Work, Energy, Momentum, and Fluid Mechanics: Structured Study Notes

Work and Energy

Work Done by a Force

Work is defined as the product of the force applied to an object and the displacement in the direction of the force. It is a scalar quantity and is measured in joules (J).

• Definition: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

• Units: 1 Joule (J) = 1 Newton meter (N·m)

• Example: If a force of 10 N is applied over a distance of 2 m at an angle of 0°, $W = 10 \times 2 \times \cos 0 = 20$ J.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion.

• Formula: $K = \frac{1}{2}mv^2$

• Units: Joules (J)

• Example: A 2 kg object moving at 3 m/s has $K = \frac{1}{2} \times 2 \times 9 = 9$ J.

Potential Energy

Potential energy is the energy stored in an object due to its position in a force field, commonly gravitational.

• Gravitational Potential Energy: $U = mgh$

• Units: Joules (J)

• Example: A 1 kg mass at a height of 5 m: $U = 1 \times 9.8 \times 5 = 49$ J.

Conservation of Energy

The total mechanical energy (kinetic + potential) in a system remains constant if only conservative forces act.

• Mechanical Energy: $E = K + U$

• Conservation Law: $E_{i} = E_{f}$ (if only conservative forces are present)

• Example: A falling object converts potential energy to kinetic energy, but the sum remains constant.

Conservative and Non-Conservative Forces

Conservative Forces

Conservative forces are those for which the work done is independent of the path taken and depends only on the initial and final positions.

• Examples: Gravitational force, spring force

• Work Done: $W_{cons} = -\Delta U$

• Properties: Energy can be fully recovered; potential energy can be defined.

Non-Conservative Forces

Non-conservative forces, such as friction, dissipate energy as heat or other forms, and the work done depends on the path.

• Examples: Friction, air resistance

• Work Done: $W_{nc} = \Delta E_{mech}$

• Properties: Energy is not fully recoverable; mechanical energy is not conserved.

Momentum and Collisions

Linear Momentum

Momentum is a measure of the motion of an object and is the product of its mass and velocity.

• Formula: $\vec{p} = m\vec{v}$

• Units: kg·m/s

• Conservation: In an isolated system, total momentum is conserved.

Impulse

Impulse is the change in momentum resulting from a force applied over a time interval.

• Formula: $\vec{J} = \Delta \vec{p} = \int \vec{F} dt$

• Units: N·s

• Example: A force of 5 N applied for 2 s gives $J = 10$ N·s.

Types of Collisions

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not.

• Perfectly Inelastic Collision: Objects stick together after collision.

Fluid Mechanics

Density

Density is the mass per unit volume of a substance.

• Formula: $\rho = \frac{m}{V}$

• Units: kg/m3

• Example: Water has $\rho = 1000$ kg/m3

Pressure

Pressure is the force exerted per unit area.

• Formula: $P = \frac{F}{A}$

• Units: Pascal (Pa) = N/m2

• Absolute Pressure: $P_{abs} = P_{gauge} + P_{atm}$

Buoyant Force and Archimedes' Principle

Archimedes' Principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

• Buoyant Force: $F_{B} = \rho_{fluid} V_{displaced} g$

• Application: Explains why objects float or sink in fluids.

• Specific Gravity: Ratio of the density of a substance to the density of water.

Fluid Statics

• Hydrostatic Pressure: $P = \rho g h$

• Atmospheric Pressure: Standard value is $1.013 \times 10^5$ Pa

• Gauge Pressure: Pressure measured by a gauge, excludes atmospheric pressure.

Summary Table: Key Equations and Concepts

Concept	Equation	Units
Work	$W = Fd\cos\theta$	J
Kinetic Energy	$K = \frac{1}{2}mv^2$	J
Potential Energy	$U = mgh$	J
Momentum	$p = mv$	kg·m/s
Impulse	$J = \Delta p$	N·s
Density	$\rho = \frac{m}{V}$	kg/m3
Pressure	$P = \frac{F}{A}$	Pa
Buoyant Force	$F_B = \rho V g$	N

Additional info:

• Some equations and definitions were expanded for clarity and completeness.

• Examples and applications were added to illustrate key concepts.

• Table entries were inferred and grouped for exam study purposes.