in this video we're going to talk about archimedes principle and buoyant force the basic idea behind archimedes principle is that the buoyant force acted on an object that's immersed in the fluid is equal to the weight of the fluid displaced by that object now this problem is going to illustrate the concept of arguments principle and you'll see it as we go through the problem but now let's talk about something let's say if you want to lift a block with a rope in air is it easier to lift it up in air or is it easier to lift up the block if it's submerged in water what would you say it turns out that it's a lot easier to lift up the block when it's submerged in water since water has a higher density than air the portion of water that's displaced by the block exerts an upward buoyant force on the block which we'll call fb and as a result the tension in the rope doesn't have to be as great so objects appear to be lighter in water than in air and this is all related to archimedes principle so let's focus on this example so we have a 10 kilogram block of aluminum attached to a rope what is the tension force in a rope if the block is in a vacuum so let's say that is the aluminum block and we have an upward tension force that's lifting up the block and then there's the downward weight force acting on the block now in a vacuum there is no upward buoyant force because there's no air molecules to lift up the weight of the block so therefore the tension force would represent the true weight of the block and the weight of the block is mg so the tension force is equal to mg so the mass is 10 kilograms and g is 9. 8 so the answer for part a is 98 newtons that's the actual weight of the block now what about part b what is the tension force if the block is in air now air is a fluid the same way as water is a fluid so as a result air is going to push up the block with an upward buoyant force so the sum of all forces in the y direction is going to be based on the two upward forces that's going to be positive t and positive fb and the weight force is in the negative y direction so we're going to put negative w in order to get the minimum force that the rope has to apply in order to keep the block stationary if we don't want the block to fall or to accelerate upward if we want it to stay in place the net force in the y direction must be zero so solving for the tension force we need to take these two terms move it to the other side so w is negative on the right side but it's going to be positive on the left and the buoyant force is going to be negative on the left so the tension force is the weight force minus the buoyant force in part a there was no buoyant force so the tension force was just the weight force but for parts b and c the tension force is going to be the difference between the weight force and the buoyant force now how do we calculate the buoyant force what can we do to find the answer now let's say if we have an object submerged in the fluid the fluid could be water or it could be air and let's use a cylinder the fluid exerts a force on the top surface of the cylinder which we'll call f1 and it also exerts a force on the bottom surface f2 the difference in these two forces is equal to the buoyant force as you go deeper in water the pressure is greater so because the pressure at the bottom is greater than the pressure at the top f2 is going to be greater than f1 the deeper you go in water the greater the force that it will exert on you and so that's why we have a net upward buoyant force because the pressure is greater at the bottom than it is at the top so that buoyant force is the difference between f2 and f1 now granted the fluid does exert a force in the x direction but these forces cancel out because they're at the same depth so we don't have to worry about that so only the force is in the y direction is what we're concerned about now force well we know that pressure is force divided by area so force is pressure times area so f2 is going to be p2 times a and f1 is p2 p1 times a now a is the same the area of the top part of the cylinder is the same as the area of the bottom part the cylinder we're assuming that we have a uniform reshaped cylinder so now what should we replace p2 and p1 with now the pressure exerted by a fluid is force divided by area the force that the fluid exerts over a given area and the force exerted by the fluid is basically the weight of the fluid and density is mass over volume so mass is density times volume so we can replace m with rho times v and volume is basically area times height so we could cancel a so the pressure exerted by a fluid is equal to the density times the gravitational acceleration times height so we can replace p with that rho g h now this is going to be the density of fluid times the gravitational acceleration times the height now the height is going to be different at these two points so h1 is the distance between the top surface of the fluid and where the force acts h2 is the difference between a surface of the fluid and this location the bottom part of the cylinder so p1 is going to be the density of the fluid times g times h1 times a now let's factor out pf g and a so the buoyant force is equal to rho ga times h2 minus h1 now if we take the difference between h2 and h1 this will give us the height of the cylinder which i'm going to call just h and now let's get rid of some stuff so the buoyant force is equal to the density of the fluid times the gravitational acceleration times the area and then let's replace h2 minus h1 of h now a times h that's the volume so here's the equation for the buoyant force it's equal to the density of the fluid times the volume times the gravitational acceleration you may want to write down this equation now the density of the fluid times the volume of the object submerged in this case the entire object is submerged that will give you the mass of the fluid as we said density is mass over volume so density times volume is mass so m is equal to rho times v so the buoyant force is equal to the mass of the fluid times the gravitational acceleration and mg is basically the weight so the upward buoyant force is equal to the weight of the fluid that's displaced by the object and so this is the basis of archimedes principle which states that the buoyant force of an object immersed in the fluid is equal to the weight of the fluid that's displaced by the object so that expression is the basis for archimedes principle and this is how we can calculate the upward point force so based on the picture that we drew we had an upward tension force a downward weight force and an upward buoyant force and we said that the tension is the difference between the weight force and the buoying force so now to answer part b we need to calculate the buoyant force in order to do so we need the volume of the aluminum block so density is mass divided by volume if you rearrange the equation let's cross multiply so m is equal to rho v and solving for v you need to divide by the density so the volume of the aluminum block is going to be the mass of the block divided by the density of the block the mass of the block is 10 kilograms and the density is 2700 kilograms per cubic meter so let's divide those two numbers so the volume of the block is point zero zero seven cubic meters so now we can calculate the buoyant force acting on the block in air so the density of the fluid in part b which is air is very small it's 1. 29 and the volume is .
0037 times the gravitational acceleration of 9. 8 and so the buoyant force is . 047 newtons so the tension of the rope in air is going to be the weight force which is 98 minus the buoyant force of 0.
047 which is really not going to make much of a difference so the tension force is slightly less it's 97. 9 newtons so if you wish to calculate the weight of an object in air it's going to be almost equal to the true weight in this case which is 98 newtons but now let's see how much is going to change in water so the volume of the object is going to be the same however the buoyant force is different because the density of the fluid is different so the buoyant force caused by the displaced water molecules is going to be the density of the fluid which the density of water is a thousand kilograms per cubic meter times the volume of the submerged object which is point zero zero three seven times g so the buoyant force is 36. 26 newtons and that's significant so the tension is going to be the weight force which is mg that's 10 times 9.
8 or 98 minus the point force of 36. 26 and so that's going to be 61. 74 so as you can see it's a lot easier to lift up the block in water than it is in a vacuum or in air in air it's almost about 98 newtons of force that's required to lift up this block in water it's only 61.
