Practice Questions

List the seven fundamental quantities in the SI system.

Critique the expression $E = mc^2 + v$, where $E$ is energy, $m$ is mass, $c$ is the speed of light, and $v$ is velocity, based on the principle of homogeneity of dimensions.

Define a base unit in the context of a system of units.

Recall the SI unit and symbol for luminous intensity.

Define the term 'dimensional formula'.

A rectangular metal sheet has a length of $4.35$ m and a width of $1.25$ m. Calculate the area of the sheet and express the result with the correct number of significant figures.

Analyze the number $0.004500 \text{ kg}$ and state the number of significant figures it contains. Justify your answer based on the rules for significant figures.

Evaluate the statement: 'Changing the system of units from SI to CGS changes the number of significant figures in a measurement.'

Examine why writing a measurement in scientific notation, such as $4.700 \times 10^3$ m instead of $4700$ m, is a better way to represent the number of significant figures.

A student reports the area of a circle with radius measured as $2.1 \text{ cm}$ to be $13.8544 \text{ cm}^2$. Justify why this reported value is scientifically inappropriate.

A student calculates the volume of a sphere. The radius is measured as $2.51 \text{ cm}$. The student reports the volume as $66.2356 \text{ cm}^3$. Justify the correct number of significant figures the final answer should have and provide the correctly rounded-off value.

Examine the following set of formulas for kinetic energy ($K$). Apply dimensional analysis to identify which ones are dimensionally incorrect. (Here, $m$ is mass, $v$ is velocity, and $a$ is acceleration). (a) $K = m^2v$ (b) $K = ma$ (c) $K = \frac{1}{2}mv^2 + ma$

A student measures three masses as $125.4$ g, $23.68$ g, and $1.2$ g. Calculate the total mass and report the result to the correct number of decimal places.

Create a new physical quantity called 'Energy Flux' defined as the rate of energy flow per unit area. Formulate its dimensional formula and propose a physical scenario where this quantity would be relevant.

The speed of light in a vacuum is approximately $3.00 \times 10^8 \text{ m/s}$. Apply this value to calculate the distance light travels in one nanosecond ($1 \text{ ns} = 10^{-9} \text{ s}$), expressing the answer in centimeters.

A car accelerates from rest, covering a distance of $120$ m in $5.0$ s. Calculate the acceleration of the car, assuming it is constant. Use the equation of motion $s = ut + \frac{1}{2}at^2$ and express your answer with the correct number of significant figures.

Calculate the volume of a sphere with a radius of $3.20$ cm, and express the answer with the correct number of significant figures. The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$. Use $\pi = 3.14159$.

Compare the numbers $4.700$ m and $4700$ m. Analyze how the number of significant figures differs between them and explain what this difference implies about the precision of the measurements.

Analyze the statement: 'Dimensional analysis can prove that an equation is physically correct.' Is this statement true or false? Justify your answer with an example.

Evaluate the dimensional consistency of the equation for the period of a torsional pendulum, $T = 2\pi \sqrt{\frac{I}{C}}$. The term $I$ is the moment of inertia with dimensions $[\text{ML}^2]$, and $T$ is the time period. Propose the dimensional formula for the torsional constant, $C$.

Justify the statement: 'A dimensionally correct equation is not necessarily a physically correct equation, but a dimensionally incorrect equation must be wrong.' Provide a unique example not found in the source text to support your justification.

Name the physical quantity represented by the dimensional formula $[\text{M L T}^{-2}]$.

State the convention for rounding off a number if the insignificant digit to be dropped is 5 and the preceding digit is even.

The pressure $P$ of an ideal gas is found to depend on its density $\rho$ and the root mean square velocity $v$ of its molecules. Formulate a possible relation between $P$, $\rho$, and $v$ using the method of dimensional analysis.

Two lengths are measured as $L_1 = 100.0 \pm 0.1 \text{ cm}$ and $L_2 = 1.00 \pm 0.01 \text{ cm}$. Evaluate which measurement has greater precision and justify your reasoning based on relative error.

The viscous force $F$ on a spherical object moving through a fluid depends on the coefficient of viscosity $\eta$, the radius of the sphere $r$, and its velocity $v$. Formulate the relationship between these quantities using dimensional analysis, assuming the relation is of the form $F = k \eta^a r^b v^c$.

A physical quantity $X$ is defined as $X = \frac{A^2 B^{1/2}}{C^3 D}$. The percentage errors in the measurement of $A, B, C,$ and $D$ are $1\%$, $2\%$, $3\%$, and $4\%$ respectively. Propose a formula to calculate the maximum percentage error in $X$ and calculate its value.

Identify the number of significant figures in the following measurements and explain the rule used for each: (a) $0.050 \text{ cm}$ (b) $2.008 \text{ kg}$ (c) $5.30 \times 10^4 \text{ J}$

Explain the principle of homogeneity of dimensions and describe its primary application in physics.

Describe the CGS, MKS, and FPS systems of units by listing the base units for length, mass, and time in each. Explain why the SI system is considered superior.

Explain the difference between fundamental units and derived units, providing one example for each.

A measurement of length is $15.42 \text{ cm}$. Another measurement is $3.1 \text{ cm}$. Recall the rule for addition with significant figures and state the sum of these two lengths.

Solve the following expression and present the answer with the correct number of significant figures: $\frac{2.51 \times 10^4 \times 3.2}{0.080}$.

Examine the equation for pressure, $P = h \rho g$, where $h$ is the height of a fluid column, $\rho$ is its density, and $g$ is the acceleration due to gravity. Apply dimensional analysis to demonstrate that this equation is dimensionally consistent.

Propose a method to determine the number of significant figures in the value of $\pi$ required for calculating the circumference of the Earth ($C = 2\pi R$) to a precision of $1$ meter, given the Earth's radius $R = 6.4 \times 10^6 \text{ m}$.

Design a hypothetical system of fundamental units where the speed of light $c$, the gravitational constant $G$, and Planck's constant $h$ are chosen as the base units. Formulate the dimensions of length, mass, and time in terms of $[c]$, $[G]$, and $[h]$.

Design an experiment to determine the density of an irregularly shaped, non-porous solid that is denser than water. Formulate an expression for the maximum possible relative error in the density measurement based on the instruments you choose.

The escape velocity ($v_e$) from a planet's surface is given by the formula $v_e = \sqrt{\frac{2GM}{R}}$, where $G$ is the gravitational constant, $M$ is the mass of the planet, and $R$ is its radius. Apply dimensional analysis to verify the consistency of this formula.

Explain why a dimensionally correct equation is not necessarily a physically correct equation. Provide a simple example.

The time period ($T$) of a simple pendulum is thought to depend on its length ($l$) and the acceleration due to gravity ($g$). Apply the method of dimensions to deduce the relationship between $T$, $l$, and $g$.

A physical quantity $P$ is given by the relation $P = \frac{a^3 b^2}{\sqrt{c}d}$. If the percentage errors in the measurement of $a, b, c,$ and $d$ are $1\%, 2\%, 4\%,$ and $2\%$ respectively, calculate the total percentage error in $P$.

A student suggests that the formula for the period of a simple pendulum is $T = 2\pi \frac{l}{g}$. The dimensions of time period $T$ are $[\text{T}]$, length $l$ are $[\text{L}]$, and acceleration due to gravity $g$ are $[\text{L T}^{-2}]$. Explain using dimensional analysis why this formula is incorrect.

Describe how the modern SI units for mass (kilogram) and time (second) are defined. Recall the names of the fundamental constants used in their definitions.

Summarize the rules for performing arithmetic operations with significant figures. Provide one example for multiplication/division and one for addition/subtraction.

A student proposes a new system of fundamental units where the unit of force is 'Newtonia' ($F$), the unit of velocity is 'Velocia' ($V$), and the unit of time is 'Timon' ($T$). Critique this choice of base quantities and formulate the dimensional formula for mass in this new system.