A famous story says Isaac Newton discovered gravity when he saw an apple fall from a tree. However, the crucial thing he realized was not why the apples fall from threes. But the fact that the reason an apple falls is the same reason why the planets orbit the sun! Centuries later, his discovery is still considered one of the most remarkable contributions to science! It allowed us to explain the planet's motion and take us to the moon. This article will take you through the definition of gravity and how to measure its effects by calculating the weight of objects in the Earth's gravitational field.
Gravity definition
Gravity is a Force of attraction between two bodies due to their mass.
Fig. 1: Gravity acting on a falling apple.
Gravity is a non-contact Force, which means it can act over two objects that are not physically touching, even at large distances! Let's see how it works in a bit more detail.
Theory of gravity
To understand how gravity works, it is useful to introduce the concept of the Centre of Mass.
We can assume that the whole mass of an object is located at a single point called the Centre of Mass. Its location depends on how the mass is distributed in the object.
The centre of mass of a sphere is located in its geometrical centre. In an average person, it is slightly below the navel.
The theory of gravity states that every mass in the universe attracts every other object by an invisible force acting along the direction of the line joining both objects' centres of mass. The force between the two objects depends on their masses and the distance between their centres of mass.
Fig. 2: Force of gravity between two masses.
The bigger the mass the stronger the force. We do not see a bunch of pencils on a table jumping towards each other because their gravity is far too small to be noticed since their mass is very small. On the other hand, this effect becomes noticeable for very massive objects like the planets and stars in our solar system. For example, the moon keeps in orbit around the earth because of the gravitational force between them.
Fig. 3: Because gravity pulls the moon toward the earth, the moon describes an approximately circular trajectory around the earth.
As mentioned above, the distance between the objects also plays an important role. If the distance increases, the magnitude of the gravitational force decreases. We will talk more about this below.
Gravity value
The strength of a gravitational field is measured in \(\mathrm{N/kg}\) and it is represented by the symbol \(g\). This means that in a gravitational field of \(1 \,\,\mathrm{N/kg}\), an object will experience a force of \(1\,\,\mathrm{N}\) per each kilogram of its mass. The strength of the earth's gravitational field can be considered to be constant, and it has an approximated value of \(9,8 \,\,\mathrm{N/kg}\). This means that we experience a force of \(9,8\,\,\mathrm{N}\) per each kilogram of mass in our bodies. In comparison, the gravitational field strength of the moon is only \(g=1,6\,\,\mathrm{N/kg}\), about six times smaller. Since the force per kilogram that we would feel on the moon is smaller, things are lighter on the moon!
The strength of the gravitational field decreases as the distance of the centres of mass increases. Then, why do we considerto be constant on earth? Shouldn't it decrease as we get higher and increase our distance from its centre? That is true. It is not really constant, and it decreases with height. However, we would have to be at a height of about \(165\,\,\mathrm{km}\) from the surface of the earth for the value of \(g\) to decrease by just \(5\%\). For heights below \(30\,\,\mathrm{km}\), the change is smaller than \(1\%\). Since the variation at lower altitudes is quite negligible, the value of \(g\) is taken as a constant.
A calibrated spring balance, also known as a Newton metre, can be used to directly measure weight. It uses a spring attached to a hook to hold the objects. Then the spring elongates at a length that is proportional to the weight of the object. A scale in the device indicates the weight in newtons.
Fig. 4: A Newton metre uses a calibrated spring to measure the weight of an object.
Gravity equation
It is very common in our everyday life to refer to the weight or the mass of an object as if they were the same thing. However, they are very different things in physics.
Mass is a scalar quantity that refers to the amount of matter of an object.
Since the mass of an object is constant irrespective of the gravitational field it is in, the mass of an astronaut is the same on the moon as it is on earth.
The weight of an object is the force acting on it due to the gravitational field that it is in.
Since the moon and the earth have different Gravitational Fields, an astronaut has a different weight on the moon compared to the earth.
The weight of an object is directly proportional to both the strength of the gravitational field it is in and its mass. The weight of an object in a gravitational field can be calculated by using the following equation:
\[W=mg\]
or in words,
\[\text{Weight}=\text{mass}\cdot \text{gravitational field strength}\]
Where \(W\) is the weight in (\(\mathrm{N}\)), \(m\) is the mass in kilograms (\(\mathrm{kg}\)), and \(g\) is the gravitational field strength in \(\mathrm{N/kg}\). Let us look at a few examples where this equation can be used.
Quantum gravity is a theory that states that gravity is a force that acts through a theoretical particle called the graviton. This theory has been difficult to prove, as the conditions required to observe its proof are unattainable and limited by today's technology.
Calculate the weight of a man whose mass is \(75\,\,\mathrm{kg}\), standing on the surface of the earth where the strength of the gravitational field is \(g=9,8\,\,\mathrm{N/kg}\). What would his weight be if he was on the moon instead (\(g_m=1,6\,\,\mathrm{N/kg}\))?
Step 1: List the given quantities.
\[m=75\,\,\mathrm{kg},g=9,8\,\,\mathrm{N/kg},W+?\]
Step 2: Calculate the weight on Earth.
The equation to calculate the weight of an object in a gravitational field is given by
\[\begin{aligned} W_e=& mg \\ W_e=& 75\,\,\cancel{\mathrm{kg}}\cdot 9,8\,\,\mathrm{N}/\cancel{\mathrm{kg}}\\W_e=&735\,\,\mathrm{N}\end{aligned}\]
Step 3: Calculate the weight on the moon.
Similarly, his weight on the moon is given by the same equation, but we change the value of \(g\)
\[\begin{aligned}W_m=&m\cdot g_m\\ W_m=&75\,\,\cancel{\mathrm{kg}}\cdot 1,6\,\,\mathrm{N}/\cancel{\mathrm{kg}}\\ W_m=&120\,\,\mathrm{N}\end{aligned}\]
The weight of the object on the moon is less due to the relatively lower strength of its gravitational field as compared to the earth.
Let's look at another example.
If a car had a weight of \(1900\,\,\mathrm{N}\) on the moon. Calculate its weight on Earth. The gravitational field strength on Earth is \(9,8\,\,\mathrm{N}/\mathrm{kg}\) and the Gravitational Field Strength on the moon is \(1,6\,\,\mathrm{N}/\mathrm{kg}\).
Step 1: List the given quantities
\[W_m=1900\,\,\mathrm{N},g_n=1,6\,\,\mathrm{N/kg},g_e=9,8\,\,\mathrm{N/kg}\]
Step 2: Equation for weight on earth
The equation to calculate the weight of an object in a gravitational field is given by
\[\begin{aligned}W_e=&m\cdot g_e \\ W_e=&m\cdot 9,8 \,\,\mathrm{N/kg} \end{aligned}\]
Step 3: Equation for weight on the moon
Similarly, the weight on the moon is
\[\begin{aligned}W_m=&m\cdot g_m \\ W_m=&m\cdot 1,6 \,\,\mathrm{N/kg} \end{aligned}\]
Step 4: Use the fact that mass does not change
Since the mass is constant irrespective of the Gravitational Fields, we can rewrite both of the above equations to isolate the mass of the object and equate them.
\[\begin{aligned} m&=\dfrac{W_e}{g_e} \\ m=& \dfrac{W_m}{g_m}\end{aligned}\]
Equate both the equations
\[\dfrac{W_e}{g_e}=\dfrac{W_m}{g_m}\]
\[W_e=\dfrac{W_m}{g_m}\cdot g_e =\dfrac{1900\,\,\mathrm{N}}{1,6\cancel{\mathrm{N/kg}}}\cdot 9,8\,\,\cancel{\mathrm{N/kg}}=11\,637,5\,\,\mathrm{N}\]
This gives us the weight of the object on Earth.
This brings us to the end of this article. Let's go through what we've learned so far.
Gravity - Key takeaways
- Gravity is an invisible force of attraction between two bodies due to their mass.
- Gravity is a non-contact force as it can act at large distances over objects that aren't in any physical touch.
- The gravitational field strength of the earth can be considered constant, and it is approximately \(g=9,8\,\,\mathrm{N/kg}\).
The weight of an object in a gravitational field is given by the following equation \(W=mg\).
The theory of gravity states that every pair of objects with mass in the universe attracts each other by an invisible force acting along the direction of the line joining both objects' centres of mass.
References
- Fig. 1- Falling apple (https://commons.wikimedia.org/wiki/File:Falling_apple_crop.jpg) by Zátonyi Sándor (https://commons.wikimedia.org/wiki/User:Fizped) is licensed by CC BY 3.0 (https://creativecommons.org/licenses/by/3.0/deed.en).
- Fig. 3- Effects of gravity on the Moon and Earth by Encyclopædia Britannica (https://www.britannica.com/science/mechanics/Circular-orbits#/media/1/371907/141791).
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