Units, Dimensions and Errors

If L, C and R are the self inductance, capacitance and resistance respectively, which of the following does not have the dimension of time?

Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Product of Pressure (P) and time (t) has the same dimension as that of coeffcient of viscosity.

Reason R : Coefficient of viscosity $=\dfrac{\text{Force}}{\text{Velocity gradient}}$

Choose the correct answer from the options given below :

Velocity ($v$) and acceleration ($a$) in two systems of units 1 and 2 are related as $v_2=\dfrac n{m^2}v_1$ and $a_2=\dfrac{a_1}{mn}$ respectively. Here $m$ and $n$ are constants. The relations for distance and time in two systems respectively are :

A student in the laboratory measures thickness of a wire using screw gauge. The readings are 1.22 mm, 1.23 mm, 1.19 mm and 1.20 mm. The percentage error is $\dfrac x{121}$%. The value of $x$ is ______.

In van dar Wall equation $\left[P+\dfrac a{V^2}\right][V-b]=RT$; P is pressure, V is volume, R is universal gas constant and T is temperature. The ratio of constants $\dfrac ab$ is dimensionally equal to :

If momentum [P], area [A] and time [T] are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :

Which of the following physical quantities have the same dimensions ?

The maximum error in the measurement of resistance, current and time for which current flows in an electrical circuit are 1%, 2% and 3% respectively. The maximum percentage error in the detection of the dissipated heat will be :

A screw gauge of pitch 0.5 mm is used to measure the diameter of uniform wire of length 6.8 cm, the main scale reading is 1.5 mm and circular scale reading is 7. The calculated curved surface area of wire to appropriate significant figures is :

[Screw gauge has 50 divisions on its circular scale]

In an experiment to determine the Young's modulus of wire of a length exactly 1 m, the extension in the length of the wire is measured as 0.4 mm with an uncertainty of ± 0.02 mm when a load of 1 kg is applied. The diameter of the wire is measured as 0.4 mm with an uncertainty of ± 0.01 mm. The error in the measurement of Young's modulus ($\Delta Y$) is found to be $x\times10^{10}$ Nm$^{-2}$. The value of $x$ is ______.

(take $g=10$ ms$^{-2}$)

In a Vernier Calipers, 10 divisions of Vernier scale is equal to the 9 divisions of main scale. When both jaws of Vernier calipers touch each other, the zero of the Vernier scale is shifted to the left of zero of the main scale and 4$^{\text{th}}$ Vernier scale division exactly coincides with the main scale reading. One main scale division is equal to 1 mm. While measuring diameter of a spherical body, the body is held between two jaws. It is now observed that zero of the Vernier scale lies between 30 and 31 divisions of main scale reading and 6$^{\text{th}}$ Vernier scale division exactly coincides with the main scale reading. The diameter of the spherical body will be :

A torque meter is calibrated to reference standards of mass, length and time each with 5% accuracy. After calibration, the measured torque with this torque meter will have net accuracy of :

The one division of main scale of Vernier callipers reads 1 mm and 10 divisions of Vernier scale is equal to the 9 divisions on main scale. When the two jaws of the instrument touch each other, the zero of the Vernier lies to the right of zero of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the zero of the Vernier scale lies in between 4.1 cm and 4.2 cm and 6$^{\text{th}}$ Vernier division coincides with a main scale division. The diameter of the bob will be ______ $\times10^{-2}$ cm.

An expression of energy density is given by $u=\dfrac\alpha\beta\sin\left(\dfrac{\alpha x}{kt}\right)$, where $\alpha$, $\beta$ are constants, $x$ is displacement, $k$ is Boltzmann constant and $t$ is the temperature. The dimensions of $\beta$ will be :

The dimensions of $\left(\dfrac{B^2}{\mu_0}\right)$ will be :

(if $\mu_0$ : permeability of free space and B : magnetic field)

Consider the efficiency of carnot's engine is given by $\eta=\dfrac{\alpha\beta}{\sin\theta}\log_e\dfrac{\beta x}{kT}$, where $\alpha$ and $\beta$ are constants. If T is temperature, k is Boltzmann constant, θ is angular displacement and $x$ has the dimensions of length. Then, choose the incorrect option :

In an experiment to find acceleration due to gravity (g) using simple pendulum, time period of 0.5 s is measured from time of 100 oscillation with a watch of 1 s resolution. If measured value of length is 10 cm known to 1 mm accuracy, The accuracy in the determination of g is found to be $x$%. The value of $x$ is ______.

Given below are two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).

Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension (S), if density of the liquid is ρ and radius of the drop is r, then $T=K\sqrt{\dfrac{\rho r^3}{S^{3/2}}}$ is dimensionally correct, where K is dimensionless.

Reason (R) : Using dimensional analysis we get R.H.S. having different dimension than that of time period.

In the light of above statements, choose the correct answer from the options given below.

A travelling microscope has 20 divisions per cm on the main scale while its vernier scale has total 50 divisions and 25 vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling microscope ?

Match List I with List II.

Choose the correct answer from the options given below: