$A$ $300 \, mm$ long tube contains $60 \, cm^3$ of a concentrated sugar solution. When placed in a polarimeter,it produces an optical rotation of $10^\circ$. If the specific rotation of sugar is $60^\circ$,calculate the amount of sugar in grams present in the tube.

Two nicols are crossed to each other. Now one of them is rotated through $60^{\circ}$. What percentage of incident light will pass through the system (in $\%$)?

Statement-$1$: The blue part of the sky is clearly visible through a calcite crystal. The intensity of the transmitted light changes as the crystal is rotated.
Statement-$2$: Light from the sky is due to the scattering of sunlight by particles in the atmosphere.

$A$ mixture of plane polarised and unpolarised light falls normally on a polarizing sheet. On rotating the polarizing sheet about the direction of the incident beam,the transmitted intensity varies by a factor of $4$. Find the ratio of the intensities $I_P$ and $I_0$,respectively of the polarized and unpolarized components in the incident beam.

$A$ polarizer and an analyzer are placed such that the intensity of the transmitted light is maximum. If the analyzer is rotated by $60^{\circ}$,find the intensity of the transmitted light.

$A$ plane polarized light is incident normally on a tourmaline plate. Its $\vec{E}$ vectors make an angle of $60^{\circ}$ with the optic axis of the plate. Find the percentage difference between initial and final intensities. (in $\%$)