Unit 2: Gravity and Force

The Force of Gravity

Gravity is a fundamental force of nature that causes mutual attraction between all objects that have mass. The more mass an object has, the stronger its gravitational pull.

Near the Earth's surface, we simplify this force. We call the force of Earth's gravity on an object its weight (W).

W = m × g

• `m` = mass of the object (in kg)
• `g` = acceleration due to gravity (a constant value of ≈ 9.8 m/s²)

Mass vs. Weight:

• Mass is the amount of "stuff" in an object. It is constant everywhere (your mass is the same on Earth and on the Moon).
• Weight is the force of gravity on that mass. It changes depending on where you are (your weight on the Moon is ~1/6th of your weight on Earth, because the Moon's `g` is smaller).

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Friction

Friction (f) is a force that opposes motion (or attempted motion) between surfaces in contact. It is a non-conservative force (it dissipates energy as heat).

Friction is caused by the microscopic imperfections (bumps and valleys) on surfaces, which lock together, and by intermolecular forces between the surfaces.

Advantages of Friction

We depend on friction for almost everything we do:

• Walking/Running: Friction between your shoes and the ground pushes you forward. Without it (like on pure ice), you can't move.
• Vehicle Brakes: Brake pads create friction against the wheels to stop the car.
• Grip: Allows you to hold a pen, turn a doorknob, or pick up a glass.
• Nails and Screws: Friction holds them in place in wood or walls.
• Lighting a match: Friction provides the heat to ignite the chemicals.

Disadvantages of Friction

Friction also causes problems:

• Wears out parts: It wears down car tires, shoe soles, and engine components.
• Wastes Energy: In any machine (like a car engine), a large amount of energy (fuel) is wasted as heat just to overcome friction.
• Reduces Efficiency: Makes it harder to push or move objects.
• Generates Heat: Can cause moving parts to overheat and fail.

Reducing Friction: We reduce friction using lubricants (like oil or grease), ball bearings, or by polishing surfaces.

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Circular Motion: Bending and Banking

When an object moves in a circle, it is constantly changing direction. A change in direction is a change in velocity (which is a vector), and a change in velocity is an acceleration. This acceleration is directed *towards the center* of the circle and is called centripetal acceleration (a_c).

By Newton's 2nd Law (F=ma), if there is an acceleration, there must be a net force. The force that causes this centripetal acceleration is called the centripetal force (F_c).

F_c = m a_c = m v² / r

Where `m` is mass, `v` is speed, and `r` is the radius of the circle. This is not a new *type* of force; it is the *job* that a real force does (e.g., gravity for a planet, friction for a car).

Bending of Cyclist

When a cyclist takes a turn, they need a centripetal force to pull them into the curve. This force is provided by friction between the tires and the road, pointing towards the center of the turn.

This friction force creates a torque that would make the cyclist topple *outwards*. To balance this, the cyclist leans inwards. By leaning, the cyclist's own weight (gravity) and the ground's normal force create a counter-torque that balances the toppling effect, allowing them to make the turn safely.

Banking of Roads

For cars at high speed, relying on friction alone is dangerous (e.g., if the road is wet). To help, curved roads are banked (tilted) inwards.

By tilting the road, a component of the normal force (which is always perpendicular to the road surface) now points horizontally towards the center of the turn. This component of the normal force provides the necessary centripetal force, reducing or eliminating the need for friction.

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Work Done by a Centripetal Force

The work done by a centripetal force is always ZERO.

Explanation:

Work is defined as `W = F × d × cos(θ)`.

• The centripetal force (F) always points *towards the center* of the circle.
• The displacement (d) of the object at any instant is *along the tangent* to the circle.
• The tangent and the radius are always perpendicular (90°) to each other.
• Therefore, the angle `θ` between the centripetal force and the displacement is 90°.
• Since cos(90°) = 0, the work done is W = 0.

This also makes sense from an energy perspective: `W = ΔK`. Since the centripetal force only changes the *direction* of the velocity, not the *speed*, the kinetic energy `(1/2)mv²` remains constant. If `ΔK = 0`, then `W = 0`.

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Buoyancy and Flight

This section deals with forces in fluids (liquids and gases).

Hot Air and Helium Balloons

This is an application of Archimedes' Principle (Buoyancy).

An object immersed in a fluid (like air) experiences an upward buoyant force (F_B) equal to the weight of the fluid it displaces.

F_B = (Density of fluid) × (Volume of object) × g

An object floats if its *average density* is less than the density of the fluid.

• Hot Air Balloon: The balloon is a large, fixed volume (V). By heating the air *inside* the balloon, its molecules spread out, making its density (ρ_hot) much lower than the density of the cold air *outside* (ρ_cold).
  The balloon's total weight is W = (m_balloon + m_{hot_air})g.
  The buoyant force is F_B = (Weight of *displaced* cold air) = (ρ_cold × V)g.
  Lift occurs when F_B > W.
• Helium Balloon: Helium (He) is a gas that is naturally much less dense than air (a mix of N₂ and O₂). A balloon filled with helium has a very low average density, so the buoyant force from the displaced air is much greater than its weight, and it rises.

Working of Airplanes

Airplanes fly using a combination of four forces:

1. Lift (Up): The force that pushes the plane up, generated by the wings.
2. Weight (Down): The force of gravity on the plane.
3. Thrust (Forward): The force from the engines pushing the plane forward.
4. Drag (Backward): Air resistance (friction) pushing the plane back.

How is Lift created? The wing (airfoil) is specially shaped: curved on top, flatter on the bottom.
This shape, combined with the "angle of attack," forces the air moving over the top to travel a *longer distance* in the *same amount of time* as the air moving under the bottom.
This means the air on top moves faster.
According to Bernoulli's Principle (see below), faster-moving fluid has lower pressure.
So, there is low pressure *above* the wing and high pressure *below* the wing. This pressure difference creates a net upward force: Lift.

Working of Helicopters

A helicopter uses its main rotor blades to fly. These blades are essentially rotating wings (airfoils).
By spinning the blades, they create "relative wind" and generate Lift in the same way an airplane wing does.
By changing the *pitch* (angle) of the blades, the pilot can control the amount of lift:

• Hover: Lift = Weight
• Ascend: Lift > Weight
• Descend: Lift < Weight

By tilting the entire rotor disc, the pilot can direct the lift force slightly forward or backward, creating Thrust to move in any direction.

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Conservation of Angular Momentum

As covered in mechanics, the Principle of Conservation of Angular Momentum states:

If no net external torque acts on a system, its total angular momentum (L) remains constant.

Angular Momentum (L) = I × ω

• `I` = Moment of Inertia (resistance to rotation, depends on mass and how it's distributed)
• `ω` = Angular Velocity (how fast it spins)

So, if L is constant: I_initial × ω_initial = I_final × ω_final

Ice Skater Spin

This is the classic example. A skater starts a spin with their arms outstretched.

• Arms Out: Mass is far from the axis of rotation. 'I' is large, so spin speed 'ω' is small. (L = I_large × ω_small)
• Arms In: The skater pulls their arms in. Mass is now close to the axis. 'I' becomes small.
• To keep L constant, 'ω' must become large. The skater spins much faster. (L = I_small × ω_large)

Planet Orbiting Around Sun

A planet (like Earth) orbits the Sun in an ellipse. The force of gravity from the Sun is a central force, meaning it always points *towards* the Sun (the axis of rotation).

This force creates zero torque (τ = r × F = 0, because F is parallel to r).
Therefore, the planet's angular momentum (L) is conserved.

• Aphelion (far from Sun): `r` is large, so `I` is large. The planet moves slowly ('ω' is small).
• Perihelion (close to Sun): `r` is small, so `I` is small. The planet moves faster ('ω' is large).

This is also the physical basis for Kepler's Second Law (equal areas in equal times).

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Bernoulli’s Theorem: Swing of a Cricket Ball

Understanding Bernoulli’s Theorem

Bernoulli's Principle (or Theorem) is a simplified form of the conservation of energy for a moving fluid. In simple terms:

Where the speed of a fluid (liquid or gas) is high, its internal pressure is low.
Where the speed of the fluid is low, its internal pressure is high.

(This assumes the fluid is at the same height).

Swing of a Cricket Ball (The Magnus Effect)

A cricket ball is "swung" (made to curve in the air) by a bowler. This is achieved by combining the ball's spin with its forward velocity, using Bernoulli's principle.

The bowler polishes one side of the ball (making it smooth) and leaves the other side rough (the "seam" side).

1. Airflow: As the ball moves forward, air flows past it on both sides.
2. The Smooth Side: The air flows smoothly and fast over this side. This is called laminar flow.
3. The Rough Side: The seam and roughness "trip" the air, causing it to become turbulent. This turbulent layer actually *clings* to the ball's surface longer, and is moving slower relative to the ball.
4. The Spin Effect: The bowler also spins the ball (e.g., seam pointing towards the batsman). Let's say the ball spins so the rough side is moving *against* the airflow and the smooth side is moving *with* the airflow.
   • On the smooth side: The air speed and surface speed add up. The air is "dragged" along, moving even faster.
   • On the rough side: The air speed and surface speed oppose. The air is "slowed down" by the rough, spinning surface.
5. The Result (Bernoulli):
   • Smooth Side: VERY FAST air = LOW Pressure
   • Rough Side: SLOW air = HIGH Pressure
6. A net force (the Magnus Force) is created, pushing the ball from the high-pressure side to the low-pressure side.

This causes the ball to "swing" (curve) in the air, deceiving the batsman.