In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of [MaLbTc]$\left[\mathrm{M}^{\mathrm{a}}\mathrm{L}^{\mathrm{b}}\mathrm{T}^{\mathrm{c}}\right]$ . If b = 3$b = 3$ , the value of c$c$ is

Related Formula

Definitions of physical quantities:

• Viscosity: F = -η A (dv)/(dx)$F = -\eta A \frac{dv}{dx}$
• Intensity: I = PowerArea$I = \frac{\text{Power}}{\text{Area}}$
• Pressure gradient: (dP)/(dx)$\frac{dP}{dx}$
• Compressibility: K = (1)/(B) = StrainStress$K = \frac{1}{B} = \frac{\text{Strain}}{\text{Stress}}$

Core Logic

Let's calculate each dimensional formula:

• Coefficient of viscosity (η$\eta$):

Matches (IV).

• Intensity of wave (I$I$):

Matches (I).

• Pressure gradient ((dP)/(dx)$\frac{dP}{dx}$):

Matches (II).

• Compressibility (K$K$):

Compressibility is the inverse of Bulk Modulus:

Matches (III).

Thus: (A)-(IV), (B)-(I), (C)-(II), (D)-(III).

Step 1: Final Conclusion

The correct option is (2).

Pattern Recognition

Target basic matching terms first: Compressibility is the reciprocal of pressure, giving [M⁻¹LT²]$[M^{-1}LT^2]$. Wave intensity has units of power per unit area, giving [MT⁻³]$[MT^{-3}]$. This immediately isolates option (2).

Chapter Mix

Class 11 Physics: Units and Measurements Class 11 Physics: Mechanical Properties of Fluids Class 11 Physics: Mechanical Properties of Solids

Related Formula

By the principle of dimensional homogeneity, terms added or subtracted must have the same dimensions:

[V] = [b]$[V] = [b]$

Core Logic

Let's find the dimensional formula of the quantities:

• Pressure P$P$:

• Volume V$V$:

Substituting these to find [a]$[a]$ and [b]$[b]$:

Now, compute the dimensions of ab⁻²$ab^{-2}$:

This matches the dimensions of pressure.

Let's evaluate the options:

• Planck's constant: [h] = M L²T⁻¹$[h] = \text{M L}^2\text{T}^{-1}$
• Compressibility: [β] = M⁻¹LT²$[\beta] = \text{M}^{-1}\text{L}\text{T}^2$
• Strain: dimensionless
• Energy density (energy per unit volume):

Step 1: Final Conclusion

Therefore, the dimension of ab⁻²$ab^{-2}$ is equivalent to that of Energy density.

Pattern Recognition

By writing the relation directly as [ab⁻²] = ([a])/([b]²)$[ab^{-2}] = \frac{[a]}{[b]^2}$, and noting [a] = [P][V]²$[a] = [P][V]^2$ and [b] = [V]$[b] = [V]$, we get [ab⁻²] = ([P][V]²)/([V]²) = [P]$[ab^{-2}] = \frac{[P][V]^2}{[V]^2} = [P]$ (Pressure). Since pressure and energy density have identical dimensions, the answer is immediately Energy density.

Chapter Mix

Class 11 Physics: Units and Measurements Class 11 Physics: Kinetic Theory of Gases

Related Formula

Formulas to find dimensional formulas:

• Boltzmann constant:

• Coefficient of viscosity:

• Planck's constant:

• Thermal conductivity:

Core Logic

Evaluate each constant individually:

Step 1: Dimensions of Boltzmann constant (kB$k_B$)

Step 2: Dimensions of Coefficient of viscosity (η$\eta$)

Step 3: Dimensions of Planck's constant (h$h$)

Step 4: Dimensions of Thermal conductivity (K$K$)

This sequence yields A-III, B-IV, C-I, D-II, matching Option (1).

Pattern Recognition

To solve matching sets efficiently, search for the most recognizable dimensions first. Planck's constant h$h$ (ML²T⁻¹$ML^2T^{-1}$) and viscosity coefficient η$\eta$ (ML⁻¹T⁻¹$ML^{-1}T^{-1}$) are highly unique and usually resolve the options instantly.

Chapter Mix

Class 11 Physics: Units and Measurements

Related Formula

For a physical quantity defined by algebraic powers C = (p^a q^b)/(r^c s^d)$C = \frac{p^a q^b}{r^c s^d}$, the maximum fractional error is calculated by summing absolute scaled fractional errors:

Expressed as percentages:

Core Logic

Given expression:

Max fractional error equation:

Step 1: Calculate the total percentage error

Substitute the individual percentage errors:

• Error in p = 1%$p = 1\%$
• Error in q = 2%$q = 2\%$
• Error in r = 3%$r = 3\%$
• Error in s = 2%$s = 2\%$

The total percentage error in C$C$ is 15%$15\%$.

Pattern Recognition

In error propagation, individual errors always combine constructively to produce the maximum possible uncertainty limit. Hence, negative powers (like division by r³$r^3$ or s1/2$s^{1/2}$) are integrated using positive coefficients during maximum absolute error summation.

Chapter Mix

Class 11 Physics: Units and Measurements

Related Formula

For a quantity Q = X^a Y^b Z^c$Q = X^a Y^b Z^c$, the maximum fractional error is:

where, (Δ X)/(X), (Δ Y)/(Y), (Δ Z)/(Z)$\frac{\Delta X}{X}, \frac{\Delta Y}{Y}, \frac{\Delta Z}{Z}$ are fractional errors of individual variables

Core Logic

Given formula: Q = X⁻² Y3/2 Z-2/5$Q = X^{-2} Y^{3/2} Z^{-2/5}$.

Identify the absolute exponents:

• |a| = |-2| = 2$|a| = |-2| = 2$
• |b| = |(3)/(2)| = (3)/(2)$|b| = \left|\frac{3}{2}\right| = \frac{3}{2}$
• |c| = |-(2)/(5)| = (2)/(5)$|c| = \left|-\frac{2}{5}\right| = \frac{2}{5}$

Now write the error expression:

Substitute the given values:

• (Δ X)/(X) = 0.1$\frac{\Delta X}{X} = 0.1$
• (Δ Y)/(Y) = 0.2$\frac{\Delta Y}{Y} = 0.2$
• (Δ Z)/(Z) = 0.5$\frac{\Delta Z}{Z} = 0.5$

Step 1: Compute Maximum Fractional Error

Calculate term by term:

Pattern Recognition

Sees: Exponential algebraic relation for errors. Trap: Exponents are negative, but maximum error is cumulative. Always take the absolute value of exponents when summing errors! ✓

Chapter Mix

Class 11 Physics: Units and Measurements