1. Units and Measurements | connecting-the-dots

Intercepted Area

The intercepted area ( $dA$ ) is the specific part of the spherical surface that is "cut out" or "blocked" by a cone originating from the center of the sphere.

Steradian

the solid angle is the ratio of the intercepted area of the spherical surface, centered at the vertex, to the square of its radius. Mathematically, it is expressed as: $d\Omega = \frac{dA}{r^2}$
$d\Omega$ : The solid angle (measured in steradians, symbol: sr).
$dA$ : The intercepted area of the spherical surface.
$r$ : The radius of the sphere.

Have units but no dimension

A quantity has units but no dimensions when it is defined as a ratio of two similar physical quantities

Physical Quantity	Formula/Definition	SI Unit	Dimensional Formula
Plane Angle ( $\theta$ )	$\text{Arc length} / \text{Radius}$	Radian (rad)	$[M^0 L^0 T^0]$
Solid Angle ( $\Omega$ )	$\text{Intercepted Area} / \text{Radius}^2$	Steradian (sr)	$[M^0 L^0 T^0]$

Significant figures

Rule Number	Description	Examples	Number of Significant Figures
1	All non-zero digits are significant.	1.625 cm	4
2	Zeros between two non-zero digits are significant, regardless of the decimal point's position.	100.05 m	5
3	Leading zeros (zeros to the left of the first non-zero digit) are not significant. They only indicate the position of the decimal.	0.007 g	1
4	Trailing zeros in a number with a decimal point are significant.	3.500 V	4
5	Trailing zeros in a number without a decimal point are generally not significant (unless they come from a measurement).	400 kg	1
6	Power of ten is irrelevant to the determination of significant figures.	$5.2 \times 10^6$	2

Luminous Intensity and Luminous Flux

Feature	Luminous Intensity	Luminous Flux
Unit	Candela (cd)	Lumen (lm)
Concept	Light "strength" in one direction.	Total light emitted in all directions.
Analogy	The brightness of a laser beam.	The total light from a lightbulb.

Distance vs Displacement

Feature	Distance	Displacement
Definition	The total path length covered by an object.	The shortest straight-line distance between the initial and final positions.
Type of Quantity	Scalar (has only magnitude).	Vector (has both magnitude and direction).
Direction	Not required; it depends on the actual path taken.	Required; it points from the starting point to the ending point.
Value	Always positive and cannot be zero if the object has moved.	Can be positive, negative, or zero.
Path Dependency	Depends on the path followed.	Independent of the path; depends only on endpoints.
Comparison	Distance $\ge$	Displacement

Linear Momentum ( $\vec{p}$ )

Linear momentum is defined as the product of the mass of an object and its velocity. It is a measure of how difficult it is to stop a moving object. $\vec{p} = m\vec{v}$

Angular Momentum ( $\vec{L}$ )

Angular momentum is the rotational equivalent of linear momentum. For a particle, it is defined as the moment of momentum relative to a specific point (usually the origin or axis of rotation).

For a particle $\vec{L} = \vec{r} \times \vec{p}$ $L = r \times p$
- Where $\vec{r}$ is the position vector and $\vec{p}$ is the linear momentum For a rigid body $L = I\omega$
- Where $I$ is the Moment of Inertia and $\omega$ is the angular velocity.

Stress and Pressure

Feature	Pressure	Stress
Origin	External force acting on a surface.	Internal restoring force developed within a body.
Nature	It is always an external effect.	It is an internal reaction to deformation.
Direction	Always acts normal (perpendicular) to the surface.	Can be normal (tensile/compressive) or tangential (shearing).
Type of Quantity	Scalar quantity.	Tensor quantity (treated as a scalar in basic NCERT contexts).
Medium	Relevant for solids, liquids, and gases.	Relevant only for solids.
Formula	$P = \frac{F_{\text{ext}}}{A}$	$\sigma = \frac{F_{\text{restoring}}}{A}$

Viscosity

1. Definition

Viscosity is the property of a fluid by virtue of which an internal force comes into play when the fluid is in motion, which opposes the relative motion between its different layers.
Just as friction opposes the motion of a solid over a surface, viscosity opposes the "shearing" or sliding of one layer of fluid over another.

2. Coefficient of Viscosity ( $\eta$ )

According to Newton's Law of Viscosity, for a regular (laminar) flow, the viscous force ( $F$ ) acting between two layers of area $A$ and velocity gradient $dv/dx$ is given by: $F = -\eta A \frac{dv}{dx}$

$\eta$ (Eta): The coefficient of viscosity.
$A$ : The contact area between layers.
$dv/dx$ : The velocity gradient (change in velocity per unit distance perpendicular to the flow).
The Negative Sign: Indicates that the viscous force acts in a direction opposite to the fluid's motion.

3. Units and Dimensions

Property	Value/Formula
SI Unit	Pascal-second (Pa s) or $\text{kg m}^{-1} \text{s}^{-1}$
Common Unit	Poise ( $1 \text{ Pa s} = 10 \text{ Poise}$ )
Dimensional Formula	$[ML^{-1}T^{-1}]$

4. Factors Affecting Viscosity

Nature of the Fluid: Honey has higher viscosity than water because its internal molecular attractions are stronger.
Temperature: * For Liquids: Viscosity decreases as temperature increases (molecules move faster and overcome cohesive forces).
- For Gases: Viscosity increases as temperature increases (more frequent molecular collisions).

Liquid	Intermolecular Attraction	Viscosity	Flow Speed (same slope)
Water	Moderate	Low	Fast
Castor Oil	High	High	Slow
Honey	Very High	Very High	Very Slow