Styrofoam has a density of 150 kg/m³ . What is the maximum mass that can hang without sinking from a 50-cm-diameter Styrofoam sphere in water? Assume the volume of the mass is negligible compared to that of the sphere.

Video transcript

Hi, everyone in this practice problem. We're being asked to determine the maximum mass that can be suspended from a cubic object in water without causing it to sink. We will have a cube shaped floating object made out of a lightweight material with a density of 100 and 20 kg per meter cube and a site length of 38 centimeter. We're being asked to determine the maximum mass that can be suspended from this cubic object in water without causing it to sink. And we will have to consider that the volume of the suspended mass is negligible compared to the cube's volume. The options given for the maximum masses are a 48. kg b 38.9 kg, C 68.6 kg and d 33. kg. So first, I will list out all the given information in our problem statement. So first, we will have the uh density of the material or the lightweight material that is used to make the cube shaped floating object. So I'm gonna write that down as row object is going to be 100 and 20 kg per meter cube. And then we will have the site length. I'm just gonna write down S L which is going to be centimeter which is going to be converted into meters for simplicity purposes. So we will have to multiply it by 10 to the power of negative two m divided by one centimeter. And that will come out to be 38 times 10 to the power of negative two m. Awesome. And then we know that the density of water or row of water is going to be 1000 kg per meter cube. And then lastly, we know the gravitational acceleration G will also just be 9. m per second squared. Awesome. So to find the maximum mass that can be suspended from the cubic object in water without actually causing it to sink, we will have to first calculate the overall volume of the cube. So the overall volume of the cube, we want to recall that the formula for volume for a cube is just L cube. So that will then be just 38 times 10 to the power of negative two m cube. And that will come out to a value of 0.5487 m cube. Awesome. Next, we want to actually um calculate the weight of the object that we have. But what we are given is the density and also the site length which can be calculated into volume. So if we recall weight weight is just mass multiplied by the gravitational acceleration. However, mass can be calculated from multiplying row width volume. And we can use row multiply it by volume multiply it by gravitational acceleration in order for us to actually get the weight of the cube shape floating object. So I'm gonna um just put a little note here that row O or row object multiplied by volume is going to be the mass of the object or M. So in this case, I am then going to pluck everything into this equation. So row O is going to be 100 and 20 kg per meter cube multiply that by the volume that we just get which is 0.5487 m cube. And then the gravitational acceleration is of course 9. m per second squared just like so awesome. So then that will actually give us the weight of the object or W O two B 64.595 Newton. Next, we want to calculate the buoyant force that will actually pre be provided by the water when the cube is going to be fully submerged. If we recall the formula to calculate Buon force or F B here will be equals to the row of the water or row W multiplied by the volume submerge and multiplied that by the gravitational acceleration. So plugging in all the values, we have the rule of water to be 1000 kg per meter cube. And then the volume to be what we just calculated previously, which is 0.5487 m cube multiply that again by the graph stational acceleration which is 9. m per second squared. And that will resulted into our boy and force which is going to be 538. Newton. Awesome. So um the next thing that we need to do is that to, we have to realize that to prevent the cube from sinking the buoyant force must be equal to the weight of the cube plus the weight of the suspended mass, which is what we're interested in finding. So that will mean F B will be equals to W cube plus W of the suspended mass. So in this case, we want to substitute the expressions for the boy and force and the weight of the cube that we just calculated, which is W O N F B. We want a substitute dose two expression into this expression right here for us to then finally found, find the weight of the suspended mass. So F B is 538.294 Newton and W Q equals W O. So in this case, I'm just gonna represent that first with W O or the object plus the uh weight of the suspended mass. So 538.294 Newton will then be equals to 64.595 Newton plus the weight of the suspended mass will be the maximum M max multiply that by G and that this will resulted in 538.294, Newton minus 64.595, Newton equals two. And max which is the mass that we're interested to find for multiplied by G which is 9.81 m per second squared. And then uh rearranging things, we will have an equation for M max which is going to equals to 538.294 Newton minus 64.595 Newton. All of that going to be divided by 9.81 m per second squared. So that will result in a max value of 48.287 kg or essentially rounding it up a little A max will equals to 48.3 kg just like that. So the maximum mass that can be suspended from the object without causing the object itself to sink is approximately 48.3 kg, which will correspond to option A in our answer choices. So option A 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 that'll be it for this one. Thank you?