The ice cube is replaced by a 50 g plastic cube whose coefficient of kinetic friction is 0.20. How far will the plastic cube travel up the slope? Use work and energy.

Video transcript

Welcome back, everyone. We are making observations about the following system here. First, we are given an incline. Let me go ahead and draw this incline, make sure it's at the right angle here and I shall give it a horizontal and a vertical as well. Now, we are told that a cast iron block which has a mass of 80 g or point oh eight kg is shot up and down. This cast iron in line here via a spring that's compressed and uncompressed at the bottom. Here, here is our block moving up and down. Now, we are told a couple of different things about the system. We are told that the incline forms a 20 degree angle with the horizontal. We are told that the spring constant is going to be newtons per meter and we are told that the cast iron incline has a coefficient of kinetic friction of 0.15. Now, per launch, we have that the spring is compressed to centimeters or 150.15 m. And we are tasked with finding how far up the incline, the cast iron block is going to travel. So how are we going to do this. Well, what we can turn to is we can turn to the work and energy theorem or the conservation of energy principle here. And here's what it states. We're essentially going to have that our final energies, the summation of our final energies is going to equal the summation of our initial energies. So what I'm writing out here is that we have our final kinetic energy plus our final gravitational potential energy plus our final spring potential energy plus our change in thermal energy is going to be equal to our initial kinetic energy plus our initial gravitational potential energy plus our initial spring potential energy plus our work done by external forces before plugging in any of these numbers here. Let's actually go over our answer choices for what the answer could be. We have a, a traveling distance of 0.59 m up the incline B four m C 1.9 m or D 0. m. Going back to our equation here, we actually have formulas for each of these terms. So let me go ahead and expand this entire equation. What we get is one half times our mass times our final velocity squared for our final kinetic energy mass times our gravitational acceleration times our final height plus one half times our spring constant times our final spring deformation squared plus our change in thermal energy, which is going to be given by UK times the normal force times our traveled distance. And this is gonna be equal to one half times our mass times our initial velocity squared plus mass times gravitational acceleration times our initial height plus one half times our spring constant times our initial deformation squared plus work done by external forces. I know that seems like a pretty large equation. But what we can do is we can actually simplify this a lot. And here's how we can do this. We know that once the block comes to a stop at some point on the incline, its final velocity is going to be equal to zero, we can also say the same thing for the initial velocity because it starts off from rest. Now the final spring deformation, well, it's just going to return back to its equilibrium point. So that means it's also going to be zero and there's no external forces acting on the block, meaning work done by external force is going to be zero. Let's say that whatever point the block starts at once the spring is fully compressed. We can just label that as our initial height. Therefore, making our initial height zero as well. When we plug these in, you'll see that it simplifies our large equation a lot here. But there's still two variables that were missing in all of this. What is the final height achieved by the block? And what is the normal force acting on the block? We have to figure out these two terms here. First, what I'm gonna do is I'm actually going to start with trying to find our normal force here. And I'm going to draw a free body diagram of our block here. So we are going to have our normal force acting in that direction. We're going to have our friction force acting to the left along or parallel to the incline. And then our gravitational force is just going to be acting straight downwards here. So let me label everything that I just said. We have our normal force up to the left, we have our friction force uh down into the left and then we have our force due to gravity right here. Now, if I draw an imaginary right triangle here, we can say that this imaginary force acting downwards is going to be equivalent to our normal force. Otherwise, the brick would either be barreling into the incline plane or flying off of it. And we know, but based on this angle right here that this angle is also degrees. So what this gives us is that our normal force is equal to our force due to gravity times the cosine of our angle 20. Wonderful. Now, what about our final height achieved fire block? It's not going to be the height of the ramp. We're not given that, but instead it's going to be whatever this height is our height of our block once its velocity is zero. Now, what is this going to be? Well, what it's going to be is it's going to be the distance traveled along the inclined plane, which is what we're trying to look for here right times the sign of our angle 20. Wonderful. So now that we've established those two variables, we have every single variable, we need to plug into this large equation up here. So I'm gonna go term by term, we'll plug in everything and we'll be able to simplify this a lot. So what do we get? What we get is zero for our uh final kinetic energy plus M G times our final height. So I'm gonna plug that in which is delta X F times the sign of 20 plus zero for our final spring potential energy plus our change in thermal energy, which is UK times our normal force, which is established to be M G cosine times our desired distance traveled. And what this is equal to is zero plus zero plus one half K times our initial deformation squared plus zero. All right. So what does this simplify to? Well, what I'm able to do is I'm able to take out our desired distance variable out of the entire left side of the equation. Then if I divide both sides by what is remaining on the left side of the equation, what we get is this, we get that our desired distance traveled is equal to our spring, constant times our initial deformation squared all divided by two times. Let's see, mass times gravity times the sign of 20 plus UK times the cosine of 20. What I'm gonna do? Let me scroll down here just a little bit and let's go ahead and plug in all of our values. So we have delta X F is equal two 20 times 200. weird divided by two times our mass of 0.8 times 9.8 for gravitational acceleration times the sign of 20 plus UK our coefficient of kinetic friction times the cosine of 20. And what we get when we plug all of this into our calculator is a total travel distance of 0.59 m up the incline plane which corresponds to our final answer. Choice of a. Thank you all so much for watching. I hope this video helped. We will see you all in the next one.