# Tutor profile: Darshak M.

## Questions

### Subject: Mechanical Engineering

Stress must increase with an increase in strain. But that's not always the case. In materials like steel, the stress decreases after the upper yield point. Why does steel have an upper as well as a lower yield point?

Materials such as steel are added with impurities like carbon to alter their properties. Dislocations exist in all of the materials but impurities block the movement of these dislocations. For unlocking the dislocations higher stress is required resulting in the upper yield point. But once the dislocations start unlocking it is a rapid process decreasing further stress resulting in the lower yield point.

### Subject: Calculus

Find the value of the integral, $$ \int_0^5 \int_0^5 \lfloor x+y \rfloor\mathrm{d}x\mathrm{d}y $$ where $$ \lfloor.\rfloor $$ indicates floor function.

I = $$ \int_0^5 \int_0^5 \lfloor x+y \rfloor\mathrm{d}x\mathrm{d}y $$ Using the laws of integration, I = $$ \int_0^5 \int_0^5 \lfloor 5-x+5-y \rfloor\mathrm{d}x\mathrm{d}y $$ = $$ \int_0^5 \int_0^5 \lfloor 10-x-y \rfloor\mathrm{d}x\mathrm{d}y $$ Adding these equations, 2I = $$ \int_0^5 \int_0^5 \lfloor x+y \rfloor\mathrm{d}x\mathrm{d}y $$ + $$ \int_0^5 \int_0^5 \lfloor 10-x-y \rfloor\mathrm{d}x\mathrm{d}y $$ 2I = $$ \int_0^5 \int_0^5 (\lfloor x+y \rfloor+\lfloor 10-x-y \rfloor)\mathrm{d}x\mathrm{d}y $$ if x+y in non integer, $$ (\lfloor x+y \rfloor+\lfloor 10-x-y \rfloor) $$ = 9 When x+y is negative the area under the curve at those points is quite negligible. So final integral becomes 2I = $$ \int_0^5 \int_0^5 9\mathrm{d}x\mathrm{d}y $$ = 225 I = 225/2

### Subject: Physics

You are walking peacefully in the street enjoying the cold, chilly weather. For some reason, you look above your head and see a bunch of mosquitoes hovering over your head. You try to run away from them but adamant as they are, mosquitoes catch you up. What do you think is the reason for the hovering of mosquitoes over the head?

Biology: Mosquitoes are cold-blooded animals. So they need an external medium to regulate their temperature for better development. Mosquitoes' body includes heat sensors around its mouth to detect the warmth of the body. But why hover over the head? Physics: Due to the temperature difference between the human body and atmosphere, heat transfer takes place. The density gradient is induced in the fluid(in this case air) due to temperature gradient resulting in a buoyant force. This is called Free/Natural convection mode of heat transfer. The human head acts as a spherical body. The boundary layer due to free convection starts to develop from the bottom increasing the thickness along the circumference to the top resulting in plume formation. This plume attracts the mosquitoes.

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