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PROBLEM:A block of mass m = 8 kg is attached to an elastie spring of force constant k = 4000 Nlm as shown in the ligure:The spring (with the block attached) stretch...

Question

PROBLEM:A block of mass m = 8 kg is attached to an elastie spring of force constant k = 4000 Nlm as shown in the ligure:The spring (with the block attached) stretched by 4 = 0.10 m and released. The friction force acting 0n the block by the surface is Fr = 40 N_ a-) Find the work done by the friction force as the spring returns from the stretched position (x = 0.10 m) to equilibrium position (x = 0): b-) How much the mechanical energy decreased as the spring returns from the stretched position (

PROBLEM: A block of mass m = 8 kg is attached to an elastie spring of force constant k = 4000 Nlm as shown in the ligure: The spring (with the block attached) stretched by 4 = 0.10 m and released. The friction force acting 0n the block by the surface is Fr = 40 N_ a-) Find the work done by the friction force as the spring returns from the stretched position (x = 0.10 m) to equilibrium position (x = 0): b-) How much the mechanical energy decreased as the spring returns from the stretched position (x = 0.10 m) to equilibrium position (x = 0)? c-) Find the speed of the block as the spring returns to equilibrium position (x = 0). Xi Xf



Answers

A block of mass $m=2.00 \mathrm{kg}$ is attached to a spring of force constant $k=5.00 \times 10^{2} \mathrm{N} / \mathrm{m}$ that lies on a horizontal frictionless surface as shown in Figure $\mathrm{P} 13.8$ . The block is pulled to a position $x_{i}=5.00 \mathrm{cm}$ to the right of equilibrium and the right of equilibrium and released from rest. Find (a) the work required to stretch the spring and (b) the speed the block has as it passes through equilibrium.

So here in this equation we will first write the equation of acceleration of the block. So excellence and a. Of the blood is equals two. Beauty minus you divided. Um Thanks this is the Excellence. Any question so we will substitute Excellence and A Z. Equals two. Really very very where E. X. This will be equals to music witness the X divided way. Um Uh huh. We can write really is equals two music minus motivated by M. X multiplayer. The way he X. So we can integrate it. Um Zero to be U. T. V. This will be consumed integrating from export to X. Music minus divided by M. X. The X. So after integration we will get this clarity very well too equals two mule Beauty X minus the excess clarity where I am minus museum X. Note minus the excellent square divided way. Let this question do whatever to him there. It will be to him To let this question question #1 Chris in the Middle one the latest substitute the values here in this equation we have final village tv is given a zero so zero divided by two Is equals to the value of new is given a 0.4 When we played with the is 9.81 x minus the value of K. Is 1 50. So it will be 1 50 Access. Were debated where to? The value of Masses five. So this minus The value of μ 0.4 multiplied by the is 9.81 Multiplied by excellent. Now the x note has given us 200 of them. So it will be 0.2 m minus K. The value of case 1 50 multiplied by X notice correct notice 0.2 m Square where there were two multiplied where MMS five. So after calculation we will get 0.18 would it? This the Kleenex is good minus three point And it's three 924 X 9 to 4 at is equals to zero. So solving the root for the ever be person we will get X is equal to 200 or 61 point six. And So now the spring has any sell a formation of 200. And this comes to completely stop at 61.67 the final diplomacy and at with the block come to rest his 61 point six mm. This is the this is the finally formation when the four missing you for missing at which block comes to it, based at which blog comes tourist.

Well, most of the block uh um is equal to 20 kilograms, 20 kilograms and sipping constant K is a cool too four Qilong Newtons per meter and the super is stretched to a distance. Um, de is a call to 10 centimeters beyond beyond its origin in the land, An initial speed of the block zero we I is equal to zero meter per second. So initial speed off the block is zero and the force of friction is a cold tour. Uh, 18 Newton's gets a call, too. 80 Newtons. Okay. No, uh, from the conservation of mechanical energy off the block, Uh, change in kinetic energy less, uh, changing potential energy. Plus, um, energy due to friction is equal to Zito and changing kinetic energy can be written is give minus K. I gave minus K II and changing potential energy is uh huh. Okay. D square minus, huh? Okay. Divan square Devens were less An energy, uh, due to friction can be written is force of friction. Uh, four. So fiction times the displacement acts and it's equal to zero now, putting the values we have kinetic energy final um minus zero. Initial lost zero therefore initial kinetic energy zero plus one. Divided by do is get one divided victor gaze common and we have the square minus d one square inside the bracket less the force of friction. Times X is equal to zero, all right. And no putting the values we have. Finally, kinetic energy. Ah, plus one divided by two multiplied by K is for killer mutants. Four multiplied by 10 to the power three, Newton's and D Uh, in this case is 0.0. It, uh, scare minus divan is equal to 0.1 square Less force of friction is 80. Newton's multiply by X is zero point there too. And it's equal to zero, all right, and therefore kinetic energy final. Um, kinetic energy final is equal to 5.6 Jules. So what? We have 5.6, Jules, uh, which is the final kinetic energy. So this was parking? No. Let's old Barbie sold part B again. Um, we have are chanting kinetic energy plus change in potential energy. Bless, uh, energy due to friction. Is it going to zero and no, the block. In this case, moves are 10 centimeters and the final elastic potential energy off the soup ring is zero because it returns to its, uh, original position. And therefore, we have, uh, g f minus. Can I, um, bless. One divided by two K. Um, here D is zero in this case. Well, zero square minus one, divided by two Gay. Do you even spare, uh, bless? Force of friction. Dimes X. Um, in this case, it's not X, but it is d I It is Ah Tiu i onda. This is equal to zero d I and no, uh, d eyes actually called 0.1. No meters. No, we have, uh, game F uh, minus zero plus zero. Um, minus one. Divided by two K um, D in the show. Sodini Shield. Oh, spare, um bless force of friction times. Ah, dee initial is equal to zero. And therefore we have, um ah, Finally, kinetic energy. Um, minus one. Divided by two. Um, multiply by our keys. Formal could like I 10 to the power three. Um Then we have the high 0.1 scare. Less course affection is 80 and the eyes equal to 0.1 and it's equal to zero. And finally be happy. Ah, Finally, kinetic energy, and it's equal to 12. Jules, now mix old Bart. See, um, well, the relationship between the gannet peak energy of the block and the distance it moves is, um, we have finally kinetic energy plus one divided by two K, uh, into d initial minus ex hold square minus one. Divided by two oh times gay times d Initial scare Bless. Ah, force of friction Times X is it cool to zero on. Duh. Get it Is equal to minus one. Divided by two. Okay. Into D I minus X whole square Bless one divided by I do. Cool. Do I, uh, square minus force of friction times eggs. All right. On DDE. No, we need to ah, differentiate this equation with respect to X. And therefore we have ah dee. Okay. Divided by D X now differentiating. Okay. After respect, tow X. We have, um, guess is equal to, um two multiplied by two. Multiplied by one. Divided by two, uh, times. Okay. Into D I minus eggs. Um, minus force of friction. Um, sins. Um, it's equal to zero, and therefore we have zero is equal to key into d I minus X minus or so friction on from this equation. Um, we have Dante 18 is equal to four multiplied by 10 to the power tree into a 0.1 minus X just putting the values in this equation. We have this equation know solving for X, we have X is equal to 0.1 minus minus 18 divided by four multiplied by 10 to the power train and therefore X is equal to 0.8 meters. So X is equal to a 0.8 uh, meters, no, uh, these X is equal to 0.8 corresponds to the maximum kinetic energy off the block. And, uh, dude, for a maximum Can I dig energy? Uh, key maximum. Is it called to minus one? Divided by two. Multiply by four multiplied by 10 to the power trade multiplied by 0.1 um minus 0.8 Holds here. Bless one divided by two. Multiply by four multiplied by 10 to the power three multiplied by 0.1 square minus 80 multiplied by zero point 08 And solving this. We have maximum kinetic energy and it's going to 12.8 jobs. And if the problem thank you for watching

For this problem on the topic of energy conservation, we're told that a block of mass one kg, is resting against a light compressed spring at the bottom of a rough plane. The plane is inclined at an angle of 30° to the horizontal, and the coefficient of kinetic friction between the block and plane is 0.1. The spring is compressed 10 cm from its equilibrium length, and when the spring is released, The block separate from the spring and slides up the incline incline, a distance of two cm beyond these springs equilibrium length before coming to rest. We want to find the change in total mechanical energy of the system and the spring constant K. Now, since this is not a conservative system that change in total mechanical energy can be related to the energy lost due to friction. This energy can be determined by calculating the work done by the force of friction, which we'll call w friction, and this is equal to the frictional force times the distance over which the friction acts, which is the distance that the block moves up the plane D. So we can write this as the coefficient of kinetic friction. UK times the component of the weight parallel to the plane. Mg. Call sign theta times D. And so the change in total mechanical energy of the system is minus the work done by friction. And so this is equal to minus UK times MG times co signed data, times deep. So this is equal to minus the coefficient of kinetic friction. 0.1 times the mass. One kg Times the acceleration due to gravity 9.81 m. The square second times The call sign of 30°. Multiplied bye. 12 times 10 To the -2 m or 12 cm which gives the change in mechanical energy To be -1 02 times 10 To the -1 jules, which is lost to friction. Now for part B from conservation of energy, the change in total energy that we determined above is equal to delta cape style to you. Now, since K is equal to zero about the initial and final points, it follows that the change in total mechanical energy is equal to the change in gravitational potential energy. You f minus you I which is equal to M G H, which is M G. D. Science data minus the initial potential energy, which is elastic, half K delta al squared. And so from here we can rearrange and solve for the spring constant K. And we get K two B to into M G. D. Scientific data minus the total change in mechanical energy divided by the compression of the spring data, L squared and all of these values are known. So this is minus two into the mass one kg times G, which is 9.81 m per square second Times the distance d, which is 0.12 m times the sine of 30 degrees minus the change mechanical energy, which is minus zero point 1019 jewels, all divided by The compression of the spring, 0.1 m squared. This gives us the spring constant of the spring to be 138 newtons per meter.

Friends. This is the problem based on conservation off energy. Here it is given a block off musk while five kg attends toe one end off the spring off spring constant 18 u turn permitted spring is fixed with the support kept over the inclined plane having the friction fine too. If the block is to be compressed so that the compression in the spring is okay. Compression in the spring is yeah, 10 centimeter that iss quite 1 m. Say this position Toby V on from this petition it is to be released from compressed position. Yeah, Block is released in the first part. We have took fire. The potential energy storing this pink in the second part were to calculate where the city off the block when spring. Neither. Be compressed. Uh huh. Extended. That is at position A and she part. We have to fight it. The position of the block. Then its velocity becomes zero. That is at this position I can draw. Had to see. Yeah, let us start solving first part potential energy. Storing the spring with me half the excess way. Cake X is the compression In the spring, Etienne took quite valid script so participative it He called a quiet for too. This is the answer apart. A now in part me. We have to find out velocity at eight. So we have to apply the contribution off. Energy initial kind of technology initial spring energy and the self gravitational energy is going toe final Kinetic energy final is spring energy final gravitational energy and worked on by friction. Can you see it in Asia? Kind of technology? Zero This Already we have calculated point for this is zero. Finally, we're finding acquisition eight. It is zero. It will be m g X 80. Okay, Yeah, yeah. Sign of theater plus new energy. X b A cause some critters Absolute Developed. Yeah, Mass off the block is 0.5 square. This is Europe and 0.5 G 9.8 XB. Appoint one sign off. 30 fellas, new point. Look. 0.59 point eight white. What? Cause of 13. So on solving it, you will get velocity at. Came to me? Yeah. Why? 5296 meter per second. At what decency? Well again, Applied contribution off energy from petition. Yeah, B to C. Yeah. Can you take energy at v Spring Energy at me reputational potential energy at me Bourbon by fiction at me It's got to do Can it technology at XY? A spring energy at sea, gravitational potential, energy, etc. And worked on by friction energy Let yeah, easy to be. It's got toe X in each planet. Energy at these Europe spring energy is quite portal potential energy zero kind of technology at seat against Europe Potential energy in the spring half k access square gravitational potential energy MGI ex compression plus x Hm. Sign off theater Mhm plus new energy Ex compression plus X cause Some theater substitute the video. Mhm. Okay, off X mask eight. Sorry. K is given 80. Yeah, 0.5 to 9.8. Quite one plus x. Sign up. 30. Why? To 559.8, right? Yeah. And do 0.1 plus x into cause off 30 The solving ever equation. What x You will get white 1694 meter. That's all. Thanks for watching


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