. This means that the planets move with different velocities depending upon their position in the orbits.

The Law of Periods

The ratio of the squares of the periods of any two planets revolving about the Sun is equal to the ratio of the cubes of their semi-major axes.

Can you derive this?

G m1 Ms / r12 = m1 (v12)/ r1

v1 = 2πr1/T1

Substituting and rearranging we get

T12/ r13 = 4π2 / G Ms

Deriving this for another planet, we can arrive at the third law.

Additional resources:

1. Gravity is more than a name - This link gives an overview of what gravity is.

2.

4. Animation of the Cavendish experiment

5. www.hyperphysics.com - From Classical Mechanics to General Relativity - This is a good description of the geometry of Newtonian gravity and how to move from classical mechanics to relativity.

6. The Value of "g" (http://www.physicsclassroom.com/Class/circles/U6L3e.cfm) - This is a good resource to study the variation of “g” at various distances above the Earth's atmosphere.

7. http://science.nasa.gov/science-news/science-at-nasa/2004/06may_lunarranging/ - This link examines the Galileo experiment and discusses if there are other possible explanations.

8. http://www.physicsclassroom.com/Class/circles/U6L4b.cfm -This website describes the mathematics of orbital motion.

9. http://spaceflight.nasa.gov/gallery/images/station/crew-9/html/iss008e21996.html

Keywords

Mass, Inertial, Gravitational, Force field, Universal law of gravitation, Acceleration due to gravity, “g”, weight, weightlessness