Study Notes on Centrifuges, Circular Motion, and Gravitation

A centrifuge uses circular motion principles to separate biological samples by density.
Denser particles experience a greater "apparent force" and move radially outward faster, while less dense particles remain closer to the center, allowing for effective separation of mixtures.
It spins samples at high speeds, achieving accelerations up to $250,000 g$ (Earth's gravity).
The force experienced by samples results in significant normal force, which can be compared to apparent weight.

Centripetal acceleration formula: $a_c = \frac{v^2}{r}$ where $v$ is the tangential velocity of the sample and $r$ is the radius of the circular path.
For an ultracentrifuge with a radius of $0.09 m$ and acceleration of $250,000 g$ , the required spin frequency is about $830$ revolutions per second, equivalent to $50,000 RPM$ . This frequency translates to an angular velocity $\omega$ where $v = r\omega$ .

Normal force dominates in centrifuge leading to high apparent weight.
This "apparent weight" is the force exerted by the container walls on the sample, pushing it towards the center, providing the necessary centripetal force rather than the actual gravitational weight.
For a sample of mass $0.003 kg$ and centripetal acceleration of $2.45 \times 10^6 \frac{m}{s^2}$ , the apparent weight is approximately $7400 N$ (approximatively $1700 lbs$ ).

All objects are attracted to each other by gravity.
Gravitational force is inversely proportional to the square of the distance between masses and directly proportional to the product of the masses.
This inverse square relationship means that as the distance between two objects doubles, the gravitational force between them decreases by a factor of four ( $2^2$ ).
$F_g = G \frac{m_1 m_2}{r^2}$
Universal gravitational constant $G = 6.67 \times 10^{-11} \frac{Nm^2}{kg^2}$ .

An object in orbit falls towards another body but maintains a curved path.
The gravitational force provides the centripetal force required for circular motion. Specifically, for a stable circular orbit, the gravitational pull is exactly equal to the centripetal force needed to keep the object in its path.
The orbital velocity formula for a body orbiting close to a planet's surface: $v = \sqrt{g \cdot r}$ , where $g$ is the local gravitational acceleration and $r$ is the radius of the orbit (effectively the planet's radius).
Time taken for one complete orbit around Earth is about $84$ minutes at the surface level (a hypothetical scenario without atmospheric drag).

The acceleration due to gravity ( $g$ ) varies on different planets and also with altitude.
It depends directly on the mass of the celestial body and inversely on the square of its radius from the center.
Example: On the Moon, $g_{moon} \approx 1.62 \frac{m}{s^2}$ , leading to different weights compared to Earth.
Calculation of gravitational acceleration on other celestial bodies involves similar principles as Earth but with their specific mass and radius.

First Law: Planets orbit the Sun in elliptical paths, with the Sun at one focus.
Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal times, implying planets move faster when closer to the Sun.
The relationship between a planet's orbital period ( $T$ ) and its average orbital radius ( $r$ ): $T^2 \propto r^3$ , often stated as $\frac{T^2}{r^3} = \text{constant}$ for all planets orbiting the same central body.
Derived from Newton's laws, establishing a connection between celestial mechanics and terrestrial physics.

All objects and systems in our universe follow the same physical laws, revealing the interconnectedness of projectile motion, gravitational forces, and circular motion.
Understanding these principles is essential for applications ranging from centrifugation to satellite motion and beyond, including designing space missions and understanding cosmic phenomena.