Silica aerogel, an extremely porous, thermally insulating...

Silica aerogel, an extremely porous, thermally insulating material made of silica, has a density of $1.00 \mathrm{mg} / \mathrm{cm}^{3}$. A thin circular slice of aerogel has a diameter of $2.00 \mathrm{~mm}$ and a thickness of $0.10 \mathrm{~mm} .$ a) What is the weight of the aerogel slice (in newtons)? b) What are the intensity and the radiation pressure of a $5.00-\mathrm{mW}$ laser beam of diameter $2.00 \mathrm{~mm}$ on the sample? c) How many $5.00-\mathrm{mW}$ lasers with a beam diameter of $2.00 \mathrm{~mm}$ would be needed to make the slice float in the Earth's gravitational field? Use $g=9.81 \mathrm{~m} / \mathrm{s}^{2}$

Silica aerogel, an extremely porous, thermally insulating material made of silica, has a density of $1.00 \mathrm{mg} / \mathrm{cm}^{3}$. A thin circular slice of aerogel has a diameter of $2.00 \mathrm{~mm}$ and a thickness of $0.10 \mathrm{~mm} .$
a) What is the weight of the aerogel slice (in newtons)?
b) What are the intensity and the radiation pressure of a $5.00-\mathrm{mW}$ laser beam of diameter $2.00 \mathrm{~mm}$ on the sample?
c) How many $5.00-\mathrm{mW}$ lasers with a beam diameter of $2.00 \mathrm{~mm}$ would be needed to make the slice float in the Earth's gravitational field? Use $g=9.81 \mathrm{~m} / \mathrm{s}^{2}$

Key Concepts

Density

Density is a fundamental material property that quantifies the amount of mass per unit volume. It is essential in converting a measured volume into a mass, which is needed to calculate forces such as weight. In many applications, understanding density helps establish how light or heavy a material is relative to its size.

Volume Calculation

Determining an object's volume is key in many physical calculations, especially when dealing with regular shapes like cylinders or spheres. By multiplying the cross?sectional area by the length or thickness, one obtains the volume, which, combined with density, is used to calculate the mass of the object.

Weight Calculation

Weight is the force exerted by gravity on an object's mass. Calculated by multiplying the mass by the gravitational acceleration, this concept is essential in understanding the influence of gravity on objects, and it plays a crucial role in balancing forces in physical systems.

Intensity of Radiation

Radiation intensity is defined as the power per unit area that a beam of light or other electromagnetic radiation carries. It quantifies how much energy is delivered to a surface, which is fundamental when analyzing the impacts of laser beams or other radiant energy sources.

Radiation Pressure

Radiation pressure arises from the momentum transfer of electromagnetic radiation when it impinges on a surface. It is typically calculated by dividing the intensity of the radiation by the speed of light, and it becomes a key factor in phenomena where light exerts mechanical forces, such as in optical manipulation or levitation.

Force Equilibrium

Force equilibrium involves the balancing of all forces acting on a system so that the net force is zero, resulting in no acceleration. In practical scenarios, this means matching opposing forces, such as upward radiation forces and downward gravitational force, to achieve a stable state like levitation.

Transcript

00:01 We have density row equals to 1 .00 milligram per centimeter cube of the silica which is equals to 1 kilogram per meter cube and diameter d equals to 1 .2 .00 mm that is equals to 2 .00 modiplat byte 10 to the power minus 3 meter and thickness t equals to 0 .10mm that is equals to 0 .10 by 10 to the power minus 3 meter.

00:30 So for the part a, we have to calculate the weight of the arrow gel slice in newton.

00:36 So weight is given by mass multiplied by gravitational acceleration and mass can be written as density multiplied by volume.

00:43 So volume will be equals to area multiplied by thickness.

00:47 So area will be given by pi multiplied by d square by d by 2 whole square multiplied by thickness t and this this has become mass and this is g so this become weight so substituting value so we get w equals to density which is one but plight by pi mudplied by d by 2 which will be 1 .00 modplared by 10 to the power minus 3 whole square multiplied by thickness 0 .10 to the power minus 3 and g is 9 .8 1.

01:19 So from here after solving we get a w equals to 3 3.

01:23 0 .08 nanon.

01:26 Okay, so this is the answer for the part a of the problem.

01:29 Now, moving to the part b in which we have to calculate the intensity and radiation pressure of a 5 .00 millivot laser beam and diameter d equals to 2 .00 mm on the sample.

01:47 Okay.

01:47 So, now we can calculate that the intensity i, it will be given by power divided by air, and area will be pi d square by four so it will be equals to 4 p divided by pi modplared by d square so now substituting value so we get 4 mudplied by p which is 5 .00 modplored by 10 to the power minus 3 divided by pi muddclad by d which is this value so 2 .00 madaplored by 10 to the power minus 3 whole square okay so from here after solving we will get that intensity i it is equals to 1 .59 multiplied by 10 to the the power 3 watt per meter square which can be written as 1 .59 kilo watt per meter square.

02:31 So this is the intensity for the question.

02:33 Now we can calculate the radiation pressure...

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