Abucket of water of mass 15.8 kg is suspended by a rope wrapped around a windlass, that is a solid cylinder with diameter 0.260 m with mass 13.0 kg . the cylinder pivots on a frictionless axle through its center. the bucket is released from rest at the top of a well and falls a distance 11.0 m to the water. you can ignore the weight of the rope. part a
what is the tension in the rope while the bucket is falling
-take the free fall acceleration to be g=9.80 m/s squared

part b
with what speed does the bucket strike the water?
-take the free fall acceleration to be g=9.80 m/s^2

part c
what is the time of fall
-take the free fall acceleration to be g=9.80m/s^2

part d
while the bucket is falling, what is the force exerted on the cylinder by the axle?
-take the free fall acceleration to be g=9.80 m/s^2

i think it because equal matrices have equal dimensions, only square matrices can be symmetric. for all indices and . every square diagonal matrix is symmetric, since all off-diagonal elements are zero. in linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space.

it is given that trillions of microbes make up to 1% to 3% of the body mass.

let's take average of this percentage: 2%

consider the average mass of the human body as 70 kg.

then, trillions of microbes would have mass  

therefore, 100 trillion micro-organisms would weigh:

this is an approximate value and could change depending on the mass of the human body and percentage of the microbes present in it.

average mass per microorganism =

bacteria that live on the ocean floor are sustained by phytoplankton. phytoplankton, microscopic marine photosynthetic organisms, have a vastly significant role to play not only in the marine food web of which they’re part of, but also on a more global scale. despite their infinitely small size in comparison to other marine organisms, these tiny creatures occupy an immensely important ecological niche: they are the foundation of the marine food web, and as primary producers, play key roles in supporting all other organisms in the marine environment, as well as in the regulation of the earth’s climate through the sequestration of carbon, oxygen production, and other related processes. phytoplankton account for roughly half of all global primary productivity; therefore, their significance extends far beyond the marine environment alone. there is an intriguing sense of irony in the realization that these tiny living beings that often live out their existence unnoticed and undetected by the rest of the world, have such a far-reaching impact on the lives of virtually all other living organisms on the planet, particularly on those in the marine environment. the world that we have become accustomed to has been and is continuously shaped by the workings of these miniscule yet vital “plants of the sea”.

hope this !