Two blocks are attached to opposite ends of a massless rope that goes over a massless, frictionless, stationary pulley. One of the blocks, with a mass of 6.0 kg, accelerates downward at 3/4g. What is the mass of the other block?

Video transcript

Hey, everyone in this problem, two buckets of sand hang on a frictionless pulley. They are connected with a massless cable such that if one bucket moves up, the other moves down the bucket moving downward has an acceleration of 0.6 G where G is the acceleration due to gravity and a mass of 14 kg. We're asked to determine the mass of the bucket moving upwards. We're given four answer choices. option a 5.6 kg option B 3.5 kg, option C 10.5 kg and option D 4 kg. Now let's start by drawing a little diagram. So we have a pulley, it's frictionless and we have a massless cable that connects two buckets. So we have a bucket we're gonna draw on the left hand side of our police and a bucket on the right hand side of the pulley, we have one moving down and one moving up. So we're gonna say that the one on the left is moving downwards, the one on the right is moving upwards. The mass of the downward bucket we're told is 14 kg. And we wanna find the mass of the upward bucket and we're gonna call that M U. Now the bucket moving downwards, we know it has an acceleration of 0.6 G. We're gonna take up to be positive in our problem. And so this acceleration because the buckets moving downwards opposite our positive direction is going to be negative 0.6 G. Now we're looking for the mass of a bucket, we have some information about the acceleration. So let's draw off pre body diagrams for our downward bucket. We have this tension of this cable acting upwards. We're gonna call this T D intention on the downward bucket. And we have the force of gravity acting downwards. We're gonna call that F G D the force of gravity on the downward bucket. And similarly, for the upward bucket, we have the tension, we're gonna call it T U acting upwards for this upwards bucket and the force of gravity acting downwards. And I'm gonna draw again our what that we're taking up to be positive? Same as our little diagram with our buckets. Now, like I said, we had information about acceleration. We've drawn some forces out here. We're looking for a mass. So let's recall Newton's second law which tells us that the sum of the forces is equal to the mass multiplied by the acceleration. OK. So this lets us relate that mass in the acceleration. So if we apply Newton's second law, we can write this for our downward bucket and our upward bucket. So the song of forces for our downward bucket, it's going to be equal to the mass of the downward bucket multiplied by the acceleration of the downward bucket. And similarly for the upward bucket, some of the forces of the upward bucket is equal to the mass of the upward bucket multiplied by the acceleration of the upward bucket. Now, which forces do we have, well in the positive direction, we have the tension acting upwards for both the downward and upward bucket. And we have the force of gravity acting downwards in the negative direction. And so our sum of forces for the downward bucket is T D minus F G D and this is equal to the mass MD multiplied by the acceleration ad and similarly for the upward bucket, the tension to U minus the force of gravity, F G U is equal to M U multiplied by A U. Now we're gonna isolate for the tension in both of these equations. The tension T D is equal to MD multiplied by AD plus F G D. And similarly, T U is equal to M U multiplied by A U plus F G U. We're gonna call the equation that is T D is equal to M MD multiplied by AD plus F G D equation one. And the corresponding equation for the upward bucket T U equals M U multiplied by A U plus F G U is going to be equation two. Now let's think about this pulley that we have. OK. We have the pulley, we have this massless cable connecting our two buckets. We have the same cable connecting both buckets, which tells us that the tension is going to be the same for both buckets. OK? It's the same cable. The tension must be the same if the tension wasn't the same, that cable would snap or break. And so T D must be equal to T U if T D is equal to T U and we can substitute our equations. So let's substitute equation one into equation two. OK. We're essentially setting these two equations equal because T D is equal to T U. We get MD multiplied by AD plus F G D is equal to M U multiplied by A U plus F G U. Now what is F G the force of gravity? They recall that the force of gravity is given by the mass multiplied by the acceleration due to gravity. So we get MD multiplied by AD plus MD multiplied by G is equal to M U multiplied by A U plus M U multiplied by G. Now, we know the mass of the downward bucket. We know the acceleration of the downward bucket. We know the acceleration due to gravity. G, we're looking for the mass of the upward bucket M U but we don't know the acceleration A U. That's the only value we don't have that we need in order to find M U. Now, just like the tension of the downward bucket was related to the tension in the upward bucket. The accelerations are related as well. They're connected again to one cable. So if the downward bucket is moving with a particular acceleration down, then the acceleration is or the upward bucket is going to be accelerating with the exact same magnitude of acceleration, but in the opposite direction. OK. So it's accelerating upwards at the same rate that the downward bucket is accelerating downwards. What this tells us is that the acceleration A U is equal to negative the acceleration ad a same magnitude opposite direction. So now we can use M or AD in our equation instead of A U since we know the value of AD, so we can write our equation as MD multiplied by AD plus MD multiplied by G is equal to M U multiplied by negative AD plus M U multiplied by G. Now we're trying to isolate for M U. OK. So let's factor on the right hand side, we have MD multiplied by AD plus MD multiplied by G is equal to M U multiplied by G minus AD. And so solving for M U, we have M U is equal to MD multiplied by AD plus MD multiplied by G divided by G minus ad. And now we can substitute in our values. And we found this expression for M U. We've done all the hard work. We're gonna substitute in what we know the mass of the downward bucket is 14 kg. So we have 14 kg multiplied by the acceleration. OK. Well, this was negative 0.6 times G which is 9.8 m/s squared plus again, the mass of the downward bucket 14 kg multiplied by G 9.8 m per second squared divided by G 9. m per second squared minus the acceleration ad which is negative 0.6, multiplied by G 9.8 m/s squared. Now, in each of these terms, we have a 9.8 m per second squared. So we can divide the entire equation by 9.8 and those terms will divide out. So we have 14 kg multiplied by negative 0.6 plus 14 kg divided by 1 -2.6. We substitute this into our calculator. We get that the mass of the upward bucket is equal to 3.5 kg. And that is what we were looking for the mass of the upward bucket in our system. We found that that mass is equal to 3.5 kg which corresponds with answer choice B. Thanks everyone for watching. I hope this video helped see you in the next one.