MHB Integration Application: Find the force on each end of the cylinder

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To find the force on each end of a horizontally positioned cylinder filled halfway with tar, first calculate the volume of the tar using the cylinder's dimensions. The volume of the tar is 2m in diameter and 1m in height, resulting in a volume of approximately 3.14 m^3. With a density of 920 kg/m^3, the mass of the tar is 2,888 kg, leading to a weight of about 28,308 N when multiplied by the acceleration due to gravity (9.8 m/s^2). The force on each end of the cylinder is equal to the weight of the tar divided by two, resulting in approximately 14,154 N per end. This approach highlights the importance of understanding fluid dynamics and pressure distribution in cylindrical containers.
andvaka
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A cylinder shaped container of tar has a diameter of 2m. If it is half full of tar with a density of 920 kg/(m^3) and is on it's side, find the force on each end.

Helpful hints:
920 kg per m^3 is the mass density not the weight density. Take g= 9.8 m/ sec^2Please assist. I'm not sure at all how to approach this or what relationships to use.​
 
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I would be interested in seeing your solution. Normally the problems involving cylinders and the density of a material within that cylinder involve calculating the work done by pumping out a certain amount.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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