Number of significant figures in 23.023 = 5
Number of significant figures in 0.0003 = 1
Number of significant figures in 2.1 × $${10^{ - 3}}$$ = 2
2.
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 $$1s,$$ then the reported mean time should be-
$$\eqalign{
& \Delta T = \frac{{\left| {\Delta {T_1}} \right| + \left| {\Delta {T_2}} \right| + \left| {\Delta {T_3}} \right| + \left| {\Delta {T_4}} \right|}}{4} \cr
& = \frac{{2 + 1 + 3 + 0}}{4} \cr
& = 1.5 \cr} $$
As the resolution of measuring clock is 1.5 therefore the mean time should be $$92 \pm 1.5$$
3.
A wire has amass $$0.3 \pm 0.003\,g,$$ radius $$0.5 \pm 0.005\,mm$$ and length $$6 \pm 0.06\,cm.$$ The maximum percentage error in the measurement of its density is
4.
A spherical body of mass $$m$$ and radius $$r$$ is allowed to fall in a medium of viscosity $$\eta .$$ The time in which the velocity of the body increases from zer to $$0.63$$ times terminal velocity $$\left( v \right)$$ is called me constant $$\left( \tau \right).$$ Dimensionally $$\tau $$ can be represented by :
A.
$$\frac{{m{r^2}}}{{6\pi \eta }}$$
B.
$$\sqrt {\left( {\frac{{6\pi mr\eta }}{{{g^2}}}} \right)} $$
5.
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
Let $${\mu _0}$$ related with $$e,m,c$$ and $$h$$ as follows.
$$\eqalign{
& {\mu _0} = k{e^a}{m^b}{c^c}{h^d} \cr
& \left[ {ML{T^{ - 2}}{A^{ - 2}}} \right] = {\left[ {AT} \right]^a}{\left[ M \right]^b}{\left[ {L{T^{ - 1}}} \right]^c}{\left[ {M{L^2}{T^{ - 1}}} \right]^d} \cr
& = \left[ {{M^{b + d}}{L^{c + 2d}}{T^{a - c - d}}{A^a}} \right] \cr} $$
On comparing both sides we get
$$\eqalign{
& a = - 2,b = 0,c = - 1,d = 1 \cr
& \therefore \left[ {{\mu _0}} \right] = \left[ {\frac{h}{{c{e^2}}}} \right] \cr} $$
6.
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 $${45^{th}}$$ division coincides with the main scale line and 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 $${25^{th}}$$ division coincides with the main scale line?
$$L.C.$$ = 50 = 0.01 $$mm$$
Zero error = 5 x 0.01 = 0.05 mm (Negative)
Reading = (0.5 + 25 x 0.01) + 0.05 = 0.80 $$mm$$
7.
In order to measure physical quantities in the sub-atomic world, the quantum theory often employs energy $$\left[ E \right],$$ angular momentum $$\left[ J \right]$$ and velocity $$\left[ c \right]$$ as fundamental dimensions instead of the usual mass, length and time. Then, the dimension of pressure in this theory is
A.
$$\frac{{{{\left[ E \right]}^4}}}{{{{\left[ J \right]}^3}{{\left[ c \right]}^3}}}$$
B.
$$\frac{{{{\left[ E \right]}^2}}}{{\left[ J \right]\left[ c \right]}}$$
C.
$$\frac{{\left[ E \right]}}{{{{\left[ J \right]}^2}{{\left[ c \right]}^2}}}$$
D.
$$\frac{{{{\left[ E \right]}^3}}}{{{{\left[ J \right]}^2}{{\left[ c \right]}^2}}}$$
Answer :
$$\frac{{{{\left[ E \right]}^4}}}{{{{\left[ J \right]}^3}{{\left[ c \right]}^3}}}$$
8.
Ina vernier callipers $$N$$ division of vernier coincide with $$\left( {N - 1} \right)$$ divisions of main scale in which length of a division is $$1\,mm.$$ The least count of the instrument in $$cm$$ is
$$\eqalign{
& L.C = \frac{{{\text{value of 1 division of main scale}}}}{{{\text{number of division on main scale}}}} \cr
& = \frac{1}{N}mm = \frac{1}{{10\,N}}cm \cr} $$
9.
Surface tension of a liquid is $$70\,dyne/cm.$$ Its value in SI is
10.
In the density measurement of a cube, the mass and edge length are measured as $$\left( {10.00 \pm 0.10} \right)kg$$ and $$\left( {0.10 \pm 0.01} \right)m,$$ respectively. The error in the measurement of density is-