A uniform 2.00-m ladder of mass 9.00 kg is leaning against a vertical wall while making an angle of 53.0° with the floor. A worker pushes the ladder up against the wall until it is vertical. What is the increase in the gravitational potential energy of the ladder?

Video transcript

Welcome back. Everyone in this problem. A uniform plank 5 m in length and weighing around 12 kg rests against a smooth surface at an angle of approximately 30 degrees to the ground. If the plank is pushed to a vertical position, how much will its gravitational potential energy increase? A says 100 and 47 jewels B two point sorry 294 jewels, C 392 jewels and D 588 jewels. Now, what are we trying to figure out here? Let's try to sketch a diagram to understand what's going on. So we have a plank. OK. That's resting against the wall. This is our plank and we're told that it's pushed to a vertical position. So we can imagine when vertical or plank looks like this. OK. And now we're trying to figure out how much its gravitational potential energy would increase by for this plank. Sorry, that is approximately at an angle of approximately 30 degrees to the ground. So how can we figure out what that gravitational potential change will be? Well, recall, OK, that gravitational potential energy U equals the mass multiplied by the acceleration due to the gravity g multiplied by the height. OK. So that means the change in gravitational potential energy is going to be equal to our mass. In this case, our mass stays the same multiplied by our gravity also a constant or gravitational or acceleration due to gravity, sorry, which is a constant multiplied by our change in height. Because that is what changes in this scenario when we move it from being from, from resting against our surface to a vertical position. So if we can figure out what that change in height will be for our plank, then we should be able to figure out its increase in gravitational potential energy. How can we find that? Well, we need to identify the initial and final positions for the center of gravity and the center of gravity for any uniform rod lies at its midpoint. Now, when our rod is resting against our wall, we could say that its midpoint is here. And when it's vertical, we could say that its midpoint is here. So really what we're trying to figure out then is our difference between the height for both of these midpoints which on this diagram here. OK. Let me put that in red. On this diagram, we can refer to as delta H No, what will those heights be? Well, we were told that our plank is 5 m in length. So for our vertical plank, we know that its height is going to be 2.5 m. And for our plank resting against our wall, we know that the height of its midpoint will be the vertical component of the right triangle that is formed here. In other words, for our midpoint, the length at our midpoint, it's going to be the Y component of that length midpoint length. So how can we figure that out? Well, we already know it's 2.5 m. And if we look at it, this length is opposite to a 30 degree angle. And since it forms a right triangle, that means the vertical component is going to be 2.5 m multiplied by the sine of 30 degrees, the sine because it's opposite to our angle, know that we have our heights, we can find the difference in heights because this tells us then if we come over here, that means our change is going to be equal to the mass multiplied by the gravity multiplied by 2.5 m minus 2.5 m multiplied by the sine of 30 degrees. Let's go ahead and substitute what we know. We know that our plank is 12 kg. We can take our acceleration due to gravity as 9.8 m per second squared and finding the difference between 2.5 m and 2.5 m multiplied by the sine of 30 degrees. OK. Then we should get our change in height to be 1.25 m. And if we throw all of this into our calculator. OK. Then we should get the increase in gravitational energy delta U to be 147 joules. Therefore, a is the correct option. Thanks for watching everyone. I hope this video helped.