10. Gravitation

Concept

• Concept: The force responsible for:
  • Objects falling towards the Earth.
  • The Moon going around the Earth.
  • Planets going around the Sun.
• Centripetal Force: The force that causes acceleration and keeps a body moving along a circular path, acting towards the center (e.g., the force the Earth exerts on the Moon). In the absence of this force, the object flies off in a straight line (tangent to the circle).

2. Universal Law of Gravitation (Section 10.1.1)

• Statement: Every object in the universe attracts every other object with a force ($\mathbf{F}$) which is:
  • Directly proportional to the product of their masses ($M \times m$).
  • Inversely proportional to the square of the distance ($d$) between them.
• Direction: The force is along the line joining the centers of the two objects.
• Formula: $F = G\frac{M \times m}{d^{2}}$
  • $G$ is the Universal Gravitation Constant.
• SI Unit of G: $\text{N m}^{2} \text{kg}^{-2}$.
• Value of G: $6.673 \times 10^{-11} \text{N m}^{2} \text{kg}^{-2}$.
• Nature of the Law: It is universal; applicable to all bodies (big or small, celestial or terrestrial).

3. Importance of the Universal Law of Gravitation (Section 10.1.2)

The law successfully explained: 1. The force that binds us to the Earth. 2. The motion of the Moon around the Earth. 3. The motion of planets around the Sun. 4. The tides due to the Moon and the Sun.

4. Free Fall and Acceleration due to Gravity ($g$) (Section 10.2)

• Free Fall: When objects fall towards the Earth under the influence of the gravitational force alone.
• Acceleration due to Gravity ($g$): The acceleration involved when an object falls towards the Earth due to the Earth's gravitational force.
• Unit of $g$: $\text{m s}^{-2}$.
• Relation between F and g: $F = m \times g$ (Gravitational Force on an object of mass $m$).
• Formula for $g$ (General): $g = G\frac{M}{d^{2}}$, where $M$ is the mass of the Earth and $d$ is the distance from the object to the center of the Earth.
• Formula for $g$ (Near the surface): $g = G\frac{M}{R^{2}}$, where $R$ is the radius of the Earth.
• Calculated Value of $g$ (on Earth): $g \approx 9.8 \text{ m s}^{-2}$.
• Variation of $g$:
  • Altitude: $g$ decreases with altitude.
  • Poles to Equator: $g$ is greater at the poles than at the equator (because the Earth's radius $R$ is smaller at the poles).
• Motion Equations for Free Fall (replacing $a$ with $g$):
  • $v = u + gt$
  • $s = ut + \frac{1}{2}gt^2$
  • $v^2 = u^2 + 2gs$
  • Sign Convention: $g$ is taken as positive for motion in the direction of velocity (downwards) and negative when opposing motion (upwards).
• Independence of Mass: The acceleration experienced during free fall ($g$) is independent of the object's mass (i.e., all objects fall at the same rate in a vacuum).

5. Mass and Weight (Sections 10.3 and 10.4)

• Weight on the Moon: The weight of an object on the Moon ($W_m$) is approximately $\frac{1}{6}$th of its weight on the Earth ($W_e$). $\frac{W_m}{W_e} \approx 0.165 \approx \frac{1}{6}$.

6. Thrust and Pressure (Section 10.5)

• Thrust: The force acting on an object perpendicular to the surface.
• Pressure ($P$): The thrust per unit area.
• Formula: $P = \frac{\text{Thrust}}{\text{Area}}$
• SI Unit of Pressure: $\text{N/m}^2$ or $\text{N m}^{-2}$, also called pascal (Pa).
• Effect: The same force (thrust) acting on a smaller area exerts a larger pressure, and a smaller pressure on a larger area (e.g., sharp knives, wide foundations).
• Pressure in Fluids: Liquids and gases (fluids) also exert pressure on the base and walls of their container.

7. Buoyancy and Archimedes' Principle (Sections 10.5.2, 10.5.3, and 10.6)

• Buoyancy/Upthrust/Buoyant Force: The upward force exerted by a fluid on an object immersed in it.
  • Its magnitude depends on the density of the fluid.
• Floating and Sinking:
  • Float: If the density of the object is less than the density of the liquid, the object floats (upthrust > weight).
  • Sink: If the density of the object is greater than the density of the liquid, the object sinks (upthrust < weight).
• Archimedes' Principle (Statement): 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.
• Applications: Designing ships and submarines, lactometers, and hydrometers.

8. Relative Density (Section 10.7)

• Definition: The ratio of the density of a substance to the density of water.
• Formula: $\text{Relative density} = \frac{\text{Density of a substance}}{\text{Density of water}}$
• Unit: It has no unit (it is a ratio of similar quantities).

Derivations of the key formulas

1. Universal Law of Gravitation ($F = G\frac{Mm}{d^2}$)

The Universal Law of Gravitation states that the force ($F$) of attraction between two objects is:

1. Directly proportional to the product of their masses ($M$ and $m$): $F \propto M \times m \quad \text{(10.1)}$
2. Inversely proportional to the square of the distance ($d$) between them: $F \propto \frac{1}{d^2} \quad \text{(10.2)}$

Combining these two proportionality statements: $F \propto \frac{M \times m}{d^2}$

Introducing $G$ (the Universal Gravitation Constant) as the constant of proportionality: $F = G\frac{M \times m}{d^2} \quad \text{(10.4)}$

2. Relation between Gravitational Force and Acceleration due to Gravity ($g = G\frac{M}{d^2}$)

The force of attraction exerted by the Earth (mass $M$) on an object (mass $m$) at a distance $d$ from the center of the Earth can be expressed in two ways:

A. Using Newton's Second Law of Motion For an object falling under the influence of gravity, the force ($F$) is the product of its mass ($m$) and the acceleration due to gravity ($g$): $F = m \times g \quad \text{(10.6)}$

B. Using the Universal Law of Gravitation The gravitational force between the Earth (mass $M$) and the object (mass $m$) at distance $d$ is: $F = G\frac{M \times m}{d^2} \quad \text{(10.4)}$

Equating the two expressions for force (Equations 10.6 and 10.4): $m \times g = G\frac{M \times m}{d^2}$

Canceling the mass of the object ($m$) from both sides: $g = G\frac{M}{d^2} \quad \text{(10.7)}$

Note: For objects on or near the surface of the Earth, the distance $d$ is replaced by the radius of the Earth, $R$, giving the specific form: $g = G\frac{M}{R^2} \quad \text{(10.9)}$

3. Relation for Weight ($W = m \times g$)

Weight ($W$) is defined as the force with which an object is attracted towards the Earth.

Starting with the relation for the gravitational force ($F$) near the Earth's surface (where $a=g$): $F = m \times g \quad \text{(10.14)}$

Since weight ($W$) is this force of attraction ($F$): $W = m \times g \quad \text{(10.15)}$

4. Ratio of Weight on the Moon to Weight on the Earth ($\frac{W_m}{W_e} \approx \frac{1}{6}$)

The weight of an object (mass $m$) on any celestial body is the gravitational force exerted by that body on the object.

A. Weight on the Moon ($W_m$) Using the universal law of gravitation, where $M_m$ is the mass of the Moon and $R_m$ is its radius: $W_m = G\frac{M_m \times m}{R_m^2} \quad \text{(10.16)}$

Substituting the values for the Moon ($M_m = 7.36 \times 10^{22} \text{ kg}$ and $R_m = 1.74 \times 10^6 \text{ m}$): $W_m \approx 2.431 \times 10^{10} G \times m \quad \text{(10.18a)}$

B. Weight on the Earth ($W_e$) Using the universal law of gravitation, where $M$ is the mass of the Earth and $R$ is its radius: $W_e = G\frac{M \times m}{R^2} \quad \text{(10.17)}$

Substituting the values for the Earth ($M = 5.98 \times 10^{24} \text{ kg}$ and $R = 6.37 \times 10^6 \text{ m}$): $W_e \approx 1.474 \times 10^{11} G \times m \quad \text{(10.18b)}$

C. Ratio Dividing Equation (10.18a) by Equation (10.18b): $\frac{W_m}{W_e} = \frac{2.431 \times 10^{10} G \times m}{1.474 \times 10^{11} G \times m}$

Canceling $G \times m$: $\frac{W_m}{W_e} = \frac{2.431 \times 10^{10}}{1.474 \times 10^{11}} = 0.165$

$\frac{W_m}{W_e} \approx 0.165 \approx \frac{1}{6} \quad \text{(10.19)}$

5. Pressure Formula ($P = \frac{\text{Thrust}}{\text{Area}}$)

Pressure is defined as the thrust (force acting perpendicular to a surface) per unit area.

$\text{Pressure} = \frac{\text{Thrust}}{\text{Area}} \quad \text{(10.20)}$

6. Relative Density Formula

Relative density is defined as the ratio of the density of a substance to the density of water.

$\text{Relative density} = \frac{\text{Density of a substance}}{\text{Density of water}}$