A physical quantity x is calculated from x = ab 2 / c .Calculate the maximum percentage error in measuring x when the percentage errors in measuring a, b, and c are 4, 2, and 3%, respectively.
The resistance of a metal is given by R = V/I, where V is potential difference and I is the current. In a circuit, the potential difference across resistance is V= (8 ± 0.5) V and current in resistance, I = (4 ± 0.2) A. What is the value of resistance with its percentage error?
Write the dimensions of a/b in the relation P = a − t 2 bx , where P is the pressure, x is the distance, and t is the time.
Acceleration due to gravity on the surface of the earth is g = GM R 2 . The gravitational constant G is exactly known. But percentage error in measurement of the mass of earth M and radius of the earth R are 1% and 2% respectively. The maximum percentage error is measurement of acceleration due to gravity on the surface of the earth in n%. Find the value of n.
The accuracy of measurement also lies in the way the result is expressed. The number of digits to which a value is to be expressed is one digit more than number of sure numbers. Rules do exist to deal with number of digits after an operation is carried out on the given values. The error can be minimised by many trials and using the correct methods and instruments.
The energy of a system as a function of time t is given as E ( t ) = A 2 exp ( − α t ) , Where α = 0.2 s − 1 . The measurement of A has an error of 1.25 % . If the error in the measurement of time is 1.50 % , the percentage error in the value of E ( t ) at t = 5 s is
Each division on the main scale is 1mm. Which of the following vernier scales will give least count equal to 0.01 mm?
If the dimensions of length are expressed as G x c y h z ; where G, c and h are the universal gravitational constant, speed of light and Planck’s constant respectively, then (x + z) is
The energy of a system as a function of time t is given as E t = A 2 e − α t , where α = 0.2 s − 1 . The measurement of A has an Error of 1.25%. If the error in the measurement of time is 1.50 % the percentage error in the value of E(t) at t = 5 s is
In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, E a n d B stand for dimensions of electric and magnetic fields respectively, while ε 0 a n d μ 0 stand for dimensions of the permittivity and permeability of free space respectively. L a n d T are dimensions of length and time respectively. All the quantities are given in SI units.
If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation z = x / y . If the errors in x, y and z are Δ x , Δ y a n d Δ z , respectively, then z ± Δ z = x ± Δ x y ± Δ y = x y 1 ± Δ x x 1 ± Δ y y − 1 The series expansion for 1 ± Δ y y − 1 , to first power in Δ y / y , is 1 ∓ Δ y / y . The relative errors in independent variables are always added. So the error in z will be Δ z = z Δ x x + Δ y y The above derivation makes the assumption that Δ x / x < < 1 , Δ y / y < < 1 . Therefore, the higher power of these quantities are neglected.
Match the following Column – I Column – II a) Boltzmann constant p) M L 2 T − 1 b) Coefficient of viscosity q) M L − 1 T − 1 c) Planck constant r) M L T − 3 K − 1 d) Thermal conductivity s) M L 2 T − 2 K − 1
The length of a simple pendulum, measured by an instrument of least count 4 mm, is found to be 40 cm. Its time period is measured by taking 200 oscillations using a watch of resolution 2s. The time period of found to be 0.4 s. Find the fractional error in the determination of the acceleration due to gravity using this data.
Pitch of a screw gauge is 1 mm and there are 100 divisions on it’s circular scale. When nothing is kept between the studs, the zero of the main scale is not visible and 14 th division of circular scale coincides with the horizontal mark on main scale. When circular scale is rotated by one complete rotation, zero mark on main scale is still not visible, but during the second rotation zero mark becomes just visible and 10th division of circular scale is coinciding with horizontal mark on main scale. When a glass plate is placed between the studs, the circular scale lies between 8 th and 9 th division of main scale and 4 th division of circular scale coincides with horizontal mark on main scale. Main scale division is equal to 1mm. Then
When a physical quantity is obtained by arithmetic combination of measured quantities, certain set of rules are applied. Idea behind those rules is the fact that precision in reporting of derived physical quantity is limited by the physical quantity which has been measured with least precision. Keeping that into mind, which of the following options are CORRECT for the arithmetic operation of measured values a=130 cm and b=121.50 cm
A vernier calipers which is used to measure length of a cylinder has 1 mm marks on the main scale. It has 10 equal divisions on the vernier scale which match with 8 marks of main scale. If main scale reading is 4 and vernier is 5 then the length of cylinder is 1.25 N × 10 − 3 m then the value of N is:
Let ε 0 denote the dimensional formula for the permittivity of the vacuum and μ 0 that of the permeability of the vacuum. If M = mass, L = length, T = time and I = electric current, then
Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows: [position] = [ X α ]; [speed] = [ X β ]; [acceleration] =[ X p ]; [linear momentum] = [ X q ]; [force] = [ X r ]. Then –
Match the physical quantities given in Column I with dimensions expressed in terms of mass (M), length (L), time (T ), and charge (Q) given in Column II. Column I Column II (a) Angular momentum (p) ML 2 T − 2 (b) Latent heat (q) ML 2 Q − 2 (c) Torque (r) ML 2 T − 1 (d) Capacitance (s) ML 3 T − 1 Q − 2 (e) Inductance (t) M − 1 L − 2 T 2 Q 2 (f) Resistivity (u) L 2 T − 2
The external and internal diameters of a hollow cylinder are measured to be (4.23 ± 0.01) cm and (3.89 ± 0.01) cm. The thickness of the wall of the cylinder is:
To estimate ‘g’ from g = 4 π 2 L T 2 , error in measurement of L is ± 2% and error in measurement of T is ± 3%. The error in estimated ‘g’ will be:
In a hypothetical set of units 1 star joule = 10 3 joule 1 star Newton = 10 4 Newton 1 star second = 10 second How many star meters should be in one meter in this hypothetical set of units?
If equation ∫ dt 3 at − 2 t 2 = a x sin − 1 t 2 a 2 − 1 , find the value of x.
A student determines a dimensionless quantity, B = e n 2 ε 0 hc . Find the value of n. (Here, e = electric charge, ε 0 = electric permittivity of vacuum, h = Planck’s constant and c = speed of light.)
The percentage errors in the measurement of mass and speed are 2% and 3% , respectively. How much will be the maximum error in the estimation of KE obtained by measuring mass and speed?
If unit of mass becomes 2 times, the unit of length becomes 4 times and the unit of time becomes 4 times in the unit of Planck’s constant. Due to this unit of Planck’s constant becomes n times. Find the value of n.
Pressure is dimensionally
Choose the correct statement(s).
The power for hovering helicopter depends on its linear size, the density of air and ( g × density of the helicopter) as p ∝ ( linear size ) x ( density of air ) y ( g × density of helicopter ) z where g is acceleration due to gravity. Find the ratio of power output of engines of two helicopters p 1 p 2 , when linear size of first helicopter is four times the linear size of the second and all other parameters are same.
During measurement of kinetic energy T, the percentage error in measurement of mass of particle and momentum of particle are 2% and 3%, respectively. The percentage error in measurement of kinetic energy is n%. Find the value of n.
E 2 μ 0 has the dimensions (E= electric field, μ 0 = permeability of free space)
E, m, J, G denote energy, mass, angular momentum and gravitational constant respectively. The dimensions of E J 2 m 5 G 2 will be that of
If ρ is the mean density of the earth, r its radius, g the acceleration due to gravity and G the gravitational constant, on the basis of dimensional consistency, the correct expression is
If the speed of light (c) acceleration due to gravity (g) and pressure (p) are taken as the fundamental quantities, then the dimensions of length will be
If the velocity of light C, the universal gravitational constant G, and Planck’s constant h are chosen as fundamental units, the dimensions of mass in this system are
If E, M, J, and G, respectively, denote energy, mass, angular momentum, and gravitational constant, then EJ 2 / M 5 G 2 has the dimensions of
The specific resistance ρ of a circular wire of radius r. resistance R, and length l is given by ρ = πr 2 R / l . Given: r = 0 .24 ± 0 .02 cm , R = 30 ± 1 Ω , and l = 4 .80 ± 0 .01 cm . The percentage error in ρ is nearly
