Units, Dimensions and Errors

In an experiment to determine the acceleration due to gravity $g$, the formula used for the time period of a periodic motion is $T=2\pi\sqrt{\dfrac{7(R-r)}{5g}}$. The values of $R$ and $r$ are measured to be $(60\pm1)$ mm and $(10\pm1)$ mm, respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57 s, 0.54 s and 0.59 s. The least count of the watch used for the measurement of time period is 0.01 s. Which of the following statement(s) is(are) true?

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 :

In the following 'I' refers to current and other symbols have their usual meaning. Choose the option that corresponds to the dimensions of electrical conductivity :

A, B, C and D are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation AD = C ln(BD) holds true. Then which of the combination is not a meaningful quantity ?

Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the circular scale moves it by two divisions on the linear scale. Then:

Planck's constant $h$, speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then the correct option(s) is(are)

The energy of a system as a function of time $t$ is given as $E(t)=A^2\exp(-\alpha t)$, where $\alpha=0.2\ \mathrm s^{-1}$. 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 terms of potential difference $V$, electric current $I$, permittivity $\varepsilon_0$, permeability $\mu_0$ and speed of light $c$, the dimensionally correct equation(s) is(are)

The period of oscillation of a simple pendulum is $T=2\pi\sqrt{\dfrac Lg}$. 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 1s resolution. The accuracy in the determination of g is :

If the capacitance of a nanocapacitor is measured in terms of a unit '$u$' made by combining the electronic charge 'e', Bohr radius '$a_0$', Planck's constant 'h' and speed of light 'c' then :

If electronic charge e, electron mass m, speed of light in vacuum c and Planck's constant h are taken as fundamental quantities, the permeability of vacuum $\mu_0$ can be expressed in units of :

A beaker contains a fluid of density ρ kg/m$^3$, specific heat S J/kg°C and viscosity η. The beaker is filled up to height h. To estimate the rate of heat transfer per unit area $(\dot Q/A)$ by convection when beaker is put on a hot plate, a student proposes that it should depend on η, $\left(\dfrac{S\Delta\theta}h\right)$ and $\left(\dfrac1{\rho g}\right)$ when Δθ (in °C) is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for $(\dot Q/A)$ is :

During Searle's experiment, zero of the Vernier scale lies between $3.20\times10^{-2}$ m and $3.25\times10^{-2}$ m of the main scale. The 20th division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between $3.20\times10^{-2}$ m and $3.25\times10^{-2}$ m of the main scale but now the 45th division of Vernier scale coincides with one of the main scale divisions. The length of the thin metallic wire is 2 m and its cross-sectional area is $8\times10^{-7}\ \mathrm m^2$. The least count of the Vernier scale is $1.0\times10^{-5}$ m. The maximum percentage error in the Young's modulus of the wire is

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog, intensity (power/area) $S$ of the light from the signal and its frequency $f$. The engineer finds that $d$ is proportional to $S^{1/n}$. The value of $n$ is

During an experiment with a metre bridge, the galvanometer shows a null point when the jockey is pressed at 40.0 cm using a standard resistance of 90 Ω, as shown in the figure. The least count of the scale used in the metre bridge is 1 mm. The unknown resistance is

The current voltage relation of diode is given by $I=\left(e^{1000V/T}-1\right)$ mA, where the applied voltage $V$ is in volts and the temperature $T$ is in degree Kelvin. If a student makes an error measuring $\pm0.01$ V while measuring the current of 5 mA at 300 K, what will be the error in the value of current in mA ?

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

An experiment is performed to obtain the value of acceleration due to gravity $g$ by using a simple pendulum of length $L$. In this experiment time for 100 oscillations is measured by using a watch of 1 second least count and the value is 90.0 seconds. The length $L$ is measured by using a meter scale of least count 1 mm and the value is 20.0 cm. The error in the determination of $g$ would be :

In terms of resistance R and time T, the dimensions of ratio $\dfrac\mu\epsilon$ of the permeability $\mu$ and permittivity $\epsilon$ is :

From the following combinations of physical constants (expressed through their usual symbols) the only combination, that would have the same value in different systems of units, is :