Universal Law Of Gravitation

Newton's groundbreaking insight led to the formulation of the **Universal Law of Gravitation**. This law applies to any two objects anywhere in the universe.

The law states: **Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them.** The direction of this force is along the line joining the centers of the two objects.

Mathematically, if two objects of masses $M$ and $m$ are separated by a distance $d$, the gravitational force $F$ between them is given by:

• $F$ is directly proportional to the product of their masses: $F \propto M \times m$
• $F$ is inversely proportional to the square of the distance between them: $F \propto \frac{1}{d^2}$

Combining these proportionalities, we get:

$$ F \propto \frac{M \times m}{d^2} $$

To convert this proportionality into an equation, we introduce a constant of proportionality, $G$, known as the **Universal Gravitation Constant**:

$$ F = G \frac{M \times m}{d^2} $$

The value of $G$ is constant throughout the universe. It was experimentally determined by Henry Cavendish. The accepted value is $G = 6.673 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$. The unit of $G$ can be derived from the formula: $G = \frac{F d^2}{Mm}$. Substituting the units of force (N), distance (m), and mass (kg), we get N m² / kg² or N m² kg⁻².

The gravitational force is generally a very weak force unless the masses of the objects involved are very large, like celestial bodies. This is why we don't feel the gravitational pull from objects around us in everyday life, although the force does exist.

The gravitational force exerted by the Earth on objects near its surface is what we commonly call **gravity**. The motion of the Moon around the Earth is a continuous fall towards the Earth due to the Earth's gravitational pull, providing the necessary **centripetal force** that keeps the Moon in its orbit rather than moving in a straight line.

According to Newton's Third Law of Motion, if the Earth attracts an apple, the apple also attracts the Earth with an equal and opposite force. However, the Earth's mass is immensely larger than the apple's mass. By Newton's Second Law ($a = F/m$), the acceleration produced in the Earth by the apple's pull is negligible compared to the acceleration produced in the apple by the Earth's pull. The same argument applies to the Earth-Moon system; the Moon accelerates significantly towards the Earth, while the Earth's acceleration towards the Moon is very small.

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Importance Of The Universal Law Of Gravitation

The Universal Law of Gravitation unified several previously unconnected phenomena, providing a single framework for understanding them. Its importance lies in successfully explaining:

• The force that holds objects (including ourselves) to the Earth.
• The motion of the Moon around the Earth.
• The motion of planets around the Sun.
• The occurrence of tides on Earth due to the gravitational pull of the Moon and the Sun.

This law was a monumental achievement in physics, demonstrating the power of a single principle to explain diverse natural phenomena.

Free Fall

**Free fall** refers to the motion of an object falling towards the Earth when it is under the influence of the Earth's gravitational force **alone**. This means other forces like air resistance are neglected.

When an object falls freely, its direction of motion is downwards, towards the Earth. However, the magnitude of its velocity changes due to the Earth's attraction. Any change in velocity implies acceleration. The acceleration experienced by an object during free fall is due to the Earth's gravitational force and is called the **acceleration due to gravity**. It is denoted by the symbol **g**.

The unit of $g$ is the same as acceleration, which is metres per second squared (m/s² or m s⁻²).

According to Newton's Second Law ($F=ma$), the gravitational force ($F_g$) acting on an object of mass $m$ in free fall can also be written as $F_g = m \times a = m \times g$.

We also know from the Universal Law of Gravitation that the force of gravity between the Earth (mass $M$, radius $R$) and an object on its surface (mass $m$) is $F_g = G \frac{M m}{R^2}$.

Equating the two expressions for the gravitational force on the object:

$$ mg = G \frac{M m}{R^2} $$

We can cancel out the mass of the object ($m$) from both sides:

$$ g = G \frac{M}{R^2} $$

This equation shows that the acceleration due to gravity ($g$) depends on the mass of the Earth ($M$), the radius of the Earth ($R$), and the gravitational constant ($G$), but it is **independent of the mass of the falling object** ($m$). This is a crucial result: in the absence of air resistance, all objects, regardless of their mass, fall towards the Earth with the same acceleration.

The Earth is not a perfect sphere; its radius is slightly larger at the equator than at the poles. According to the formula $g = G \frac{M}{R^2}$, a larger radius $R$ means a smaller value of $g$. Therefore, the value of $g$ is slightly less at the equator than at the poles. However, for practical purposes involving objects near the Earth's surface, $g$ is often considered constant.

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To Calculate The Value Of G

Using the formula $g = G \frac{M}{R^2}$ and accepted values for $G$, $M$ (mass of Earth), and $R$ (mean radius of Earth), we can calculate the average value of $g$ on the Earth's surface.

• $G = 6.673 \times 10^{-11}$ N m² kg⁻²
• $M = 6 \times 10^{24}$ kg
• $R = 6.4 \times 10^6$ m

$$ g = (6.673 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}) \times \frac{6 \times 10^{24} \text{ kg}}{(6.4 \times 10^6 \text{ m})^2} $$

$$ g = (6.673 \times 10^{-11}) \times \frac{6 \times 10^{24}}{40.96 \times 10^{12}} \text{ m s}^{-2} $$

$$ g = (6.673) \times \frac{6}{40.96} \times 10^{(-11+24-12)} \text{ m s}^{-2} $$

$$ g \approx (6.673) \times (0.1465) \times 10^{1} \text{ m s}^{-2} $$

$$ g \approx 0.9775 \times 10 \text{ m s}^{-2} \approx 9.775 \text{ m s}^{-2} $$

Rounding off, the average value of acceleration due to gravity on the Earth's surface is approximately **9.8 m/s²**.

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Motion Of Objects Under The Influence Of Gravitational Force Of The Earth

In the absence of air resistance, all objects falling towards the Earth experience the same uniform acceleration $g$. Therefore, the equations of motion derived for uniformly accelerated motion (Chapter 7) are applicable to objects in free fall, by replacing the general acceleration 'a' with 'g'.

The equations for free fall are:

1. $v = u + gt$
2. $s = ut + \frac{1}{2}gt^2$
3. $v^2 = u^2 + 2gs$

Where:

• $u$ = initial velocity
• $v$ = final velocity
• $g$ = acceleration due to gravity ($~9.8$ m/s²)
• $t$ = time taken
• $s$ = distance covered (often vertical height)

When using these equations, it is important to use proper sign conventions. We can choose the downward direction as positive and the upward direction as negative, or vice versa. If downward is positive, then $g$ is positive ($+9.8$ m/s²). If upward is positive, then $g$ is negative ($-9.8$ m/s²). Velocity and displacement signs should be consistent with this choice.

Activity 9.3 illustrates the effect of air resistance. A sheet of paper falls slower than a stone dropped from the same height due to greater air resistance acting on the larger surface area of the paper. In a vacuum (where there is no air), the paper and the stone would indeed fall at the same rate, as predicted by the constant value of $g$.

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Mass

As discussed in Chapter 8, the **mass** of an object is a measure of its inertia. It quantifies the amount of matter in the object and its resistance to changes in motion. Mass is a **scalar quantity** and is a fundamental property of an object.

A key characteristic of mass is that it is **constant and does not change from place to place**, whether the object is on Earth, on the Moon, or in outer space. The mass of an object remains the same regardless of the gravitational environment.

Weight

The **weight (W)** of an object is the **force with which the Earth attracts the object** towards its center. Since weight is a force, it is a **vector quantity**, acting vertically downwards, and its SI unit is the same as force, which is **newton (N)**.

From Newton's Second Law, Force ($F$) = mass ($m$) $\times$ acceleration ($a$). In the case of gravity, the force is weight ($W$) and the acceleration is acceleration due to gravity ($g$).

$$ W = m \times g $$

Since mass ($m$) is constant for a given object, the weight ($W$) of the object is directly proportional to the acceleration due to gravity ($g$) at that location ($W \propto g$). Because the value of $g$ varies slightly from place to place on Earth (and significantly on other celestial bodies), the weight of an object also varies depending on its location.

On Earth, since $g$ is approximately constant at a given location, weight is often used as a measure of mass (e.g., a "1 kg weight" actually refers to an object with a mass of 1 kg, which has a weight of $1 \text{ kg} \times 9.8 \text{ m/s}^2 = 9.8 \text{ N}$). However, mass and weight are fundamentally different concepts.

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Weight Of An Object On The Moon

The weight of an object on the Moon is the force with which the Moon attracts that object. Since the Moon has significantly less mass and a smaller radius than the Earth, the acceleration due to gravity on the Moon ($g_m$) is much smaller than on the Earth ($g_e$).

Let $W_m$ be the weight of an object of mass $m$ on the Moon, and $W_e$ be its weight on the Earth.

$W_m = m \times g_m$ and $W_e = m \times g_e$.

From the formula $g = G \frac{M}{R^2}$, the value of $g$ for the Moon ($g_m$) and Earth ($g_e$) are:

$$ g_m = G \frac{M_{Moon}}{R_{Moon}^2} \quad \text{and} \quad g_e = G \frac{M_{Earth}}{R_{Earth}^2} $$

Using the masses and radii of the Earth and Moon (Table 9.1 in the text), the ratio of $g_m$ to $g_e$ is found to be approximately 1/6.

$$ \frac{g_m}{g_e} \approx \frac{1}{6} $$

Since $W = mg$, the ratio of weights is:

$$ \frac{W_m}{W_e} = \frac{m \times g_m}{m \times g_e} = \frac{g_m}{g_e} \approx \frac{1}{6} $$

Thus, the weight of an object on the Moon is approximately **one-sixth (1/6)** of its weight on the Earth.

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Thrust And Pressure

Understanding why a camel can walk easily in a desert or why trucks have wide tyres involves the concepts of force applied over an area. The effectiveness of a force can depend not just on its magnitude but also on the area over which it acts.

**Thrust** is defined as the **force acting on an object perpendicular to a surface**. It is a specific direction of force.

**Pressure** is defined as the **thrust per unit area**. It quantifies how concentrated the force is over a given surface.

$$ \text{Pressure} = \frac{\text{Thrust}}{\text{Area}} $$

The SI unit of thrust (being a force) is Newton (N). The SI unit of area is square metre (m²). Therefore, the SI unit of pressure is Newton per square metre (N/m² or N m⁻²). In honour of scientist Blaise Pascal, the SI unit of pressure is also called **pascal (Pa)**. So, 1 Pa = 1 N/m².

Examples illustrating pressure:

• Pushing a drawing pin into a board: You apply a force (thrust) on the head of the pin over a relatively large area. However, this same force is transmitted to the sharp point of the pin, which has a very small area. The small area at the tip experiences a very high pressure, allowing it to penetrate the board easily.

• Standing vs. lying on sand: When you stand on loose sand, your weight (thrust) is distributed over the small area of your feet, resulting in high pressure that causes your feet to sink deep. When you lie down, your weight (the same thrust) is distributed over the larger contact area of your whole body. This results in lower pressure, so your body does not sink as deep.

These examples show that for the same applied force (thrust), a smaller area results in larger pressure, and a larger area results in smaller pressure. This principle is used in designing pointed nails, sharp knives (to apply high pressure for cutting), and wide foundations for buildings (to reduce pressure on the ground).

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Pressure In Fluids

Liquids and gases are collectively called **fluids**. Like solids, fluids have weight. Fluids exert pressure on the base and walls of the container they are in. A key property of fluids is that pressure applied to any confined mass of fluid is transmitted equally and undiminished in all directions (Pascal's Law).

Objects immersed in fluids also experience pressure from the fluid, which acts perpendicular to the surface of the object at every point.

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Buoyancy

Have you noticed that objects seem lighter when submerged in water? Or that a bucket of water is easier to lift when it is in the water than when it is out? This is due to an upward force exerted by the fluid on the immersed object, known as the **buoyant force** or **upthrust**.

When an object is placed in a fluid, the fluid exerts pressure on the object from all directions. The pressure exerted by the fluid increases with depth. Since the bottom surface of a submerged object is deeper than its top surface, the upward pressure on the bottom is greater than the downward pressure on the top. This difference in pressure creates a net upward force – the buoyant force.

Activity 9.4 demonstrates buoyancy: an empty plastic bottle floats because the upward buoyant force from the water is greater than the bottle's weight. Pushing it down requires applying a downward force to overcome the increasing buoyant force. When released, it bounces back up because the buoyant force is still greater than its weight while submerged.

The magnitude of the buoyant force depends on the **density of the fluid** and the **volume of the fluid displaced** by the object.

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Why Objects Float Or Sink When Placed On The Surface Of Water?

Whether an object floats or sinks in a fluid depends on the balance between the downward gravitational force (weight of the object) and the upward buoyant force exerted by the fluid. This balance is related to the **density** of the object compared to the density of the fluid.

Density is defined as mass per unit volume ($\rho = m/V$).

• Consider placing an iron nail on the surface of water (Activity 9.5). The nail sinks. Its weight (downward force) is greater than the maximum upward buoyant force exerted by the water it displaces when fully submerged. This happens because the density of iron is **greater than the density of water**.
• Consider placing a piece of cork on the surface of water (Activity 9.6). The cork floats. Its weight (downward force) is balanced by the upward buoyant force. This is because the density of cork is **less than the density of water**. The cork sinks only until it displaces a volume of water whose weight is equal to the cork's weight. At that point, the buoyant force equals the weight, and the cork floats.

General rule: An object with a density **less than** the density of the fluid will **float**. An object with a density **greater than** the density of the fluid will **sink**. If the density of the object is equal to the density of the fluid, it will remain suspended within the fluid.

This principle explains why objects made of iron and steel (which are denser than water) like ships can float. Although the material itself is denser than water, the ship's overall design is such that it encloses a large volume of air. The average density of the ship (total mass divided by the total volume, including the hull and the air inside) is less than the density of water, allowing it to displace a volume of water large enough to generate a buoyant force equal to its total weight.

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Archimedes’ Principle

Archimedes' Principle quantifies the magnitude of the buoyant force. Activity 9.7, involving gradually lowering a stone tied to a spring balance into water, demonstrates that the stone's apparent weight decreases as it is immersed. This decrease in apparent weight is equal to the upward buoyant force. The buoyant force increases as more of the stone is submerged, until it is fully immersed, after which the buoyant force remains constant (as the volume of water displaced no longer increases).

**Archimedes’ Principle** states: **When a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it.**

Mathematically, Buoyant Force ($F_B$) = Weight of the fluid displaced = $(\text{Volume of displaced fluid}) \times (\text{Density of fluid}) \times g$.

$$ F_B = V_{displaced} \times \rho_{fluid} \times g $$

Where:

• $V_{displaced}$ is the volume of the fluid displaced by the object (which is equal to the volume of the submerged part of the object).
• $\rho_{fluid}$ is the density of the fluid.
• $g$ is the acceleration due to gravity.

Archimedes' Principle has numerous practical applications, including:

• Designing ships and submarines (ensuring the buoyant force is sufficient for floating and controlling depth).
• In instruments like lactometers (to measure the purity of milk based on its density) and hydrometers (to measure the density of liquids).

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