Key Points

Every object in the universe attracts every other object with a force $F$ that is directly proportional to the product of their masses ($M$ and $m$) and inversely proportional to the square of the distance ($d$) between their centers. The formula is $F = G \frac{M \times m}{d^2}$.

The constant $G$ is called the universal gravitational constant. Its value is the same for all pairs of objects in the universe. The accepted value of G is $6.673 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$.

The universal law of gravitation explains several key phenomena, including the force that binds us to the earth, the motion of the moon around the earth, the motion of planets around the sun, and the formation of tides.

When an object falls towards the earth under gravitational force alone, it is in free fall. This motion involves a constant acceleration called acceleration due to gravity, denoted by $g$. On Earth, its average value is $g \approx 9.8 \text{ m s}^{-2}$.

The value of $g$ on the surface of a celestial body can be calculated using the formula $g = G \frac{M}{R^2}$, where $M$ is the mass and $R$ is the radius of the body. This shows that $g$ is independent of the mass of the falling object.

The value of $g$ is not constant on the surface of the Earth. It is greatest at the poles and decreases towards the equator because the Earth's radius is smaller at the poles.

For an object in free fall, the standard equations of motion apply with acceleration $a$ replaced by $g$. The equations are $v = u + gt$, $s = ut + \frac{1}{2}gt^2$, and $v^2 = u^2 + 2gs$. We take $g$ as positive for downward motion and negative for upward motion.

Mass is the amount of matter in an object and is constant everywhere, measured in kilograms (kg). Weight is the gravitational force on an object, calculated as $W = m \times g$, and it varies with location, measured in Newtons (N).

Due to the moon's smaller mass and radius, its gravitational force is weaker than Earth's. The weight of an object on the moon is approximately one-sixth of its weight on Earth, or $W_{\text{moon}} \approx \frac{1}{6} W_{\text{earth}}$.

Thrust is the force acting perpendicular to a surface. Pressure is the thrust per unit area, given by the formula $P = \frac{\text{Thrust}}{\text{Area}}$. The SI unit of pressure is the Pascal (Pa), where $1 \text{ Pa} = 1 \text{ N/m}^2$.

When an object is immersed in a fluid (a liquid or a gas), it experiences an upward force called the buoyant force or upthrust. This force opposes the object's weight and makes it feel lighter.

Archimedes' principle states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object. This principle is used in designing ships and submarines.

An object floats or sinks in a liquid based on its density relative to the liquid. An object floats if its density is less than the liquid's density, and it sinks if its density is greater than the liquid's density.

Density is defined as mass per unit volume. The formula is $\rho = \frac{m}{V}$, where $\rho$ is density, $m$ is mass, and $V$ is volume. The SI unit for density is $\text{kg/m}^3$.