A cube 5.0 cm on each side is made of a metal alloy. After you drill a cylindrical hole 2.0 cm in diameter all the way through and perpendicular to one face, you find that the cube weighs 6.30 N. (a) What is the density of this metal?

Video transcript

Hey everyone welcome back in this problem. We have a cylinder made of unknown material that has a diameter of eight cm and a height of 10 cm. An engineer wishes to house an axle at the center of the cylinder. Therefore, they drill a hole of diameter four cm at the center of the cylinder, The engineer measures the weight of the hollowed cylinder to be 25 newtons and were asked to determine the density of the unknown material. So let's just draw like a bird's eye view. So we have this cylinder, okay, we don't know the material. It has a diameter of eight centimeters. And then an engineer is going to drill a hole into the middle of it with a diameter of four centimeters. Alright, so we're looking for density and recall that the density is going to be equal to the mass divided by the volume. So we need to figure out the mass of this material. We need to figure out the volume of this material in order to find the density. Now for the mass, recall that the mass is equal to the weight divided by the acceleration due to gravity, or we often see this as weight is equal to MG. We can rewrite that as M is equal to W over G. Now, the way we were told is 25 news, The acceleration due to gravity, we know is 9.8 m/s square. And so we have that our mass is going to be 25 newtons over 9.8 m per second squared. And we're gonna leave it like that for now, Instead of rounding and approximating a messy value when we do this division, we're just gonna leave it like that until we go to our density equation. So we have our masks. Now we need to find the volume. Now we're talking about volume, we're gonna need to use all of these diameters and heights were given those in centimeters. We want to convert them to our standard units of meters. So let's go ahead and do that first. Um so that we have that all out of the way. Now when we're calculating the volume, what we have is we have the volume of that solid cylinder initially that we had, we also have the volume of the center of the cylinder. And the volume that we want to calculate is kind of this ring around the outside and I'm gonna draw it in blue. This hollowed out cylinder because we're told that that's the way we have the weight is of the hollowed cylinder. So, if we have the weight of the hollow cylinder, that means we have the mass of the hollow cylinder. And so in order to find the density, we need to have the volume that relates to that same amount of material. Okay, so we want to find the volume of this hollowed cylinder in blue. So we need to take the volume of that solid cylinder that we had initially and subtract the volume of the center of that cylinder that was removed. Alright so let's say that R. S. Is going to be the radius of the solid cylinder. We're told that it has a diameter of 8cm. So this is going to be equal to eight cm divided by two. This gives us four centimeters. We want to convert this to meters. So this is going to be four centimeters times one m per 100 centimeters. Okay so we divide by 100 to get meters. The unit of centimeter divides out. And we're left with 0.4 m. We can do the same for the radius of the center. We're told that the diameter is four centimeters. So we're gonna have four centimeters divided by two which gives us two centimeters. We multiply by one m per 100 centimeters. Okay this is like dividing by 100. We get a value of 0.02 m and the final one is the height. Were given a height of 10 centimeters. Again we want to convert this to meters and so we get the height is equal to 10 cm times one m per 100 centimeters. The unit of centimeter divides out. And we're left with 0.1 m. Alright so let's get back to our volume equation. Now the volume of the solid, that's the cylinder. The outside cylinder recall it. The volume of a cylinder as pie R squared H. In this case this is R. S. And we're told that they take a whole out of the center of the cylinder. And so the portion that's removed from the center is going to be pie R squared C times H as well, same height. We're just taking that piece through the center all the way through. Alright, So plugging in the values that we just found in meters. We have pie times 0.4 m squared Time, m minus pi times 0.2 m squared times 0.1 m. And this is going to give us a volume of pi times 0. m. Cute. Alright. And we're gonna leave it like this so that we don't have to approximate it just like we did with maths until we get to our density calculation. Okay, remember what we're trying to find, we're trying to find the density of this unknown material. We found the mass. We found the volume. Now the density is given by mass over volume, which is going to be equal to 25 newtons divided by 9.8 m per second squared over five times 0.00012 meters cubed. And this is going to give us a density of 6,766. approximately kg per meter cubed. All right. If we look at our answer choices, we see that they're rounded to the nearest and so rounding this to the nearest 10, we're gonna have 6770 kg per meter cubed. That is going to correspond with B and that is the density of the unknown material. Thanks everyone for watching. I hope this video helped see you in the next one.