A 100 g block attached to a spring with spring constant 2.5 N/m oscillates horizontally on a frictionless table. Its velocity is 20 c/m when 𝓍 = ─5.0 cm c. What is the block's position when the acceleration is maximum?

Video transcript

Hey, everyone in this problem. A box having a mass of 250 g is undergoing oscillations on a frictional surface. The box is connected to a spring with the spring constant of three newtons per meter and the speed of the box is 25 centimeters per second when X is equal to negative eight centimeters. And this question is asking us to calculate the displacement of the box when the acceleration is maximum. Now we're given four answer choices. Option A 10.8 centimeters, option B 11.6 centimeters, option C negative 10.8 centimeters and option D both A and C. So let's think about this. We wanna find a displacement. When we have a maximum acceleration, we have oscillations on a frictionless surface. We have some simple harmonic motion here. So let's recall that we're gonna have maximum acceleration. OK. Occurring when X is equal to A or X is equal to negative A. So when our position is equal to the amplitude or negative the amplitude, we get that maximum acceleration. What that means is that our displacement is going to be A or negative A? OK. Remember that displacement is a vector quantity. So we have to consider the direction, which means considering the sign. So we need to consider both cases. All right. So we know where this maximum acceleration is gonna occur. We know what the displacement will be in terms of our variables. OK. Now, we need to actually calculate this amplitude A. So let's write out what we've been given. We were given a mass of 250 g. Let's convert to our standard unit. So we're gonna multiply by one kg divided by g. OK? Because we know that there are 1000 g in every kilogram. So the numerator and denominator are equivalent. It's like multiplying by one, the unit of gram is gonna divide out. And what we're doing is essentially dividing by 1000 to get to kilograms, which will be 0.25 kg. We're given the spring constant three Newton meters. So that's gonna be our K value K is equal to three newtons per meter. But we're also given a speed V of 25 centimeters per second. Again, converting to our standard unit, we multiply by one m divided by 100 centimeters because there are 100 centimeters in every meter, the unit of centimeter divides out, we're essentially dividing by 100 to get 0. m per second. OK? And this speed occurs when we have a particular position when X is equal to negative eight centimeters and just like that last conversion, multiplying by one m divided by 100 centimeters will give us zero whoops, negative 0.8 m. All right. So have these oscillations. Well, let's think about our conservation of mechanical energy because we know we have particular information about the speed at a particular position and we can work our amplitude into this equation by looking at a maximum position. So conservation of mechanical energy tells us that the initial mecha er initial kinetic energy plus the initial potential energy is equal to the final kinetic energy plus the final potential energy. Now, our potential energy is due to the spring. OK. This is on a surface. So we have no gravitational potential to worry about. And so we are gonna write you not S N U F S to indicate that we're talking about spring potential energy here. Now let's recall what each of these variables is. OK. Our kinetic energy is gonna be one half M V squared and our spring potential energy is gonna be one half K X squared. So what we get one half M V not squared plus one half K X not squared is equal to one half M V F squared plus one half K X F scored. Now, we need to figure out what our time points are gonna be. What is the initial time point gonna be? What is this final time point gonna be? Now we're gonna take the initial time point represented by these not subscripts to be when the box is moving 25 m per second and the displacement is negative eight centimeters, right? That information we were given that's gonna be our time point initially. In this final time point we're gonna take at a maximum amplitude. OK. When we have a maximum amplitude recall that the speed is gonna be equal to zero. OK? Because we're gonna reach that maximum amplitude and the box is gonna start going the other way and momentarily as it switches directions, the speed is gonna come to a stop. OK. So what that tells us is that the final kinetic energy is gonna go to zero because the speed is equal to zero. OK. On the left hand side, we get one half multiplied by the mass which is 0.25 kg. We converted that unit above multiplied by the initial velocity. And this is that velocity where we're told 0.25 m per second all squared. And then we're adding the initial spring potential energy, we have one half K which is three newtons per meter multiplied by the position at that point which is negative 0.8 m all squid. This is gonna equal one half multiplied by K which is three newtons per meter multiplied by the final position. And we said that that final position was our amplitude. We're looking at that maximum position where the velocity is zero. And so X F is actually gonna be equal to the amplitude A and that is squared. All right, let's simplify. So we can simplify everything. On the left hand side, we just have numbers, we can work this out on our calculator. We get 0. kg meter squared per second squared. OK. Because we um squared our meters per second. In this second unit, we have Newton divided by meter. Can you recall that a Newton is a kilogram meter per second squared? So this leaves us with kilogram per second squared and then we multiply it by meter squared. So we get that exact same unit. On the right hand side, we have 1.5, we had newtons divided by meters. So again, that's gonna be kilograms per second squared and that's multiplied by A squared. We're gonna divide it by 1.5 kg per second squared to isolate our amplitude squared. We get that A squared is equal to 0.116083. And we're left with the unit of meter squared. Now, when we take this square root, OK, we're gonna get two values, we're gonna get the positive or the negative root. OK? So we get plus or minus 0. m. OK? Now, when we're talking about amplitude, we're talking about just the magnitude. OK? So if we were looking for the actual amplitude, we would be taking the positive route. In this case, we are going to want both of them because we want that displacement. OK. So we're looking at both the positive and negative roots. OK. So we are going to get a maximum acceleration when the displacement delta X is equal to 10.774 centimeters or when it's equal to negative 10.774 centimeters. And you can see I've written this in centimeters or answer choices were in centimeters. Um When we did our calculation, we got meters, we have already converted from centimeters to meters. Now we're doing it backwards. So all we're doing is multiplying by 100 to get from our meters to our centimeters. Mhm And so we have these two possible displacements that will give us that maximum amplitude we were looking for. If we go up to our answer choices, we're gonna round to three significant digits. We can see that the correct answer is gonna be D OK. Both 10.8 centimeters and negative 10.8 centimeters will give us that max acceleration. Thanks everyone for watching. I hope this video helped see you in the next one.