What is the minimum hose diameter of an ideal vacuum cleaner that could lift a 10 kg (22 lb) dog off the floor?

Video transcript

Everyone in this practice problem, we will have a duct which is going to be at interstellar pressure of B equals to 4.0 times 10 to the power of negative 22 ATM. We're being asked to find the circumference of a piston connected that suspends a 20 kg TV set in the air on the earth's surface. The options given for the circumferences are a 2.48 centimeter B 7.8 centimeter C 15.6 centimeter and D 4.97 centimeter. So we'll consider forces on the piston. The first one is going to be the suction force from the vacuum directed upward. And the second one is going to be the weight of the TV set directed downward. So with those two weights or those two forces, I mean, we will then set up a Newton Second Law equation. We will have Sigma F equals to M multiplied by A, the sigma F as I have described previously is going to be F suction minus the weight of the TV set because the uh the suction force is going to be directed upwards. And in our convention, anything directed upward is going to be positive. So we'll have F suction or the suction force minus the weight of the TV set, which is M multiplied by G equals to M multiplied by A, the suction force is going to actually be delta P multiplied by A delta P is going to be in absolute value here just because we only care about the magnitude and we're assuming that it is going to be positive. So delta P multiplied by A will uh equals to M multiplied by A plus M multiplied by G. In this case, we will assume that their acceleration is going to equals to zero. So therefore, the absolute value of delta P multiplied by A will equals to M multiplied by G and rearranging this in order for us to get an equation for a or area which will help us to get the circumference will then get A to be equals to M multiplied by G divided by the absolute value of delta B. The absolute value of delta P can be calculated by taking or subtracting the P of the duct with P ATM. So P ATM minus P duct. And in this case, um the P duct is given to be four times 10 to the power of negative 22 ATM which is really small compared compare to P ATM. So in this case, P duck will, I will just say it is going to be negligible. So I will just equal that to zero. So in this case, our delta p absolute value is going to equals to P ATM just like so awesome. So now I am going to uh substitute that into our equation for A. So P ATM and then I will take that equation and um expand on the A to B pi multiplied by R squared equals to M multiplied by G divided by P ATM. And from this equation right here, we will then be able to find the equation for R which will be helpful in finding our circumference. So R rearranging this equation will be then equals to the square root of M multiplied by G divided by uh pi multiplied by P ATM just like. So, so we know all the values of uh given or needed in this equation for R right here. So we want to substitute everything into our equation. So then R will equals to the square root of M is going to be 20 kg which is the weight of the TV set given in the problem statement. And then RG is going to be 9.81 m per second squared. Our pi is just a normal pi and RP A PM is going to be a constant which is 1.013 times 10 to the power of five pascal, not necessarily constant, but it's more of a known fact. And in this case, this will govern an R value of 2.483 times 10 to the power of negative 2 m. And we wanna um actually convert that into centimeter, which will be then equals 22.483 centimeter just like. So we are uh multiplying essentially by 100 centimeter and dividing it by 1 m and they'll give us 2.483 centimeter. So lastly what we need to do left is just finding the circumference. And what we call the formula for the circumference is just two pi R. So then circumference will equals to two pi multiply by 2.483 centimeter. And they will give us the circumference of 15.6 centimeter. So the circumference will be 15.6 centimeter which will actually correspond to option C in our answer choices. So option C will be the answer to this particular practice problem and that'll be it for this video. If you guys still have any sort of confusion, please make sure to check out our adolescent videos on similar topics and I'll be at for this one. Thank you.