A rope of length L and mass m is suspended from the ceiling. Find an expression for the tension in the rope at position y, measured upward from the free end of the rope.

Video transcript

Hey, everyone in this problem, a free crane cable has a length of L hanging from a pulley. The L wong section has a mass M. We're asked to work out an expression for tension at a length H measured from the lower end of the cable. We're told to assume that the cable is even. We're given four answer choices. Option A H multiplied by M multiplied by G multiplied by L. Option B H multiplied by G multiplied by L divided by M. Option C H multiplied by M multiplied by G divided by L and option D M multiplied by G. We're gonna start just draw a quick diagram. So we have our pulley and we have a cable hanging from our pulley. And this cable has a link L and then mass M. Yeah. What we want to find is the tension at a length from the lower end. So you can imagine from the lower end up to some length age. Now, if we draw a pre body diagram and so we have the element at exactly that length, we have the tension acting upwards that we're trying to find and we have the force of gravity acting downwards. Now we know that this cable is stationary. OK. It's hanging, it's not moving. But then tells us is we have the, the force of the, some of the forces is equal to zero. And if we say the upward direction is positive and the some of the forces is going to be the tension minus the force of gravity and this is equal to zero. We're in our equilibrium condition. Now, in this problem, we're talking about tension at a particular length age. So let's write the tension as a function of the height H OK. Because it changes depending on where we're at. And this is going to equal our force of gravity. And recall that the force of gravity is given by the mass multiplied by the acceleration due to gravity. We know the mass of the entire cable, but that's going to be different than the mass of just this length H section. So the force of gravity is also a function of the height H. OK. So I've drawn brackets here and these brackets indicate a function of and they're not being used to indicate multiplication in this case. All right, now, we're called the force of gravity at high H is going to be the mass multiplied by the gravitational acceleration. We're gonna call the mass of our height H section little M. So the mass of the entire cable of length L is big M, the mass of this section with length H is going to be little M and so our force of gravity is gonna be little M multiplied by G. Now, how do we find the mass of that section? Hm. We can write the density of this cable. OK. The cable is even. So we can write the density of the cable as the mass divided by the link. OK. So the density of our cable row is equal to the mass big M divided by the length L. So if we're looking for the little mass, I recall that mass is equal to density times length. And we're gonna call it little L here. Now this is the same cable. So the density will be the same M divided by big L. And the length of the section we have is age. And so our mass little M of this H long section will be M big M divided by L multiplied by H. We put this into our equation for the tension we have the detention is equal to big M divided by L multiplied by H multiplied by G. And we can rearrange this to match with one of the answer traces. We can write this as H multiplied by big M multiplied by G divided by L. And this is the expression for the tension at a length H measured from the lower end of the cable. H multiplied by big M multiplied by G divided by L which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.