Physics Chapter 1 One Shot - Class 11 Haryana Board Physics

Units and Measurement

Units and Measurement form the foundational basis of physics, enabling the quantification and comparison of physical quantities. Understanding the types of quantities, their units, and the relationships between them is essential for solving problems and analyzing physical phenomena. This topic covers the definitions, systems, and applications of units, as well as the use of dimensional analysis for checking equations, deriving relations, and converting units.

Physical Quantities and Their Measurement

A physical quantity is any quantity that can be measured using an instrument. Only those quantities which can be measured are considered physical quantities; emotions or…

Units and Measurement

Units and Measurement form the foundational basis of physics, enabling the quantification and comparison of physical quantities. Understanding the types of quantities, their units, and the relationships between them is essential for solving problems and analyzing physical phenomena. This topic covers the definitions, systems, and applications of units, as well as the use of dimensional analysis for checking equations, deriving relations, and converting units.

Physical Quantities and Their Measurement

A physical quantity is any quantity that can be measured using an instrument. Only those quantities which can be measured are considered physical quantities; emotions or feelings, for example, are not measurable and thus not physical quantities. To measure any physical quantity, two things are required: a numerical value and a unit.

• Physical Quantity: Any quantity that can be measured by an instrument.
• Measurement Expression: Physical quantity $= n \times u$, where $n$ is the numerical value and $u$ is the unit.
• Both numerical value and unit must be specified for a meaningful measurement (e.g., height = 180 cm).

Relationship Between Numerical Value and Unit

The numerical value and unit of a physical quantity are inversely related when the quantity is kept constant. If the unit is larger, the numerical value is smaller, and vice versa. This relationship is represented graphically as a rectangular hyperbola.

• Key Relation: For a constant physical quantity, $n \propto \frac{1}{u}$.
• Graph: The graph between $n$ and $u$ is a rectangular hyperbola.
• Example: $1$ meter $= 100$ centimeters; if the unit changes from meter to centimeter, the numerical value increases.

Systems of Units

There are three main systems of units, each with its own base units for length, mass, and time. These systems are used in different regions and contexts.

• MKS (Meter-Kilogram-Second): Also called the French system.
  • Length: meter (m)
  • Mass: kilogram (kg)
  • Time: second (s)
• CGS (Centimeter-Gram-Second): Also called the Gaussian system.
  • Length: centimeter (cm)
  • Mass: gram (g)
  • Time: second (s)
• FPS (Foot-Pound-Second): Also called the British system.
  • Length: foot (ft)
  • Mass: pound (lb)
  • Time: second (s)

SI System and Types of Physical Quantities

In 1971, the International System of Units (SI) was established, defining seven fundamental quantities and two supplementary quantities. Physical quantities are classified as fundamental, derived, or supplementary.

• Fundamental Quantities: Do not depend on other quantities.
  • Mass (kg), Length (m), Time (s), Temperature (K), Electric Current (A), Amount of Substance (mol), Luminous Intensity (cd)
• Derived Quantities: Depend on fundamental quantities (e.g., area, volume, velocity, density, force, work, pressure).
• Supplementary Quantities: Dimensionless quantities with units but no dimensions.
  • Plane angle (radian), Solid angle (steradian)

Dimensions and Dimensional Formula

The dimension of a physical quantity is the power to which the fundamental units are raised to represent it. Dimensional formulas are written using the first letter of the English name of the fundamental quantity, raised to the appropriate power.

• Dimension: Power of the unit of a fundamental quantity in the expression of a physical quantity.
• Dimensional Formula: Expressed as $M^a L^b T^c \ldots$
  • Mass: $M$
  • Length: $L$
  • Time: $T$
  • Temperature: $K$ or $\Theta$
  • Electric Current: $A$
  • Amount of Substance: $mol$
  • Luminous Intensity: $cd$
• Example: Area = length × breadth; dimensional formula = $L^2$.

Dimensional Formulas of Common Derived Quantities

Derived quantities are expressed in terms of the dimensions of fundamental quantities. The transcript provides explicit derivations for several key quantities.

• Area: $L^2$
• Volume: $L^3$
• Velocity/Speed: $LT^{-1}$
• Density: $ML^{-3}$
• Acceleration: $LT^{-2}$
• Force: $MLT^{-2}$
• Work/Energy: $ML^2T^{-2}$
• Pressure: $ML^{-1}T^{-2}$
• Linear Momentum: $MLT^{-1}$
• Planck’s Constant ($h$): $ML^2T^{-1}$
• Torque: $ML^2T^{-2}$
• Gravitational Constant ($G$): $M^{-1}L^3T^{-2}$

Supplementary and Dimensionless Quantities

Supplementary quantities are those with units but no dimensions. Trigonometric, exponential, and logarithmic functions are also dimensionless.

• Plane Angle: Unit is radian (rad), dimensionless.
• Solid Angle: Unit is steradian (sr), dimensionless.
• Trigonometric Functions: $\sin\theta$, $\cos\theta$, $\tan\theta$ are dimensionless.
• Exponential and Logarithmic Functions: $e^x$, $\log x$ are dimensionless.

Uses of Dimensional Analysis

Dimensional analysis is a powerful tool for checking the correctness of equations, deriving relations among physical quantities, and converting units from one system to another.

• Checking Correctness of Equations: Compare dimensions on both sides of an equation.
  • If dimensions match, the equation is dimensionally correct.
  • Example: For $v = 2gh$, dimensions do not match; for $v = \sqrt{2gh}$, dimensions match.
• Deriving Relations: Assume the required quantity depends on other quantities, assign powers, and equate dimensions to solve for powers.
  • Example: Time period of a simple pendulum $t \propto m^a l^b g^c$; solve for $a$, $b$, $c$ using dimensional comparison.
• Converting Units: Use the relation $n_1u_1 = n_2u_2$ for the same physical quantity.
  • For conversion, $n_2 = n_1 \left(\frac{u_1}{u_2}\right)^{\text{dimension}}$.
  • Example: $980$ cm/s$^2$ (CGS) $= 9.8$ m/s$^2$ (MKS).

Principle of Homogeneity

The principle of homogeneity states that only quantities with the same dimensions can be added, subtracted, or equated. This principle is used to check the validity of equations and to find the dimensional formula of unknown constants.

• If $v = a + bt^2 + ct$, then the dimensions of $a$, $bt^2$, and $ct$ must all be equal to the dimensions of $v$.
• For $A$ and $B$ in $V = A + BV$, if $V$ is volume and $P$ is pressure, then $A$ has the dimensions of $PV^2$ and $B$ has the dimensions of $V$.

Limitations of Dimensional Analysis

Dimensional analysis can only be used when the dependency is on three or fewer quantities. If a quantity depends on more than three variables, dimensional analysis cannot determine all the powers.

• Only applicable for up to three independent quantities.
• Cannot determine dimensionless constants.

Exam-Focused Takeaways

• Always specify both numerical value and unit when expressing a measurement.
• Know the three main systems of units and their base units.
• Memorize the seven SI fundamental quantities and their units.
• Understand the difference between fundamental, derived, and supplementary quantities.
• Be able to write and interpret dimensional formulas for common physical quantities.
• Use dimensional analysis to check equations, derive relations, and convert units.
• Apply the principle of homogeneity to validate equations and find unknown dimensions.
• Recognize the limitations of dimensional analysis in cases with more than three variables or dimensionless constants.