Level 1

Which of the following pairs does not have similar dimensions?

Which of the following units denotes the dimensions $\frac{M{L}^2}{Q^2}$, where $Q$ denotes the electric charge?

External and internal diameters of a hollow cylinder are measured to be $(4.23\pm 0.01) \mathrm{cm}$ and $(3.89\pm 0.01) \mathrm{cm}$. The thickness of the wall of the cylinder is

In a vernier callipers, the least count of main scale is $1 \mathrm{mm}.$ $20$ division of vernier scale are matching with $19$ divisions of main scale. The thickness of the object measured by the vernier callipers will be :

An object falls for a time of $0.25 \mathrm{s}$. The acceleration of free fall is $9.81 \mathrm{m} {\mathrm{s}}^{-2}$. The displacement is calculated. Which of the following gives the correct number of significant digits for the calculated value of the displacement of the object?

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is $0.5 \mathrm{mm}$ and there are $50$divisions on the circular scale. The reading on the main scale is $2.5 \mathrm{mm}$ and that on the circular scale is $20$ divisions. If the measured mass of the ball has a relative error of $2\%$, the relative percentage error in the density is
 

A square shape plate has area $\left(400\pm 0.4\right) {\mathrm{m}}^2$. The side length of the plate is

The main scale division and pitch of a screw gauge is $0.05 \mathrm{cm}$. Number of circular scale division in the circular scale is $100$. When no object is placed between the studs then the $0$ of main scale is not visible and $5^{\mathrm{th}}$ division of circular scale coincides with the main scale. When a wire is placed in between the studs then the reading on main scale is $4.6 \mathrm{mm}$ and ${52}^{\mathrm{nd}}$ division on circular scale matches with reference line on main scale. The actual diameter of wire is

In a vernier calliper, vernier scale has $20$division which matches with ${19}^{\mathrm{th}\mathrm{ }}$ division of main scale. The least count of main scale is $1 \mathrm{mm}$. When two jaws of vernier calliper is in contact, zero of main scale is visible and ${14}^{\mathrm{th}\mathrm{ }}$ vernier scale division matches with a main scale division. In measuring the length of iron rod by this instrument, main scale reads $35$division and $8^{\mathrm{th}\mathrm{ }}$ division of vernier scale coincides with a main scale division. Find the ten times value of the length of rod in $\mathrm{mm}$.

A spectrometer gives the following reading when used to measure the angle of the prism:
main scale reading $=58.5$ degree, vernier scale reading $=9$ division. Given that one division on main scale corresponds to $0.5$ degree, $30$ division of vernier scale matches with $29$ division of main scale. The angle of prism from above data:

The vernier of a circular scale is divided into $50$ divisions which coincides with $49$ main scale divisions. Each main scale division is $0.5°.$ The least count of the instrument is

If $E=$ energy, $G=$ gravitational constant, $I=$ impulse and $M=$ mass, the dimension $\frac{G{I}^{2}M}{E^2}$ is same as that of,

If $A=B+\frac{C}{D+E},$ the dimensions of $B$ and $C$ are $\left[{\mathrm{M}}^0{\mathrm{LT}}^{-1}\right]$ and $\left[{\mathrm{M}}^0{\mathrm{LT}}^0\right],$ respectively. Find the dimensions of $A\mathit{,}\mathit{ }D$ and $\mathrm{E}$.

In the determination of Young's modulus $\left(Y\mathit{=}\frac{4MLg}{\mathrm{\pi }l{d}^{\mathit{2}}}\right)$ by using Searle's method, a wire
of length $L=2 \mathrm{m}$ and diameter $\mathrm{d}=0.5 \mathrm{mm}$ is used. For a load, $M=2.5 \mathrm{Kg},$ an extension $l=0.25 \mathrm{mm}$ in the length of the wire is observed. Quantities $d$ and $l$ are measured using a screw gauge and a micrometer, respectively. They have the same pitch of $0.5 \mathrm{mm}.$ The number of divisions on their circular scale is $100.$ The contributions to the maximum probable error of the measurement are

If $R$ and $L$ denote resistance and inductance respectively which of the following has dimension of time?

A force $F$ is given by, $F=at+b{t}^2$, where $t$ is time. What are the dimensions of $a$ and $b$?

The dimensional equation for the electrical resistance of a conductor is

The angular diameter of the sun is $1920"$. If the distance of the sun from the earth is $1.5\times {10}^{11} \mathrm{m},$ then the linear diameter of the sun is

To determine Young's modulus of a wire, the formula is $Y=\frac{F}{A}\times \frac{L}{\Delta L}$, where, $L=$length, $A=$area of cross-section of the wire, $\Delta L=$change in length of the wire when stretched with a force $F$. The conversion factor to change it from $\mathrm{CGS} \mathrm{to} \mathrm{MKS}$ system is

$\int \frac{dx}{\sqrt{2ax-x^2}}=a^{\mathrm{n}}{\sin }^{-1}\left[\frac{x}{a}-1\right]$

Find the value of $n$. (Solve using dimensional analysis)