Question 1

A body has a measured mass of 5.00 ± 0.05 kg and a measured volume of 1.00 ± 0.05 m³. What is the maximum possible percentage error in its density?

Accepted Answer

6%. For density = mass/volume, the maximum fractional error adds: (dm/m)+(dV/V) = 0.05/5.00 + 0.05/1.00 = 0.01 + 0.05 = 0.06 = 6%.

Question 2

A substance has a density of 4 g/cm³ in the CGS system. If a new unit system is used where the unit of length is 10 cm and the unit of mass is 100 g, what is the numerical value of the density in that system?

Accepted Answer

40 unit. Density has dimensions M/L^3, so n2 = 4 * (1 g / 100 g) * (10 cm / 1 cm)^3 = 4 * (1/100) * 1000 = 40 units.

Question 3

Which of the following physical quantities has dimensions that do not match the other three?

Accepted Answer

Angular momentum. Energy density, force per unit area (pressure) and (voltage*charge)/volume = energy/volume all have dimensions [M L^-1 T^-2]. Angular momentum has dimensions [M L^2 T^-1], which does not match the other three.

Question 4

If the percentage uncertainties in measuring mass M, length L, and time T are 1%, 1.5%, and 3% respectively, what is the percentage error in a quantity having dimensions M L⁻¹ T⁻¹?

Accepted Answer

5.5%. The percentage error in a derived quantity can be calculated by summing the percentage uncertainties of its constituent measurements, weighted by their respective dimensions. In this case, the quantity M L⁻¹ T⁻¹ has a percentage error of 1% from mass, 1.5% from length (which is inverted, thus it contributes negatively), and 3% from time, resulting in a total of 5.5%.

Question 5

If E, m, J and G denote energy, mass, angular momentum and the gravitational constant, respectively, then the dimensional formula of EJ²/(m⁵G²) is identical to that of which quantity?

Accepted Answer

angle. The dimensional formula of EJ²/(m⁵G²) simplifies to that of an angle because the units of energy, angular momentum, mass, and the gravitational constant combine in such a way that they ultimately yield a dimensionless quantity, which is characteristic of angles.

Question 6

A rod was found to have a measured length of 3.50 cm. Which measuring device was used for this reading?

Accepted Answer

A vernier caliper in which 10 vernier divisions coincide with 9 main-scale divisions, and the main scale has 10 divisions in 1 cm. A reading of 3.50 cm is given to 0.01 cm (0.1 mm). For the vernier where 10 VSD = 9 MSD and 1 MSD = 1 mm, least count = 1 MSD/10 = 0.1 mm = 0.01 cm, matching the reading. The screw gauges give 0.001 cm or 0.002 cm (three decimals), and a metre rule only 0.1 cm.

Question 7

In a screw gauge, two complete rotations of the circular scale advance the spindle by 1 mm on the main scale, and the circular scale has 50 equal divisions. The instrument is known to have a zero error of -0.03 mm. When the diameter of a fine wire is measured, the main scale reading is 3 m

Accepted Answer

3.38 mm. Two rotations advance 1 mm so pitch = 0.5 mm, and with 50 divisions least count = 0.5/50 = 0.01 mm. Observed reading = 3 + 35*0.01 = 3.35 mm; correcting for zero error of -0.03 mm: 3.35 - (-0.03) = 3.38 mm.

Question 8

If a capacitor has capacitance C and the charge on its plate is Q, what is the dimensional formula of Q²/C?

Accepted Answer

[L²MT⁻²]. Energy stored in a capacitor is U = Q^2/(2C), so Q^2/C has the dimensions of energy = [M L^2 T^-2]. The correct option is [L^2 M T^-2].

Question 9

What are the numbers of significant figures in 23.023, 0.0003, and 2.1 × 10⁻³, respectively?

Accepted Answer

5, 1, 2. The number 23.023 has five significant figures because all non-zero digits and the zeros between them count. The number 0.0003 has one significant figure, as leading zeros do not count. Lastly, 2.1 × 10⁻³ has two significant figures, as only the digits in the coefficient are counted.

Question 10

What are the dimensions of mobility?

Accepted Answer

M⁻¹T²A. The correct option reflects the relationship between mass, time, and area in the context of mobility, which is defined as the ability to move freely and is typically expressed in terms of mass per unit time and area.

Question 11

Which pair of physical quantities does not share the same dimensions?

Accepted Answer

Momentum and Planck’s constant. Torque and work are both M L^2 T^-2; stress and Young's modulus are both M L^-1 T^-2; speed and (mu0 eps0)^(-1/2) = c are both L T^-1. But momentum has dimensions M L T^-1 while Planck's constant has M L^2 T^-1, so this pair does not match.

Question 12

In a measurement experiment, four quantities a, b, c and d have percentage errors of 1%, 2%, 3% and 4%, respectively. If a derived quantity is given by P = a³ b² / (c d), what is the percentage error in P?

Accepted Answer

14%. For P = a^3 b^2 /(c d), %error = 3*1 + 2*2 + 1*3 + 1*4 = 3 + 4 + 3 + 4 = 14%.

Question 13

In an experiment, the density of a solid sphere is to be found. Its diameter is measured using a screw gauge with pitch 0.5 mm and 50 divisions on the circular scale. The main scale reading is 2.5 mm and the circular scale reading is 20 divisions. If the measured mass of the sphere has a r

Accepted Answer

3.1%. Least count = 0.5/50 = 0.01 mm; diameter = 2.5 + 20*0.01 = 2.7 mm with error 0.01 mm, so dd/d = 0.01/2.7 = 0.37%. Density ~ m/d^3, so error = dm/m + 3*dd/d = 2% + 3*0.37% = 2% + 1.11% = 3.1%.

Question 14

In an experiment with a simple pendulum, the largest possible error in measuring its length is 2%, while the largest possible error in measuring its time period is 3%. What is the maximum percentage error in the calculated value of acceleration due to gravity, g?

Accepted Answer

8%. From g = 4*pi^2 L / T^2, the fractional errors add as (dg/g) = (dL/L) + 2(dT/T). So maximum % error in g = 2% + 2(3%) = 8%.

Question 15

A nanocapacitor’s capacitance is expressed in a unit ‘u’ formed using the electric charge e, Bohr radius a0, Planck’s constant h, and the speed of light c. Which relation for u is correct?

Accepted Answer

u = e² a0 / (hc). The correct option expresses capacitance in terms of fundamental constants, where the combination of charge, length, and constants correctly relates to the physical dimensions of capacitance, aligning with the principles of electromagnetism and quantum mechanics.

Question 16

An angle-measuring device is such that 29 main-scale divisions line up exactly with 30 vernier-scale divisions. If one main-scale division equals 0.5°, what is the least count of the instrument?

Accepted Answer

1 minute. The least count is determined by the difference between one main-scale division and one vernier-scale division. Since 29 main-scale divisions equal 30 vernier divisions, the value of one vernier division is 0.5°/29, which is approximately 0.01724°. The least count, being the smallest measurable angle, is then calculated as 1 main-scale division (0.5°) minus 1 vernier division (0.01724°), resulting in a least count of 1 minute.

Question 17

The physical quantities not having same dimensions are

Accepted Answer

momentum and planck’s constant. Momentum is a measure of motion and has dimensions of mass times velocity, while Planck's constant relates energy to frequency and has dimensions of action, which is mass times distance squared per time. Therefore, these two quantities do not share the same dimensional formula.

Question 18

Out of the following pair, which one does NOT have identical dimensions?

Accepted Answer

Moment of inertia and moment of a force. Moment of inertia is a measure of an object's resistance to rotational motion and has dimensions of mass times distance squared, while the moment of a force (torque) has dimensions of force times distance, which are different.

Question 19

The dimensions of magnetic field in M, L, T and C (coulomb) is given as

Accepted Answer

[MT⁻¹C⁻¹]. The magnetic field is defined in terms of force per unit charge and velocity, which leads to the dimensional formula of mass (M), time (T) to the power of -1, and charge (C) to the power of -1, resulting in [MT⁻¹C⁻¹].

Question 20

A body of mass m = 3.513 kg is moving along the x-axis with a speed of 5.00 ms⁻¹. The magnitude of its momentum is recorded as

Accepted Answer

17.6 kg ms⁻¹. The momentum of an object is calculated using the formula momentum = mass × velocity. By multiplying the mass of 3.513 kg by the speed of 5.00 m/s, the result is approximately 17.565 kg m/s, which rounds to 17.6 kg m/s, making option A the correct choice.

Question 21

Two full turns of the circular scale of a screw gauge cover a distance of 1 mm on its main scale. The total number of divisions on the circular scale is 50. Further, it is found that the screw gauge has a zero error of - 0.03 mm. While measuring the diameter of a wire, a student notes the

Accepted Answer

3.38 mm. To find the diameter of the wire, we first calculate the least count of the screw gauge, which is 0.02 mm (1 mm/50 divisions). The reading from the main scale is 3 mm, and the circular scale reading contributes 0.70 mm (35 divisions x 0.02 mm). Adding these gives 3.70 mm, and accounting for the zero error of -0.03 mm results in a final measurement of 3.38 mm.

Question 22

In an experiment the angles are required to be measured using an instrument, 29 divisions of the main scale coincide with 30 divisions of the vernier scale. If the smallest division of the main scale is half-a-degree ( = 0.5°), then the least count of the instrument is:

Accepted Answer

one minute. The least count of the instrument is determined by the formula: Least Count = Value of one main scale division - Value of one vernier scale division. Since 29 divisions of the main scale coincide with 30 divisions of the vernier scale, the value of one vernier scale division is smaller than one main scale division, leading to a least count of one minute.

Question 23

The respective number of significant figures for the numbers 23.023, 0.0003 and 2.1 × 10⁻³ are

Accepted Answer

5, 1, 2. The number 23.023 has five significant figures because all non-zero digits and the zero between them count. The number 0.0003 has only one significant figure, which is the '3', as leading zeros do not count. Lastly, 2.1 × 10⁻³ has two significant figures, which are '2' and '1', as the scientific notation indicates that these digits are significant.

Question 24

A screw gauge gives the following reading when used to measure the diameter of a wire. Main scale reading: 0 mm Circular scale reading: 52 divisions Given that 1 mm on main scale corresponds to 100 divisions of the circular scale. The diameter of wire from the above data is

Accepted Answer

0.052 cm. The diameter of the wire is calculated by combining the main scale reading and the circular scale reading. The main scale reading is 0 mm, and the circular scale reading of 52 divisions corresponds to 0.52 mm (since 1 mm equals 100 divisions, 52 divisions equal 0.52 mm). Converting this to centimeters gives 0.052 cm.

Question 25

A spectrometer gives the following reading when used to measure the angle of a prism. Main scale reading: 58.5 degree Vernier scale reading: 09 divisions Given that 1 division on main scale corresponds to 0.5 degree. Total divisions on the Vernier scale is 30 and match with 29 divisions of

Accepted Answer

58.65 degree. Least count = (1 MSD - 1 VSD) = (1 - 29/30)*0.5 deg = 0.5/30 deg per division. Angle = 58.5 + 9*(0.5/30) = 58.5 + 0.15 = 58.65 deg.

Question 26

A student recorded the length of a rod as 3.50 cm. Which measuring instrument was most likely used?

Accepted Answer

A vernier caliper in which 10 vernier divisions coincide with 9 main-scale divisions, and the main scale is marked in cm with 10 divisions per cm. A reading of 3.50 cm is precise to 0.01 cm (0.1 mm), so the instrument's least count must be 0.1 mm. For the vernier where 10 VSD = 9 MSD with 1 MSD = 1 mm, LC = MSD/10 = 0.1 mm, which exactly matches. The screw gauges (LC = 0.01 mm or 0.02 mm) would give three decimal places in cm, more precision than recorded.

Question 27

For a simple pendulum, the time period is given by T = 2π√(L/g). The length L is measured as 20.0 cm with an uncertainty of 1 mm, and the time taken for 100 oscillations is recorded as 90 s using a wristwatch having 1 s resolution. What is the percentage uncertainty in the calculated value

Accepted Answer

3%. g = 4*pi^2*L/T^2, so dg/g = dL/L + 2*dT/T. dL/L = 0.1/20.0 = 0.5%. For 100 oscillations in 90 s with 1 s resolution, dT/T = 1/90, so 2*dT/T = 2.22%. Total = 0.5% + 2.22% = 2.7% -> approximately 3%.

Question 28

A student measures the time period of 100 oscillations of a simple pendulum four times. The data set is 90 s, 91 s, 95 s, and 92 s. If the minimum division in the measuring clock is 1 s, then the reported mean time should be:

Accepted Answer

92 ± 1.5 s. Mean = (90+91+95+92)/4 = 92 s. Mean absolute deviation = (2+1+3+0)/4 = 1.5 s. Reported value = 92 +/- 1.5 s.

Question 29

A screw gauge with a pitch of 0.5 mm and a circular scale with 50 divisions is used to measure the thickness of a thin sheet of Aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the 45th division coincides with the mai

Accepted Answer

0.80 mm. Least count = pitch/divisions = 0.5/50 = 0.01 mm. With jaws closed the 45th division coincides and main-scale zero is just visible -> negative zero error = -(50-45)*0.01 = -0.05 mm. Observed reading = 0.5 + 25*0.01 = 0.75 mm; true thickness = 0.75 - (-0.05) = 0.80 mm.

Question 30

The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively 1.5% and 1%, the maximum error in determining its density is:

Accepted Answer

4.5%. Relative error in density = (error in mass) + 3*(error in length) = 1.5% + 3*1% = 4.5%.

Question 31

In the density measurement of a cube, the mass and edge length are measured as (10.00 ± 0.10) kg and (0.10 ± 0.01) m, respectively. The error in the measurement of density is:

Accepted Answer

0.31 kg/m³. The density of a cube is calculated using the formula density = mass/volume. Given the uncertainties in both mass and edge length, the propagation of errors leads to a combined relative error that results in a total error of 0.31 kg/m³ for the density measurement.

Question 32

The dimensions of B² / (2μ0), where B is magnetic field and μ0 is the magnetic permeability of vacuum, is:

Accepted Answer

ML−1T−2. B^2/(2*mu0) is magnetic energy density, which has the dimensions of pressure: energy/volume = (ML^2T^-2)/(L^3) = ML^-1T^-2.

Question 33

Let g_E and g_M denote the acceleration due to gravity at the surfaces of the Earth and the Moon, respectively. If Millikan’s oil-drop experiment is carried out on both surfaces, what would be the ratio of the electronic charge measured on the Moon to that measured on the Earth?

Accepted Answer

1. The charge on an electron is a fundamental constant, independent of local gravity. Millikan's experiment yields the same e on Moon and Earth, so the ratio is 1.

Question 34

A total resistance of 400 Ω is obtained by connecting four resistors of 100 Ω each, every resistor having a tolerance of 5%. What is the tolerance of the resulting combination?

Accepted Answer

5%. The total resistance of 400 Ω is achieved by connecting the resistors in series, where the tolerance of the combination remains the same as that of the individual resistors, which is 5%. Since the resistors are identical and in series, their tolerances do not compound, thus the overall tolerance remains at 5%.

Question 35

The unit ‘rad’ is used to express the measurement of which of the following?

Accepted Answer

the amount of energy deposited by radiation in a material. The rad (radiation absorbed dose) measures the amount of energy deposited by radiation per unit mass of material (1 rad = 0.01 J/kg). Biological impact is measured by the rem/sievert, and disintegration rate by the becquerel/curie.

Question 36

Let [ε0] denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then

Accepted Answer

[ε0] = [M⁻¹ L⁻³ T⁴ A²]. The correct option accurately represents the dimensional formula of permittivity of vacuum, which is derived from the relationship between electric field, charge, and force in electromagnetic theory, confirming that it has dimensions of mass to the power of -1, length to the power of -3, time to the power of 4, and electric current to the power of -2.

Question 37

The current voltage relation of diode is given by I = (e^(1000V/T) − 1) mA, where the applied voltage V is in volts and the temperature T is in degree Kelvin. If a student makes an error measuring ± 0.01 V while measuring the current of 5 mA at 300 K, what will be the error in the value of

Accepted Answer

0.2 mA. The error in current can be calculated using the derivative of the current with respect to voltage, which indicates how sensitive the current is to changes in voltage. At the given conditions, a small change of ±0.01 V in voltage results in a change of approximately ±0.2 mA in current, making option D the correct choice.

Question 38

A student measured the length of a rod and wrote its as 3.50 cm. Which instrument did he use to measure it ?

Accepted Answer

A vernier calliper where the 10 divisions in vernier scale matches 9 division in main scale and main scale has 10 divisions in 1 cm. The correct option is a vernier caliper because it allows for precise measurements to the hundredth of a centimeter, which is necessary to accurately measure and record a length of 3.50 cm.

Question 39

The period of oscillation of a simple pendulum is T = 2π√(L/g). Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. The accuracy in the determination of g is

Accepted Answer

3 %. The accuracy in determining g is influenced by the uncertainties in both the length L and the time measurement for the oscillations. Given the precision of L and the time taken for 100 oscillations, the derived value of g has a combined uncertainty that results in an overall accuracy of approximately 3%.

Question 40

A student measures the time period of 100 oscillations of a simple pendulum four times. The data set is 90 s, 91 s, 95 s and 92 s. If the minimum division in the measuring clock is 1 s, then the reported mean time should be:

Accepted Answer

92 ± 2 s. Mean = (90+91+95+92)/4 = 92 s. Mean absolute deviation = (2+1+3+0)/4 = 1.5 s, which with least count 1 s is reported as 2 s. So the result is 92 +/- 2 s.

Question 41

A screw gauge with a pitch of 0.5 mm and a circular scale with 50 divisions is used to measure the thickness of a thin sheet of Aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the 45th division coincides with the mai

Accepted Answer

0.80 mm. To find the thickness of the sheet, we first account for the zero error, which is -0.05 mm (since the 45th division coincides with the main scale line). The main scale reading is 0.5 mm, and the circular scale reading is 25 divisions, which corresponds to 0.5 mm (25/50 of the pitch). Adding these values gives 0.5 mm + 0.5 mm - 0.05 mm = 0.80 mm.

Question 42

The percentage errors in quantities P,Q,R and S are 0.5 %, 1%, 3% and 1.5 % respectively in the measurement of a physical quantity A = P³Q²/√RS. The maximum percentage error in the value of A will be -

Accepted Answer

6.5 %. The maximum percentage error in a calculated quantity is determined by summing the percentage errors of the individual components, weighted by their respective powers in the formula. For A = P³Q²/√RS, the contributions from P, Q, R, and S lead to a total maximum percentage error of 6.5%.

Question 43

The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants G, h and c. Which of the following correctly gives the Planck length ?

Accepted Answer

(Gh/c³)^(1/2). The Planck length is the unique combination of G, h, c with dimension of length: l_P = sqrt(Gh/c^3) (approx 1.6e-35 m). So option (Gh/c^3)^(1/2) is correct.

Question 44

In a screw gauge, 5 complete rotations of the screw cause it to move a linear distance of 0.25 cm. There are 100 circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of 4 main scale divisions and 30 circular scale divisions. Assuming negligible zer

Accepted Answer

0.2150 cm. The thickness of the wire can be calculated using the formula: thickness = (main scale reading + circular scale reading) × least count. The least count is determined by dividing the total movement of the screw (0.25 cm) by the number of circular scale divisions (100), which gives 0.0025 cm per division. Thus, the total thickness is (4 × 0.025 cm) + (30 × 0.0025 cm) = 0.2150 cm.

Question 45

The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively 1.5 % and 1%, the maximum error in determining the density is -

Accepted Answer

4.5 %. Density = mass/length^3, so relative error = (Δm/m) + 3(ΔL/L) = 1.5% + 3(1%) = 4.5%.

Question 46

If speed (V), acceleration (A) and force (F) are considered as fundamental units, the dimension of Young's modulus will be:

Accepted Answer

V⁻⁴ A² F. Young's modulus Y = [M L^-1 T^-2]. With V=[LT^-1], A=[LT^-2], F=[MLT^-2], solving the exponents gives Y = V^-4 A^2 F.

Question 47

Expression for time in terms of G (universal gravitational constant), h (Planck constant) and c (speed of light) is proportional to:

Accepted Answer

√(Gh/c⁵). With [G]=M^-1 L^3 T^-2, [h]=M L^2 T^-1, [c]=L T^-1: Gh = L^5 T^-3, and Gh/c^5 = L^5 T^-3 / (L^5 T^-5) = T^2. So time ~ sqrt(Gh/c^5) (the Planck time).

Question 48

If surface tension (S), M moment of Inertia (I) and Plank's constant (h), were to be taken as the fundamental units, then the dimensional formula for linear momentum would be

Accepted Answer

S¹/2 I¹/2 h⁰. Write p=S^a I^b h^c with [M L T^-1]. Matching M: a+b+c=1; L: 2b+2c=1; T: -2a-c=-1 gives a=1/2, b=1/2, c=0. So momentum = S^(1/2) I^(1/2) h^0.

Question 49

In the formula X = 5Y Z², X and Z have dimensions of capacitance and magnetic field, respectively. What are the dimensions of Y in SI units?

Accepted Answer

[M⁻³ L⁻² T⁸ A⁴]. The correct option is right because, in the equation, the dimensions of X (capacitance) and Z (magnetic field) must be balanced with Y. By analyzing the dimensions of capacitance and magnetic field, we can derive the necessary dimensions for Y, which results in [M⁻³ L⁻² T⁸ A⁴].

Question 50

In SI units, the dimensions of √(ε0/μ0) is -

Accepted Answer

A² T³ M⁻¹ L⁻². sqrt(eps0/mu0) is the reciprocal of impedance, i.e. conductance with SI units of siemens = A^2 T^3 M^-1 L^-2.

JEE Main Physics: Units and Measurements questions with solutions

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