I will double-check each derivation before writing the question.The time period (T) of oscillations of a simple pendulum is measured as T = 2.00 s, and the length (L) of the pendulum is 1.00 m. The maximum percentage error in the measurement of T is 1% and in L is 2%. The acceleration due to gravity (g) is given by g = (4π² L) / T². What is the maximum percentage error in the measurement of g?

The dimensional formula for gravitational potential is:

The speed of light in vacuum is approximately 3.00 x 10⁸ m/s. The number of significant figures in this value is:

Which of the following quantities is a fundamental quantity in the SI system?

If the error in measuring the length of a side of a cube is 1%, then the percentage error in the measurement of the volume of the cube is:

Which of the following physical quantities has dimensions different from the rest?

The least count of a screw gauge is 0.01 mm, and there are 50 divisions on its circular scale. The pitch of the screw gauge is:

If the value of a physical quantity in the new system of units is related to its value in the old system by X_new = k X_old, then k would be 1/2. The unit itself becomes 2 times larger. Let's make sure the question and option are perfectly aligned. Let's rephrase the question to avoid ambiguity. Question 45 (Revised): If the unit of length is doubled and the unit of mass is halved, while the unit of time remains the same, how does the *magnitude* of power (numerical value) for a given physical quantity change?

If the unit of length is doubled and the unit of mass is halved, while the unit of time remains the same, how does the unit of power change?

Which of the following quantities has the same dimensions as that of pressure?

The dimensional formula for specific resistance (resistivity) is:

A student performs an experiment to measure the period of oscillation of a simple pendulum. The lengths of the pendulum are 100.0 cm, 90.0 cm, 80.0 cm, and 70.0 cm. The corresponding periods of oscillation are 2.00 s, 1.90 s, 1.79 s, and 1.69 s. Which measurement has the best precision?

If the value of atmospheric pressure is 10⁵ Pa, what is its value in CGS units?

The dimensional formula for electric field strength (E) is:

The density of a cube is measured by measuring its mass and the length of its side. If the maximum error in the measurement of mass is 4% and in the measurement of length is 3%, what is the maximum percentage error in the measurement of density?

Which of the following physical quantities has dimensions [M⁰ L⁰ T^-1]?

If the unit of mass is 1 kg, unit of length is 1 m, unit of time is 1 s, and unit of temperature is 1 K, then the dimensional formula for Boltzmann constant (k) is:

What is the number of significant figures in the calculation of 3.4 x 10³ + 2.5 x 10²?

The maximum percentage error in the measurement of the area of a circle, if the error in measuring its radius is 2%, is:

If a quantity is given by X = (F * L) / (v²), where F is force, L is length, and v is velocity, the dimensions of X are:

Which of the following measurements is the most precise?

The dimensional formula for universal gas constant (R) is:

If voltage (V) and current (I) are measured with percentage errors of 2% and 1% respectively, then the percentage error in power (P = VI) is:

The dimensional formula for specific heat capacity is:

If the unit of force is 1 kN, the unit of length is 1 km and the unit of time is 1 min, what is the unit of mass?

Which of the following pairs of physical quantities has different dimensions?

The number of significant figures in the measurement 5.000 x 10⁴ kg is:

If a physical quantity P is given by P = (x² y³) / (z sqrt(w)), where x, y, z, w are measured with percentage errors 1%, 2%, 3%, and 4% respectively, then the percentage error in P is:

The dimensional formula for the coefficient of viscosity (η) is:

Which of the following is a dimensionless quantity?

The maximum percentage error in the measurement of Young's modulus Y is given by the formula Y = (F L) / (A ΔL). If percentage errors in F, L, A, and ΔL are 1%, 2%, 3%, and 4% respectively, what is the maximum percentage error in Y?

Given a = 12.3 cm, b = 2.15 cm. Calculate (a+b) with appropriate significant figures.

The dimensional formula for surface tension is:

If velocity (v), force (F), and energy (E) are chosen as fundamental units, what are the dimensions of mass?

Which of the following is not a unit of time?

A student measures the diameter of a wire using a screw gauge. The reading on the main scale is 2 mm, and the 45th division on the circular scale coincides with the main scale line. The least count of the screw gauge is 0.01 mm. The diameter of the wire is:

The physical quantity which has the same dimensions as Planck's constant is:

Question 30 (Revised): If the universal gravitational constant G, Planck's constant h, and the speed of light c are taken as fundamental units, what is the dimension of Length in this new system?

If Planck's constant (h) and the speed of light in vacuum (c) are taken as fundamental units, then the dimension of time (T) is:

A physical quantity X has dimensions [M L^-1 T^-2]. Which of the following can X represent?

The precision of a measurement is determined by:

Which of the following physical quantities is NOT correctly matched with its SI unit?

Question 26 (Revised): If the fundamental units are chosen as Force (F), Velocity (V), and Time (T), then the dimensional formula for Length (L) is:

A physical quantity has dimensions of [M^a L^b T^c]. If its fundamental units are expressed in terms of energy (E), velocity (V), and force (F), then:

The least count of a screw gauge is 0.01 mm, and there are 100 divisions on its circular scale. The pitch of the screw gauge is:

If L and R denote inductance and resistance respectively, the dimensional formula for L/R is:

The equation P + a/V² = bT/V represents a modified equation of state. If P is pressure, V is volume, T is temperature, then the dimensions of 'a' are:

Which of the following operations will result in a number with the least number of significant figures, assuming all numbers are measurements?

The dimensional formula for Young's modulus (Y) is the same as that of:

In a vernier caliper, 10 divisions on the vernier scale (VSD) coincide with 9 divisions on the main scale (MSD). If 1 MSD = 1 mm, then the least count of the vernier caliper is:

The dimensional formula for electrical resistance (R) is:

If two measurements are (10.5 ± 0.2) m and (2.5 ± 0.1) m, the sum of the measurements with proper error limits is:

The dimensional formula for permittivity of free space (ε₀) is:

Which of the following units is not a fundamental unit in SI system?

A physical quantity X is represented by the formula X = [M^a L^b T^c]. If the percentage error in the measurement of M, L, and T are α, β, and γ respectively, then the maximum percentage error in X is:

The equation of a wave is given by y = A sin(kx - ωt). The dimensions of (ω/k) are:

If energy (E), density (d), and power (P) are fundamental units, what is the dimension of time?

What is the number of significant figures in the sum of 2.65 x 10³ and 2.1 x 10²?

The dimensional formula for magnetic flux is:

The radius of a sphere is measured as (2.1 ± 0.05) cm. The percentage error in its volume is:

The number of significant figures in 0.00500 is:

If E, M, J, G, respectively, denote energy, mass, angular momentum, and gravitational constant, then the quantity E J² / (M⁵ G²) has the dimensions of:

The maximum percentage error in the measurement of length is 2% and that in mass is 3%. What is the maximum percentage error in the measurement of density if the body is a cube?

Which of the following physical quantities has dimensions different from the other three?

In an experiment, the values of current and voltage are I = 2.0 A ± 0.1 A and V = 10.0 V ± 0.2 V. The percentage error in the calculation of resistance R = V/I is:

If the density of a material is 8.4 g/cm³, its value in a system of units where the unit of length is 10 cm and the unit of mass is 100 g will be:

The equation for the velocity of a particle is given by v = at + bt² + c/t, where v is velocity, t is time, and a, b, c are constants. The dimensions of a, b, c respectively are:

Consider a quantity P = (E² h) / (c³ G), where E is energy, h is Planck's constant, c is speed of light, and G is gravitational constant. The dimensions of P are:

Which of the following physical quantities is dimensionless?

A physical quantity X is given by X = F / (v * A), where F is force, v is velocity, and A is area. The dimensions of X are:

Two quantities A and B have different dimensions. Which of the following mathematical operations is dimensionally incorrect?

If the velocity of light (c), the gravitational constant (G), and Planck's constant (h) are chosen as fundamental units, then the dimensions of mass in this new system are:

A physical quantity P is related to four observables a, b, c and d as P = a³ b² / (sqrt(c) d). The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2% respectively. What is the percentage error in the quantity P?

Which of the following pairs has different dimensions?

The length of a cylinder is measured with a vernier caliper as 3.25 cm. Its diameter is measured with a screw gauge as 1.48 mm. The volume of the cylinder in appropriate significant figures is:

If force (F), acceleration (A), and time (T) are taken as fundamental units, then the dimensions of energy will be:

Let's try another expression for Q8. A physical quantity X is given by X = F / (v * A), where F is force, v is velocity, and A is area. The dimensions of X are:

Let's try a different expression. Question 8 (Revised): A physical quantity X is given by X = (F * v) / A, where F is force, v is velocity, and A is area. The dimensions of X are:

Let's re-do Question 8 and make sure it is correct and has a good answer. A physical quantity X is given by X = (F * v²) / P, where F is force, v is velocity, and P is power. The dimensions of X are:

Let's try another one. A quantity X is given by (E² h) / (c³ G), where E is energy, h is Planck's constant, c is speed of light, and G is gravitational constant. The dimensions of X are:

A physical quantity X is given by X = (F v²) / L, where F is force, v is velocity, and L is angular momentum. The dimensions of X are:

If a unit of length is chosen such that the speed of light in vacuum is unity, and a unit of time is chosen such that the period of oscillation of a certain atomic clock is unity, what is the dimension of mass in this system if the unit of energy is chosen such that Planck's constant is unity?

The time period (T) of oscillations of a simple pendulum is measured as T = 2.00 s, and the length (L) of the pendulum is 1.00 m. The maximum percentage error in the measurement of T is 1% and in L is 2%. The acceleration due to gravity (g) is given by g = (4π² L) / T². What is the maximum percentage error in the measurement of g?

If the unit of energy is E, the unit of force is F, and the unit of time is T. What will be the unit of mass in this new system?

The equation of a wave is given by y = A sin(omega t - kx). The ratio of (omega/k) represents:

The dimensions of (L/R), where L is inductance and R is resistance, are:

Which of the following quantities is dimensionless but has a unit?

A student measures the time period of 100 oscillations of a simple pendulum with a stopwatch having a least count of 1 s. The total time recorded is 200 s. The maximum possible percentage error in the measurement of the time period is:

A light-year is a unit of:

If energy (E), velocity (V), and time (T) are chosen as the fundamental quantities, the dimensional formula for surface tension will be:

The length and width of a metal sheet are 3.124 m and 3.002 m respectively. The area of the sheet to the correct number of significant figures is:

The dimensions of the coefficient of thermal conductivity (k) are (from Q = -k A dT/dx):

Which of the following equations is dimensionally incorrect? (v = velocity, u = initial velocity, t = time, a = acceleration, s = displacement)

A physical quantity P is related to four observables a, b, c and d as P = (a³ b²) / (sqrt(c) d). The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2% respectively. What is the total percentage error in P?

The dimensions of the constant 'a' in the equation P = (a - t²) / bx, where P is power, x is distance, and t is time, are:

In the equation F = Ax + Bx², where F is force and x is distance, the dimensions of A/B are:

The quantity having dimensions [M L² T^-3 A^-2] is:

The length of a side of a cube is measured as (2.0 +/- 0.02) cm. The mass of the cube is measured as (20.0 +/- 0.4) g. What is the percentage error in the density of the cube?

The dimensions of (h c / G), where h is Planck's constant, c is the speed of light, and G is the universal gravitational constant, are the same as:

Assertion: A measurement that is more accurate is always more precise. Reason: Accuracy is the closeness of a measurement to the true value, while precision is the resolution of the measurement.

The escape velocity (v_e) from a planet depends on its mass (M), radius (R), and the gravitational constant (G). Using dimensional analysis, the correct expression for escape velocity is:

If speed (v), acceleration (a), and force (F) are chosen as fundamental units, the dimension of Young's modulus will be:

The dimensions of angular momentum are:

In Van der Waals equation (P + a/V²)(V - b) = RT, the constant 'b' represents:

Two masses are measured as 10.2 g and 10.23 g. The sum of the masses should be reported to the correct number of significant figures as:

What are the dimensions of (mu_0 epsilon_0)^(-1/2), where mu_0 is the permeability of free space and epsilon_0 is the permittivity of free space?

The equation for the amount of heat Q generated in a circuit is given by Q = I²Rt. If I represents current, R is resistance, and t is time, which of the following expressions is dimensionally consistent with Q?

The length of a cylinder is measured as (4.50 +/- 0.01) cm. Its diameter is measured as (2.00 +/- 0.01) cm. The absolute error in the calculated volume of the cylinder is approximately:

The dimensions of specific resistance (resistivity) are:

Assertion: The precision of a measurement depends on the least count of the measuring instrument. Reason: A more precise instrument will have a smaller least count.

Which of the following pairs of physical quantities have different dimensions?

A quantity P is given by P = (A³ B²) / (sqrt(C) D). If the percentage errors in A, B, C, and D are 1%, 2%, 6%, and 4% respectively, the maximum percentage error in P is:

The physical quantity which has the dimensions of (Energy / Mass) is:

In CGS system, the value of the Stefan-Boltzmann constant (sigma) is sigma_cgs. If the new unit of length is x cm, mass is y g, and time is z s, then in the new system, the value of sigma will be:

The dimensions of energy density (energy per unit volume) are the same as:

Which of the following combinations of dimensions represents Magnetic Permeability (mu_0)?

The dimensions of the coefficient of viscosity (eta) in the formula F = eta A (dv/dx), where F is force, A is area, and dv/dx is velocity gradient, are:

The resistance R is given by R = V/I, where V = (100 +/- 5) V and I = (10 +/- 0.2) A. The percentage error in R is:

The dimensions of specific heat capacity are:

Consider the equation y = A sin(kx - omega t), where y is displacement, A is amplitude, x is distance, t is time. Which of the following statements is dimensionally incorrect?

If the unit of force is 100 N, the unit of length is 10 m, and the unit of time is 100 s, then the unit of mass in this new system will be:

If the universal gravitational constant (G), speed of light (c), and Planck's constant (h) are chosen as fundamental units, what is the dimension of mass in this new system?

The dimensions of pressure gradient (dP/dx) are:

The length (L) and breadth (B) of a rectangular sheet are (10.0 +/- 0.1) cm and (5.0 +/- 0.1) cm respectively. The percentage error in the area of the sheet is:

What are the dimensions of (epsilon_0 * h * c), where epsilon_0 is the permittivity of free space, h is Planck's constant, and c is the speed of light?

A physical quantity X is given by X = (Fv²)/(WL²), where F is force, v is velocity, W is work, and L is length. The dimensions of X are:

Assertion: The value of universal gravitational constant G is different in different systems of units. Reason: Units of G in SI are N m² kg^-2, and in CGS are dyne cm² g^-2.

The quantity (1/RC), where R is resistance and C is capacitance, has the dimension of:

A new system of units is proposed in which unit of mass is 'a' kg, unit of length is 'b' m, and unit of time is 'c' s. What will be the value of 1 Joule in this new system?

In the equation (P + a/V²)(V - b) = RT, where P is pressure, V is volume, T is temperature, and R is the gas constant, the dimensions of 'a' are:

If the error in the measurement of radius R of a sphere is 2% and the error in its mass M is 3%, what is the maximum percentage error in the density of the sphere?

The dimension of the ratio (h/G), where h is Planck's constant and G is the universal gravitational constant, is:

Corrected Answer and Explanation for Q50:

The period of oscillation of a simple pendulum is T = 2pi sqrt(L/g). The measured length L = (20.0 +- 0.1) cm and time for 100 oscillations is (90.0 +- 1.0) s. The maximum percentage error in the measurement of g is:

Which of the following is a dimensionless quantity that also has a unit?

The fundamental unit of luminous intensity in SI system is:

If 'x' is distance, 't' is time, and 'a' is acceleration, the expression x = at + a²t³ is dimensionally incorrect. Which term needs correction to make it dimensionally correct for x?

The least count of a screw gauge is 0.01 mm. The diameter of a wire measured is 2.24 mm. The maximum percentage error in the measurement is:

Which of the following fundamental physical constants has the dimension of [ML²T^-1]?

If a physical quantity 'Q' is measured as Q = (10.0 +- 0.2) unit, and another quantity 'R' is measured as R = (5.0 +- 0.1) unit. What is the value of Q/R with error limits?

The velocity of sound in a medium depends on the modulus of elasticity E and density rho. The relationship can be expressed as:

Corrected Q52 (renumbering to Q51 to fit 50 questions): A physical quantity S is given by S = (F L²) / (M T²), where F is force, L is length, M is mass, and T is time. The dimensions of S are:

A physical quantity S is given by S = (F L²) / (M T²), where F is force, L is length, M is mass, and T is time. The dimensions of S are:

The side of a cube is measured to be 2.2 cm. The volume of the cube, reported with the correct number of significant figures, is:

An equation is dimensionally correct if:

Which of the following statements correctly defines a femtometer?

The heat generated in a current-carrying wire is given by H = I²Rt, where I is current, R is resistance, and t is time. If the fundamental units are Mass (M), Length (L), Time (T), and Electric current (I), the dimensions of resistance R are:

If the error in the measurement of radius of a circle is 1%, then the error in the calculation of its circumference is:

In the formula y = A sin(omega t - kx), where y is displacement, t is time, x is distance. The dimensions of omega/k are:

What is the SI unit of electric potential (voltage)?

Dimensional analysis cannot determine:

Which of the following describes the closeness of a measurement to the true value?

Given that the electric field E is proportional to (1/epsilon_0) (Q/r²), where epsilon_0 is permittivity of free space, Q is charge, and r is distance. The dimensions of epsilon_0 are:

Which pair of physical quantities has the same dimensions?

In Van der Waals equation (P + a/V²)(V - b) = RT, if the unit of pressure is 10⁵ Pa and volume is 10^-3 m³, then the unit of 'a' in terms of SI units (kg, m, s) would be:

Convert 1 picofarad (pF) into microfarad (uF).

A student records 5 readings for the diameter of a wire: 2.01 mm, 2.03 mm, 2.00 mm, 2.02 mm, 2.04 mm. The average diameter should be reported as:

The dimensions of coefficient of viscosity (eta) are:

Which of the following statements is true about a dimensionless quantity?

Two resistances R1 = (100 +- 3) ohm and R2 = (150 +- 2) ohm are connected in series. What is the equivalent resistance with error limits?

If the critical velocity (v_c) of a liquid flowing through a tube is expressed as v_c = (eta^x r^y rho^z) where eta is the coefficient of viscosity, r is the radius of the tube, and rho is the density of the liquid, then the values of x, y, and z are:

The unit of impulse is:

The ratio of the dimensions of Planck's constant (h) to the moment of inertia (I) is equivalent to the dimensions of:

Subtract 15.6 cm from 20.35 cm and express the result with the correct number of significant figures.

If force (F), length (L) and time (T) are fundamental units, what is the dimensional formula for mass?

The universal gas constant R has the same dimensions as:

Assertion (A): Dimensional analysis cannot be used to determine the value of dimensionless constants. Reason (R): Dimensionless constants have no units and thus do not carry any information about fundamental dimensions.

In the equation, P = P0 exp(-alpha t²), where P is pressure, t is time, and P0 is initial pressure. The dimensions of alpha are:

The period of oscillation of a simple pendulum is T = 2pi sqrt(L/g). If L is measured with an error of 2% and g with an error of 4%, what is the maximum percentage error in T?

Which of the following is a derived unit in the SI system?

How many astronomical units (AU) are there in one parsec? (Given: 1 AU = 1.496 x 10¹1 m, 1 parsec = 3.086 x 10¹6 m)

The mass of an object is 4.237 g and its volume is 2.5 cm³. The density of the object, to the correct number of significant figures, is:

The velocity v of waves in a string depends on the tension T and linear mass density mu (mass per unit length). The relation can be expressed as:

In the equation F = Ax + Bx², where F is force and x is distance, the dimensions of A and B are respectively:

If the speed of light (c), acceleration due to gravity (g), and pressure (P) are taken as fundamental units, then the dimensions of a quantity having the same dimensions as the unit of length would be:

Which of the following equations is dimensionally incorrect? (Where v=velocity, u=initial velocity, a=acceleration, t=time, s=displacement)

A student measures the length of a rod using a meter scale with least count 1 mm. The length is recorded as 4.35 m. The maximum percentage error in the measurement is:

Which of the following units is NOT a base SI unit?

If the time period (T) of oscillation of a simple pendulum depends on its mass (m), length (l), and acceleration due to gravity (g), then the expression for T can be derived as (k is a dimensionless constant):

A sphere has a measured radius R = (2.00 +- 0.01) cm. The percentage error in its volume is:

If the velocity v of a particle depends on time t as v = At² + Bt + C, where v is in m/s and t in seconds, what are the units of A, B, and C respectively?

In a new system of units, the unit of mass is taken as 'm' kg, the unit of length 'l' meter, and the unit of time 't' second. What is the value of 1 Joule in this new system?

Add 3.45 g, 20.0 g, and 0.534 g, and express the sum with the correct number of significant figures.

If the gravitational force F, velocity v, and time T are taken as the fundamental units, what are the dimensions of mass?

The equation of motion of a particle is given by x = at + bt², where x is displacement and t is time. The dimensions of 'b' are:

Which of the following statements about dimensional analysis is INCORRECT?

The length and breadth of a rectangle are (5.7 +- 0.1) cm and (3.4 +- 0.2) cm respectively. Calculate the area of the rectangle with proper error limits.

If energy (E), velocity (V), and time (T) are chosen as the fundamental quantities, the dimensional formula for surface tension would be:

The equation of state for a real gas is given by (P + a/V²)(V - b) = RT, where P is pressure, V is volume, T is temperature, and R is the gas constant. The dimensions of the constant 'a' are:

A physical quantity P is related to four observables a, b, c and d as P = (a³ b²) / (sqrt(c) d). The percentage errors in the measurements of a, b, c and d are 1%, 3%, 4% and 2% respectively. What is the percentage error in the quantity P?

If the velocity of light (c), Planck's constant (h), and Gravitational constant (G) are chosen as fundamental units, then the dimensions of length in this new system would be: