Unit And Dimensions Set-3

Q1: In a book, the answer for a particular question is expressed as $b = \frac{m a}{k} \left[ \sqrt{1 + \frac{2 k l}{m a}} \right]$ here $m$ represents mass, $a$ represents acceleration, $l$ represents length. The unit of $b$ should be

Q2: A simple pendulum is being used to determine the value of gravitational acceleration $g$ at a certain place. The length of the pendulum is 25.0 cm and a stop watch of 1s resolution measures the time taken for 40 oscillations to be 50 s. The accuracy in $g$ is:

Q3: In a simple pendulum experiment for determination of acceleration due to gravity ($g$), time taken for 20 oscillations is measured by using a watch of 1 second least count. The mean value of time taken comes out to be 30 s. The length of pendulum is measured by using a meter scale of least count 1 mm and the value obtained 55.0 cm. The percentage error in the determination of $g$ is close to

Q4: The least count of the main scale of a screw gauge is 1 mm. The minimum number of divisions on its circular scale required to measure 5 μm diameter of a wire is

Q5: The pitch and the number of divisions, on the circular scale for a given screw gauge are 0.5 mm and 100, respectively. When the screw gauge is fully tightened without any object, the zero of its circular scale lies 3 divisions below the mean line. The readings of the main scale and the circular scale for a thin sheet are 5.5 mm and 48 respectively, the thickness of this sheet is

Q6: A screw gauge with a pitch of 0.5 mm and a circular scale with 50 divisions is used to measure the thickness of a thin sheet of aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the 45th division coincides with the main scale line and that the zero of the main scale is barely visible. What is the thickness of the sheet, if the main scale reading is 0.5 mm and the 25th division coincides with the main scale line?

Q7: 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

Q8: A student measured the length of a rod and wrote it as 3.50 cm. Which instrument did he use to measure it?

Q9: The diameter of a cylinder is measured using a vernier calipers with no zero error. It is found that the zero of the vernier scale lies between 5.10 cm and 5.15 cm of the main scale. The vernier scale has 50 division equivalent to 2.45 cm. The 24th division of the vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is

Q10: In the determination of Young's modulus $\frac{4MLg}{\pi d^2 l}$ by using Searle's method, a wire of length $L = 2$ m and diameter $d = 0.5$ mm is used. For a load $M = 2.5$ kg, an extension $l = 0.25$ 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 mm. The number of divisions on their circular scale is 100. The contributions to the maximum probable error of the Y measurement is

Q11: 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 mm and there are 50 divisions on the circular scale. The reading on the main scale is 2.5 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

Q12: A vernier calipers has 1 mm marks on the main scale. It has 20 equal divisions on the vernier scale which match with 16 main scale divisions. For this vernier calipers, the least count is

Q13: A student performs an experiment to determine the acceleration due to gravity $g$ using the simple pendulum method. He commits an error of $\Delta \ell$ in measuring the length and $\Delta T$ in measuring the time period. Then the fractional error $\left( \frac{\Delta g}{g} \right)$ in the measurement of $g$ is maximum when:

Q14: The period of oscillation of a simple pendulum is $T = 2\pi \sqrt{\frac{L}{g}}$. 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 1 s resolution. The accuracy in the determination of $g$ is:

Q15: In an experiment the angle of a prism is measured as $60^\circ$ with least count of $1^\circ$. The percentage error in the determination of its refractive index $\mu$ is $x\%$. If the minimum deviation produced by the prism is $30^\circ$ with least count $1^\circ$, then $x$ is:

Q16: The relative error in the determination of the surface area of a sphere is $\alpha$. Then the relative error in the determination of its volume is:

Q17: The dimensional formula for magnetic flux is:

Q18: The velocity $v$ of a particle at time $t$ is given by $v = at + \frac{b}{t + c}$, where $a, b$ and $c$ are constants. The dimensions of $a, b$ and $c$ are respectively:

Q19: If $E$ and $G$ respectively denote energy and gravitational constant, then $\frac{E}{G}$ has the dimensions of:

Q20: The dimension of stopping potential $V_0$ in photoelectric effect in units of Planck's constant $h$, speed of light $c$ and Gravitational constant $G$ and ampere $A$ is:

Q21: The dimensional formula for $\frac{1}{2} \epsilon_0 E^2$, where $\epsilon_0$ is permittivity of free space and $E$ is electric field, is:

Q22: If speed $V$, area $A$ and force $F$ are chosen as fundamental units, then the dimension of Young's modulus will be:

Q23: If the capacitance of a nanocapacitor is measured in terms of the unit $U$ made by combining the electronic charge $e$, Bohr radius $a_0$, Planck's constant $h$ and speed of light $c$ then:

Q24: In the formula $X = 3YZ^2$, $X$ and $Z$ have dimensions of capacitance and magnetic induction respectively. The dimensions of $Y$ in MKSQ system are:

Q25: If $p = \frac{a t^2}{b x}$ where $p$ is pressure, $x$ is distance and $t$ is time. Then the dimensional formula for $\frac{a}{b}$ is:

Q26: If $A = B^n C^m$, where $A$ has dimensions $[L T]$, $B$ has dimensions $[L^2 T^{-1}]$ and $C$ has dimensions $[L T^2]$, then the values of $n$ and $m$ are respectively:

Q27: The dimensions of $\frac{B^2}{2\mu_0}$, where $B$ is magnetic field and $\mu_0$ is the permeability of free space, is:

Q28: The quantities $x = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$, $y = \frac{E}{B}$ and $z = \frac{l}{CR}$ are defined where $l$-length, $C$-capacitance, $R$-resistance. Then:

Q29: 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:

Q30: The dimensional formula for $\frac{\mu_0}{\epsilon_0}$ is: