A spring with mass of 4 kg has damping constant 14,and a force of 3.6 N is required to keep the spring stretched 0.3 m beyond its natural length Find the mass that ...

A mass weighing 0.4 Ib stretches a spring by 3 inches. The damping constant is $c=0.4 .$ External vibrations create a force of $F(t)=0.2 e^{-t / 2}$ ib. The spring is set in motion from its equilibrium position with a downward velocity of $1 \mathrm{ft} / \mathrm{s}$. Find an equation for the position of the spring at any time $t$

Okay. Question given us we have a spring with mass sub tender tweet. It's two kg so we have spring with a mass M has to k g and damping constant c just 14. So let's say we have you damper that ISS Sequels to 14 and the force required to stretch the spring by 0.5 m is six newton. So we know force is K x so forth given us six gave the need an excess 0.5. So we get ks 64.5 It is too well, so that's que is 12. Okay. And some initial condition is given, which is required when will solve for the solution. Okay, so let's start so we know the general equation for such kind of system. Okay, so let's have this system is m the square X by D t square, where x represent displacement nt is time plus three times the x ray ditty plus okay, X equals two zoo. Okay, so now we can replace this entire thing. Buy some new variable. Let's ticketus, Let's the X by. DT is our okay. And this one will be our square since we need a solution Engine. Okay, and we're giving em as too. So this becomes two times are square. Plus, he's 14 times are plus K, which is too well. And since it's X so we don't have to substitute, so it will become one. Okay, let's do zero. So now we have a quarter take equation, which can be sold easily. Stuff us will divide by two. Since all the numbers are multiple of too. So this will become our square 14 by two is seven and 12 or 26 close to zero. So seven can be split ID. Such that addition give seven and multiplication gives six. So that is six plus one and six into one is six hold, uh, plus six. And our plus one is equals toe zero. Okay, so we give artist minus six and our s minus one. So therefore a general solution will be X with respect to time will be C one. It is two minus 60 plus C two. It is too minus one b. Okay, so now we have a solution. General one. So we need value of C one and C two. So two conditions air given first eggs At T equals to zero is point. You did so much. It's one. So that's 1 m and velocity at T equals to zero is zero okay. And a lot of cities. Nothing but different station off displacement with respect to Thanks. Okay, so let's substitute for differentiate this equation. So we live X dash, which is Seamus. The X Y deity is C one times it is to minus 60. Different station is it is to minus 60 times minus six. Similarly, here we'll have C two times It is two minus tee times minus one. Now we'll substitute this boundary conditions the first one at the zero axis 1 m. So we'll substitute that here. So it equals to zero xs one see one. It is 20 since zero time 60 and see to it is 20 So you get one is C one plus C two and second boundary condition. Here, so exist is velocity which is zero at time T equals to zero. So this will be minus six. See one It is 20 and minus C two it is 20 So this is zero minus six C one since it is 20 is one. And this is C two. Okay, so we have to candy sessions or two equations. We can solve that simultaneously and we'll get C one is minus one or five and C two s six or five. So therefore, a solution. Eggs. Respect of T is minus 1/5 years to minus 60 plus 6/5. It is two minus t and that's a final solution.

Okay, A force of three newtons is required to keep a spring exerted that 1/4 meter beyond its equilibrium and were asked for the spring constant. So I want you to consider our equilibrium and then we are extending it by 1/4 of a meter. Now, um there's the equation for force that was given earlier and that was force equals negative K. X. Well that's the springs force as you pull the force the for the spring tries to um go back to equilibrium. So the spring is pulling to the left, but you're pulling to the right in the kind of in the same direction as displacement. So we can make that a positive because we're talking about the force required to keep the spring from recoiling. Okay, so that means three equals K times 1/4. So if we multiply by four we can find K. K. Is going to equal 12. And then the figure out the units you really can go ahead and um really just consider that you have the um force divided by meters and so you have a newton perimeter

Question to a van is law, and we have like or okay, eaten 32. There you are, K Eat 32. Divided by four. I have It's coming. Hey, have a seat. He won. All right. Through my equation his eight x doesn't 1 80 x no and me eaten. See Rome factor. The and ours Here I said eight spears equals zero are spared, equals negative. Can r equals Means it's you. Um so that gives me the Asian. He he will see he's a C two someone. Except this is 01 sign of zero is zero. So I see one best. When I take the derivative you It's prime tea like negative c one truly because 10 we're 10. Sign He's the sea too. Something for my very good. I'm fine, so but I put in a one, but I'm sort of throw in my tea. No sign of zero. So this has to equal more for the meal I had. This is suitable. Have seat too. Finds this weird of 10 times one. So I divide this word of him and end up with C two equals one of us. So the General

Uh, with a C equaling two times eight times, two times the square root of K times m. Okay, equal Teoh 16 £16 divided by 1/4 of foot, which is need hold 64 would be each pulled Teoh 16 kind of my 32 people have. So this is equal to 200 square with three to just gonna be a Times Square.