A 12 kg weather rocket generates a thrust of 200 N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 550 N/m, is anchored to the ground. (a) Initially, before the engine is ignited, the rocket sits at rest on top of the spring. How much is the spring compressed?

Video transcript

Hey, everyone in this problem, a 1.25 kg cylinder rests at the upper free end of a vertical spring. The bottom of the ideal spring is tied to the earth. We're asked to determine the spring's compression. If the spring constant is 375 newtons per meter. We're given four answer trass. Option A 2.1 centimeters, option B 3.3 centimeters, option C 3.9 centimeters and option D 31 centimeters. So we're given some information about the mass of this spring, the spring constant. We wanna find the compression. Let's start with a free body diagram and get an idea of what forces we have acting here. So we have our cylinder and acting down on our cylinder is gonna be the force of gravity, ok? Pulling this towards the earth and acting up. We're gonna have that spring force. OK? That wants to push this cylinder upwards. We're gonna take up to be our positive direction. Now, we're told that this cylinder is at rest and if it's at rest, what that means is that the, some of the forces is going to be equal to zero when I say the forces, in this case, we're talking in the Y direction, that's the only direction we have forces. So what is the sum of forces? Well, in the positive direction, we have the spring force F S and in the negative direction, we have the force of gravity. So we get F S minus F G is equal to zero. That tells us that our spring force is equal to that force of gravity. Well, how can we write the spring force? They were called the spring force is gonna be K. And in this case, delta Y because the spring is compressing in the Y direction. OK. So we have the spring constant or force constant multiplied by the stretch of compression. And this is equal to the force of gravity which we know is the mass times the gravitational acceleration G. Now we want to solve for the spring's compression and that is a delta Y value. So delta Y is going to be equal to, we can divide by the spring constant K and write this as delta Y is equal to M G divided by K substituting in the values we know and we have a mass of the cylinder is 1.25 kg multiplied by the acceleration due to gravity 9.8 m per second squared on earth divided by the spring constant K, which we're told is 375 newtons per meter. This is gonna give us a value if you work it out on your calculator of 0. repeated meters. Ok? Remember a Newton is a kilogram meter per second squared. So the unit of kilogram cancels the unit of seconds squared cancels and we're left with a unit of meter. Now remember that the answer traces we read out were given in centimeters. We have meters. OK? We have delta Y is equal to 0. repeated meters. We're gonna multiply that and we know that there are centimeters in every meter. So we have 100 centimeters per one m. The unit of meter divides out all we're doing is essentially multiplying by 100 to get our value in centimeters, which is 3.266 centimeters. And so we found that the sprints compression is gonna be 3.266 repeated centimeters if we compare this to our answer choices and we round to two significant digits. We see that this corresponds with the answer choice B 3. centimeters. That's it for this one. Thanks everyone for watching. I hope this video helped.