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Photo 3: Spring 2 Force and DistanceForce vs Distance of Spring 20,40,60,81028.35X 2E-062 8 20253035Distance (m)...

Question

Photo 3: Spring 2 Force and DistanceForce vs Distance of Spring 20,40,60,81028.35X 2E-062 8 20253035Distance (m)

Photo 3: Spring 2 Force and Distance Force vs Distance of Spring 2 0,4 0,6 0,8 10 28.35X 2E-06 2 8 20 25 30 35 Distance (m)



Answers

$\bullet\bullet$ (a) At what position is the magnitude of the force on a mass in a mass-spring system minimum: $(1) x=0$, (2) $x=-A,$ or $(3) x=+A ?$ Why? $(b)$ If $m=0.500 \mathrm{~kg}$ $k=150 \mathrm{~N} / \mathrm{m},$ and $A=0.150 \mathrm{~m},$ what are the magnitude of the force on the mass and the acceleration of the mass at $x=0,0.050 \mathrm{~m},$ and $0.150 \mathrm{~m} ?$

So there is an object off and this problem object off mass m and it disconnected by a spring. It is kept like this. Sorry for this. On the let's face, it ties to a fixed support. And then they're saying that this support, which is Jed, just fixed. The object is fixed for the spring and the massive spooled at a distance. So the mass has pulled to some distance and the masses pulled by a distance. It say's stretch initially ex not from this initial position. So it philosophy it back and forth and simple harmonic motion. Obviously, at this point, at this point, the spring will have elastic potential energy, which is maximum half gain or X squared. And this object will come moment, Lee to arrest them for the kinetic energy is going to be zero. So kinetic energy is zero. And the elastic potential energy at this point is this. When it comes to the middle point here, it will have zero elastic pretension and evil. Kinetic energy is going to be half m B square and then it will come to some place here and we'll continue our starting back and forth. So we need to draw the potential energy. So since potential energy is given by half key X squared, sir, it has got a parabolic function. So well, Joy draft for this here like this. So the graph is going to look like this, You know, a parable like function here and this is you affects and on this side is going to be displacement, so they're just resumed. Their disappoint is X not, which is the maximum displacement. So now we can have the object starting from here. And then let's say somewhere in the middle, you know this position will be kinetic energy, and then this will reflect there's potential energy. So the object then released will gain kinetic energy and loose potential energy until it races. They could be in position at this point where it will have maximum kinetic energy and maximum speed. Then it continues to move toward the left. You know, in this election losing can it take energy and gaining potential energy until unless it Lee reached the same point on the left, which is X, not our disposition. And the motion reverses. So it will continue in this austerity motion between exit goes to zero and X equals two extensions in the graph. You know, off the object, the potential energy you versus exits looks like

So we're first going is apply the conservation of momentum and we can say that I'm someone v sub one plus amps up to visa to is equaling zero. Now, here we know that then visa to is going to be equaling two negative these of one multiplied by EMS of one divided by m sub too. And so we know that the surface is frictionless and so we can apply the conservation of energy where we can say the kinetic energy sub one final plus the kinetic energy sub to final Plus, the potential energy of the spring final is equaling kinetic energies of one initial class kinetic energies of two initial and then again, plus the potential energy of the spring initial. Now we know that the Connecticut and G er sub one initial is equal in Connecticut and Jesus to initial and equaling potential energy of the spring final. These are all zero, and so we can then substitute and say 1/2 m someone v sub one square plus 1/2 m sub to Visa two squared, equaling 1/2 times k The spring constant times d squared d being the compression of the spring and so we can then cancel out the haves and say that, um, solving we can say that. Ah, rather First, we're going to factor out m sub one so we can say m someone multiplied by one plus m sub one over m sub two multiplied by V, someone squared eyes gonna equal k d squared. And so we're going to say that we're gonna choose the negative sign because we know that M someone is going to recoil to the left work mother recoil two left for m someone And so we can say that then v sub one is gonna be equaling negative d The negative compression multiplied by the square root of K the spring constant. This would be divided by m sub one multiplied by one plus m sub one divided by m sub too, and so weakened then say that solving the sub one with the equaling negative 9.9 point eight centimeters or 9.8 times 10 to the negative second meters, This would be multiplied by the square root of 280 newtons per meter, divided by 0.56 kilograms multiplied by one plus 0.56 divided by 0.88 and we find that visa one is going to be equaling negative 1.7 meters per second or 1.7 meters per second to the left. We can highlight this and so then plugging this into visa to for the equation For a visa to here we can get that visa too is gonna be equaling negative negative 1.7 meters per second, multiplied by 0.56 kilograms, divided by 0.88 kilograms. And this is going to be equaling two positive, 1.1 meters per second, of course, 1.1 meters per second to the right. This would give us these up to So this would be our final answer. Our second final answer. That is the end of the solution. Thank you for watching

Block off Mass. Amazon passed to the end of the spring. So I'll drive dagger off the situation first. So we have. But this isn't the UN stress position right now, and it is gonna force constant gate. The mass is given an initial displacement. So well, stretch the mask. Some displacement here. And the displacement is X not. And a disposition. The block has given some velocity of you. Not so it time will come when the blockers release that the block is going to go to good point off maximum stretch, which is X max, and it is going to oscillate back and forth in simple harmonic oscillator. So our disposition, the energy is kinetic energy send. It is no stretch dislocation here. The energy is Canada. Bless. We have elastic potential energy on a dislocation. We have only elastic potential energy. And in part one, it's a CZ. You need to find out an equation for the max speed. So we need to see and wish location. We have the max speed. It's very obvious that the max speed is that this location because the only energy of the block here it says this is candidate energy and energy is not shared with any potential energy. So therefore, what we can do is that we can equate the energy one with energy too. So total energy at one you know is e one is equal to the total energy at two, which is you too. So energy one is equals two. The kinetic energy bitches equals to half M b Max squid and energy to which is the energy or dislocation is equals to have and Venus squared less Half came It's not square. If you solve that half and half pencils from all the sides and we are left with an equation M v Max squad is equals to em Venus squired, escape extorted, squared Since we need to find an equation for VUE Max, we divide the entire thing. But I am And our equation is V Max is equals two. I just write V Max. There's equals to a square root off. We not squared as again. See, Eminem cancels vino Dsquared. Bless Cabe. I am ex not squared. So that is our equation for V Max. All right. For part two, it says I need to find an equation for the maximum stretch So in this case, you know, we need to equate the energy at location to and location three. So hee too is equals two. Petri at 82 will have the energy half M V, North squared less half and key X squared. This is three. And at e three, we have the maximum stretch, which is equal to have game if X Max Quaid again have and 1/2 gassed from both the sides. I need to find an equation for ex Max squared so we can divide this whole equation by him. But first of all, and to simplify the question and be no risk wired plus kxan disquiet is equals. Two key ex Max squared divide this whole equation by K. And they get an equation. Ex Max squared These equals to extort squared because the scale and the scale cancel this one and we just left over with I am by K V Nord Square. So therefore, equation is ex Max is equal to squire route off. Excellent squared. Yes. And by K being r squared

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


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