Modern Physics

# Gravity acceleration: what it is, how it is measured and exercised

The acceleration of gravity or gravitational acceleration is defined as the intensity of the gravitational field of the earth. That is, the force it exerts on any object, per unit of mass.

It is indicated by the familiar letter g and its approximate value near the Earth’s surface is 9.8 m / s 2 . This value can suffer small variations with geographic latitude and also with altitude in relation to sea level.

Astronaut on spacewalk on Earth’s surface. Source: Pixabay

The acceleration of gravity, in addition to having the aforementioned magnitude, has direction and meaning. In fact, it is aimed vertically towards the center of the earth.

Earth’s gravitational field. Source: Source: Sjlegg [Public domain]

The Earth’s gravitational field can be represented as a set of radial lines pointing to the center, as seen in the previous figure.

## What is the acceleration of gravity?

The value of the acceleration of gravity on Earth or any other planet is equivalent to the intensity of the gravitational field it produces, which does not depend on the objects that are around it, but only on its own mass and radius.

The acceleration due to gravity is often defined as the acceleration experienced by any object in free fall in the vicinity of the Earth’s surface.

In practice, this is almost always the case, as we will see in the following sections, in which Newton’s Law of Universal Gravitation will be used.

Newton is said to have discovered this famous law while meditating on falling bodies under a tree. When he felt the apple hit his head, he knew immediately that the force that makes the apple fall is the same force that makes the Moon orbit around the Earth.

### The Law of Universal Gravitation

Whether the apple legend is true or not, Newton realized that the magnitude of the force of gravitational attraction between two objects, for example, between the Earth and the Moon, or the Earth and the apple, should depend on the masses of these objects. :    #### Gravitational force characteristics

Gravitational force is always attractive; that is, the two affected bodies attract each other. The opposite is not possible, since the orbits of celestial bodies are closed or open (comets, for example) and a repulsive force can never produce a closed orbit. So the masses always attract, no matter what.

Related:   Parabolic Shot: Characteristics, Formulas and Equations, Examples

A good approximation of the actual shape of the Earth (m 1 ) and the Moon or apple (m 2 ) is to assume that they are spherical in shape. The following figure is a representation of this phenomenon.

Newton’s Law of Universal Gravitation. Source: I, Dennis Nilsson [CC BY 3.0 (https://creativecommons.org/licenses/by/3.0)]

Here are represented the force exerted by m 1 in m 2 and the force exerted by m 2 in m 1 , both of equal magnitude and directed along the line between the centers. They are not canceled as they are applied to different objects.

In all of the following sections, objects are assumed to be homogeneous and spherical; therefore, its center of gravity coincides with its geometric center. You can assume all the mass concentrated there.

## How is gravity measured on different planets?

Gravity can be measured with a gravimeter, a device used to measure gravity used in geophysical gravity surveys. These days they are much more sophisticated than the originals, but in the beginning they were based on the pendulum.

The pendulum consists of a thin, light and inextensible string of length L. One end is fixed to a support and the other has a mass m.

When the system is in equilibrium, the mass locks vertically, but when it is separated from it, it begins to oscillate in an alternative movement. Gravity is responsible for this. For all that follows, it is valid to assume that gravity is the only force acting on the pendulum.

The period T of the pendulum swing for small swings is given by the following equation:  ### Experience to determine the value of g

#### Materials

– 1 spherical metal

– Rope of several different lengths, at least 5.

– Measuring tape.

– Conveyor

– Stopwatch

– A support for repairing the pendulum.

– Graph paper or computer program with spreadsheet.

Related:   Magnetic permeability: constant and table

#### Procedure

1. Select one of the strings and mount the pendulum. Measure the length of the string + the radius of the sphere. This will be the length L.
2. Remove the pendulum from the balance position about 5 degrees (measure with the transporter) and let it swing.
3. Simultaneously start the timer and measure the time of 10 oscillations. Write down the result.
4. Repeat the procedure above for the other lengths.
5. Find the time T it takes the pendulum to perform an oscillation (dividing each of the previous results by 10).
6. Square each value obtained, obtaining T 2
7. On graph paper, plot each value of T 2 on the vertical axis against its value of L on the horizontal axis. Be consistent with the units and don’t forget to factor in the misjudgment of the instruments used: tape measure and stopwatch.
8. Draw the best line that fits the plotted points.
9. Find the slope m of this line using two points that belong to it (not necessarily experimental points). Add the trial error.
10. The above steps can be performed with a spreadsheet and the option to create and fit a straight line.
11. From the slope value to clear the value of g with the respective experimental uncertainty.

### Default value of g on Earth, Moon and Mars

The default value of gravity on Earth is: 9.81 m / s 2 , at 45° north latitude and at sea level. Since the Earth is not a perfect sphere, the values ​​of g vary slightly, being higher at the poles and lower at the equator.

Those who want to know the value in their locality can find it updated on the website of the German Institute of Metrology PTB ( Physikalisch-Technische Bundesanstalt ), in the Gravity Information System (GIS) section.

### gravity on the moon

The Moon’s gravitational field was determined by analyzing radio signals from space probes orbiting the satellite. Its value on the lunar surface is 1.62 m / s 2

### gravity on mars

The value of P for a planet depends on its mass M and its radius R, as follows: Therefore: The following data are available for the planet Mars:

M = 6.4185 x 10 23 kg

Related:   Kepler’s Laws: explanation, exercises, experiment

R = 3390 km

G = 6.67 x 10-11 Nm 2 / kg 2

With these data, we know that the gravity of Mars is 3.71 m / s 2 . Of course, you can apply the same equation to data from the Moon or any other planet and thus estimate its gravity value.

## Exercise solved: the falling apple

Suppose the Earth and an apple have a spherical shape. The Earth’s mass is M = 5.98 x 10 24 kg and its radius is R = 6.37 x 10 6 m. The mass of the apple is m = 0.10 kg. Suppose there is no other force except gravity. In Newton’s Law of Universal Gravitation, find:

a) The gravitational force that the Earth exerts on the apple.

b) The acceleration experienced by the apple when released from a certain height, according to Newton’s Second Law.

#### Solution

a) The apple (supposedly spherical, like the Earth) has a very small radius compared to the Earth’s radius and is immersed in its gravitational field. The following figure is obviously not to scale, but there is a diagram of the gravitational field g and the force F exerted by the earth on the block:

Schematic showing the fall of the apple in the vicinity of the Earth. Both the size of the apple and the height of the fall are negligible. Source: own elaboration.

When applying Newton’s Law of Universal Gravitation, the distance between the centers can be considered approximately the same value as the Earth’s radius (the height from which the apple falls is also negligible compared to the Earth’s radius). Therefore:

b) According to Newton’s Second Law, the magnitude of the force exerted on the apple is:

F = ma = mg

Whose value is 0.983 N, according to the previous calculation. Equalizing both values ​​and then clearing the magnitude of acceleration you get:

mg = 0.983 N

g = 0.983 N / 0.10 kg = 9.83 m / s 2

This is a very good approximation of the default gravity value.