An ore sample weighs 17.50 N in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 11.20 N. Find the total volume and the density of the sample.

Video transcript

Hey everyone in this problem, we have an impure metal block with a weight of 50.4 Nunes and air. When the block is fully submerged in oil, the strength tension required to keep it suspended in the oil is newtons. And were asked to determine the volume and the density of the block. So let's go ahead and draw a free body diagram because we're given some information about forces here and so we have our block. We know that we always have the weight acting downwards. Now, Archimedes principle tells us that because this is submerged in oil, the fluid, the oil is going to exert an upward force on the object equal to the weight of the fluid displaced. And so we have this upward buoyant force FB and we also have tension acting upwards keeping this suspended. Alright, so let's take up to be the positive direction. We know in equilibrium we have the some of the force is equal to zero. In this case we have forces in the Y direction only. And so the sum of the forces in the Y direction is going to be equal to zero. Now, what are the forces that we have in the Y direction? Well, we have the buoyant force, F. B. We have the force of tension T and we have in the negative y direction the weight W. And so we have F B plus t minus W is equal to zero. Now recall that the buoyant force F B is given by the density rho. Okay, and this is the density of the fluid. In this case we are in oil times G gravitational acceleration times V. The volume of the object that is submerged in fluid and in this case the metal block is fully submerged. And so when we're talking about the volume here, this is going to be the volume of the block. And then we have the tension t minus the weight. W. And this is all equal to zero. So the density of oil. We can look this up in a table in our textbook or that our professor provided And we get kilograms per meter cubed times of gravitational acceleration, 9.8 m per second squared times the volume. Well we don't know the volume but that's something we want to find. The volume of a block. We're told that the tension is 33 newtons and that the weight of the block is 50.4 newtons. This is all equal to zero. Alright, so if we isolate for V, we can move 33 newtons minus 50.4 newtons to the right hand side. We're gonna have 17.4 newtons. And then we're going to divide by 820 kg per meter cubed times 9.8 m per second squared. We have 8036 kg per meter squared second squared. And this is going to give us a volume of 0. 165 m cubed. And so we're done with the first part of the problem, we found the volume of our cube using this summer forces the equilibrium condition on our force. Now we're asked to find the density while recall that the density is the mass over the volume. We've just found the volume. We have information about the weight which is gonna allow us to find the mass and so we can find our density. So the density of the block is going to be equal to the mass of the block divided by the volume. Recall that the mass is going to be the weight divided by the gravitational acceleration. So we have W over G divided by V. The weight given in the problem is 50.4 newtons divided by 9.8 m per second squared. And all of this divided by the volume. We just found 0.2165 m cubed, which gives a density of the block of 2375. kilograms per meter. Cute. And that's it. We found the volume and we found the density. Now let's go back up to our answer traces. And we found that the volume was approximately 0.002165 m cubed. And we can write this as 2.17 times 10 to the negative three m cube. So we're looking at option either A or C. And then for the density we round to the nearest 10, we found the density to be approximately 2000 kg for meter cubed. And so we're looking at option C. Thanks everyone for watching. I hope this video helped see you in the next one.