Forces, Terminal Velocity, and Circular Motion Study Notes

In the absence of air resistance, a falling object undergoes constant acceleration of $9.8\,m/s^2$ .
As an object accelerates through the air, the air resistance opposing its movement increases as its speed rises.
The increasing air resistance reduces the object's acceleration, causing the acceleration to no longer remain constant.
Eventually, the upward air resistance equals the downward weight of the object.
When gravitational force balances frictional drag, the resultant force on the object becomes zero ( $0\,N$ ).
At this point, the object falls at a constant speed termed its terminal velocity, the value of which depends directly on the object's size, shape, and weight.
Small, dense objects (such as a steel ball-bearing) exhibit a high terminal velocity and fall over a considerable distance with a constant acceleration of $9.8\,m/s^2$ before air resistance equals their weight.
Light objects (such as raindrops) or objects possessing a large surface area (such as parachutes) have a low terminal velocity, accelerating over a comparatively short distance before air resistance balances weight.
A skydiver reaches a terminal velocity exceeding $50\,m/s$ (equivalent to $180\,km/h$ ) prior to opening their parachute.

Synchronised skydivers

Objects falling through liquids demonstrate behavior similar to objects falling through air.

Keeping an object moving along a circular path requires a force directed continuously towards the centre of the circle.
Everyday examples of circular motion include:
- Funfair rides.
- Clothes spinning inside a washing machine drum.
- Planets revolving around the Sun.
- The Moon circling the Earth.
- A vehicle turning a corner along a circular arc.
- Whirling and releasing a hammer in the sport of 'throwing the hammer' practiced at Highland Games in Scotland.
Speed is a scalar quantity possessing only size, whereas velocity is a vector quantity defined by both size (speed) and a stated direction.
When an object travels in a circle, its direction of motion changes continuously at every point along the path.
At any instant (such as point $A$ or point $B$ along the circumference), the direction of motion is along the tangent to the circle at that specific location.
Because the direction of motion changes, the object's velocity changes continuously, even if its speed remains constant.
Acceleration is defined as a change in velocity; therefore, an object undergoing circular motion is accelerating.
According to Newton's first law of motion, an accelerating body must have a resultant force acting upon it to cause that acceleration.
The force responsible for circular motion acts perpendicular to the direction of motion at all times, directed towards the fixed centre of the circle.

The centre-directed force that maintains a body in a circular path is known as the centripetal force (meaning centre-seeking force).
The magnitude of the centripetal force ( $F$ ) required to maintain circular motion increases under the following conditions:
- The speed $v$ of the object is increased (while mass $m$ and radius $r$ remain constant).
- The radius $r$ of the circular path is decreased (while mass $m$ and speed $v$ remain constant).
- The mass $m$ of the object is increased (while speed $v$ and radius $r$ remain constant).
If the required centripetal force exceeds what can be supplied (for example, if a string breaks because the force exceeds its maximum strength), the object ceases circular motion.
When the restraining force fails, Newton's first law predicts that the object flies off at a steady speed in a straight line along the tangent at the precise moment of release, rather than being thrown directly outwards.

In physical systems, centripetal force is supplied by specific existing forces:
- Whirling a ball on a string: The inward pull of string tension exerted on the ball provides the centripetal force.
- Throwing the hammer: The pull exerted by the athlete's arms on the handle towards the centre of the path provides the centripetal force.

Throwing the hammer

Car rounding a bend: Friction exerted inwards by the road surface on the vehicle's tyres provides the centripetal force.
Planets and Moons: Gravitational attraction towards the central body provides the centripetal force.

For a satellite of mass $m$ orbiting Earth at a radius $r$ with orbital speed $v$ , the required centripetal force $F$ is supplied by Earth's gravitational attraction.
To insert an artificial satellite into orbit at a specific altitude above Earth, it must achieve the exact required orbital speed.
Earth's gravitational attraction decreases as distance/height above Earth increases; if a satellite enters orbit at an incorrect speed, gravitational force will not equal the centripetal force needed to maintain that altitude.
Communication Satellites and Geostationary Orbits:
- Geostationary satellites travel in orbits directly above Earth's equator at an altitude of $36000\,km$
- They orbit at the exact same rotational speed as Earth, causing them to appear completely stationary over a fixed point on Earth's surface.
- The orbital period of a geostationary satellite is $24\,hours$ .
- These satellites are used to transmit television, intercontinental telephone, and data signals.
- Geostationary satellites must maintain adequate spacing to prevent signal interference with one another; the geostationary orbit capacity is approximately $400$ satellites.