Units, Dimensions and Errors

If current (I), kinetic energy (K) and charge (Q) are taken as the fundamental quantities. The dimensional representation of power will be :

The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is

A physical quantity $\vec S$ is defined as $\vec S = (\vec E \times \vec B)/\mu_0$, where $\vec E$ is electric field, $\vec B$ is magnetic field and $\mu_0$ is the permeability of free space. The dimensions of $\vec S$ are the same as the dimensions of which of the following quantity(ies)?

The workdone by a gas molecule in an isolated system is given by, $W=\alpha\beta^2e^{-\frac{x^2}{\alpha kT}}$, where $x$ is the displacement, k is the Boltzmann constant and T is the temperature. α and β are constants. Then the dimensions of β will be :

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] $= [X^{\alpha}]$; [speed] $= [X^{\beta}]$; [acceleration] $= [X^{p}]$; [linear momentum] $= [X^{q}]$; [force] $= [X^{r}]$. Then

Two capacitors with capacitance values $C_1 = 2000 \pm 10$ pF and $C_2 = 3000 \pm 15$ pF are connected in series. The voltage applied across this combination is $V = 5.00 \pm 0.02$ V. The percentage error in the calculation of the energy stored in this combination of capacitors is ______.

A current carrying wire heats a metal rod. The wire provides a constant power $(P)$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $(T)$ in the metal rod changes with time $(t)$ as

$$T(t) = T_0\left(1 + \beta t^{\frac14}\right),$$

where $\beta$ is a constant with appropriate dimension while $T_0$ is a constant with dimension of temperature. The heat capacity of the metal is,

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $L$, which of the following statement(s) is/are correct?

An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens is ______.

The relation between $[E]$ and $[B]$ is

The relation between $[\epsilon_0]$ and $[\mu_0]$ is

Consider the ratio $r = \dfrac{(1-a)}{(1+a)}$ to be determined by measuring a dimensionless quantity $a$. If the error in the measurement of $a$ is $\Delta a$ $(\Delta a/a \ll 1)$, then what is the error $\Delta r$ in determining $r$?

In an experiment the initial number of radioactive nuclei is 3000. It is found that $1000 \pm 40$ nuclei decayed in the first 1.0 s. For $|x| \ll 1$, $\ln(1+x) = x$ up to first power in $x$. The error $\Delta\lambda$, in the determination of the decay constant $\lambda$, in $\mathrm{s^{-1}}$, is

Consider an expanding sphere of instantaneous radius $R$ whose total mass remains constant. The expansion is such that the instantaneous density $\rho$ remains uniform throughout the volume. The rate of fractional change in density $\left(\frac{1}{\rho}\frac{d\rho}{dt}\right)$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to

A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\delta T = 0.01$ seconds and he measures the depth of the well to be $L = 20$ meters. Take the acceleration due to gravity $g = 10\ \mathrm{ms^{-2}}$ and the velocity of sound is $300\ \mathrm{ms^{-1}}$. Then the fractional error in the measurement, $\delta L/L$, is closest to

A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of :

Time (T), velocity (C) and angular momentum (h) are chosen as fundamental quantities instead of mass, length and time. In terms of these, the dimensions of mass would be :

A physical quantity P is described by the relation
$$P=a^{1/2}b^2c^3d^{-4}$$
If the relative errors in the measurement of a, b, c and d respectively, are 2%, 1%, 3% and 5%, then the relative error in P will be :

A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ of a dielectric material, Boltzmann constant $(k_B)$, the absolute temperature $(T)$, the number per unit volume $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particles. Which of the following expression(s) for $l$ is(are) dimensionally correct?

There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The Vernier scale of one of the calipers $(C_1)$ has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper $(C_2)$ has 10 equal divisions that correspond to 11 main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by calipers $C_1$ and $C_2$, respectively, are