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So let us look at an example where we apply the principle of work and energy.

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So what I've drawn here is a spring that's attached to the ground and on top of the spring, this little

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bucket and in the bucket is a marble say.

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And when it's unstitched this spring and the length of 100 millimetres, then we push down the spring

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until it has a compressed length of 50 millimeters, and then we suddenly leave it and it shoots up

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and the spring pushes the marble out of the bucket and the marble leaves the bucket at a certain velocity.

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And we want to know what is that velocity with which the marble leaves the bucket.

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So this is an ideal problem to be solved with a principle of work and energy because the spring applies

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work to the model and gravity also applies.

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Work to the marble and we can then plug that work done on the marble into our principle of work and

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energy equation and then we can get the final velocity.

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So we know the mass of the marble is 20 grams.

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We know the spring constant of the spring is three kilometres per metre.

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And of course we know the acceleration of gravity is nine eight one meters per second squared.

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So let us write down our equation for the principle of work and energy, that is one half of one squared,

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plus some of the work done is one of M.V. two squared.

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Therefore, one half M.V., one squid, and I've written the work done on this particle in read the

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work done by the spring minus of K X two squared minus X one squared minus.

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Now we have the work done by the gravity M.G. Y two minus one, and that is equal to our final kinetic

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energy.

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Now let's plug in our values and let's note that we use the right size for the work done.

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One of the masses, zero point zero two kilograms, the initial velocity is zero before we leave the

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spring to shoot the mobile out of the bucket.

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It's at a zero velocity, so that is zero.

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And note that I take the mass and KG.

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I want to do everything in AICI units so that the units are consistent.

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Now we have the work done by the spring nets minus one half K at like three thousand Newton Burm Kenesaw

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units.

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And now I want to.

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Take the distance over which this spring force acts, OK, now the spring moves upward and the marble

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also moves upward.

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So the spring as a positive work that it does on the marble.

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So because in our standard form, we have a minus before the work equation for a spring, this value

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in the bracket needs to be negative so that we get a positive work.

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And so therefore we have zero zero five squared, minus zero point one squared.

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So it's the initial position that's 15 mm, minus 100 millimetres.

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So that's the distance over which the spring does the work.

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Minus an hour at the work done by the gravity zero point zero two.

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That's the mass multiplied by G that's nine eight one multiplied by now in the standard form we have

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a minus in front of the work equation for gravity and gravity works downwards, our marble move upwards,

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marble moves upwards.

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So the work done by the gravity is negative.

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And so because we already have a negative in front of the equation, the value in the bracket has to

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be positive so that all work done stays negative.

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So what is the distance over which gravity does the work?

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It does it over a distance of zero comma one miner zero zero five.

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That is the final position with marble leaves the bucket minus the initial position where the spring

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started to push it up, and that is equal to our second kinetic energy.

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So that is one half multiplied by zero point zero two, multiplied by V two squared and V two.

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That is our velocity with which the marble leaves the bucket.

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That is what we are interested in.

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So calculating the values solving for V two squared receive two squared is one thousand one hundred

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and twenty four cm zero one nine meters squared per second squared.

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And if you take the square root of that, we get a velocity of thirty three point five meters per second.

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And I didn't mention this in the beginning, but I hope you thought of some of the assumptions that

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we've made in this problem, because there are some assumptions that we make in order for this to work

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out.

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The first assumption we made is that the spring is massless and that the bucket in which the model is

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put is also massless because in effect, the spring also does work on the bucket because it moves the

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bucket upwards.

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But we've ignored that for simplicity's sake.

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Then we also ignored air resistance on the back it because the resistance of air will also remove some

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energy from the system.

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It would also do negative work on the bucket as the bucket moves upwards.

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Right.

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So we ignored the force of air resistance.

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I'm going to leave it to you.

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Think about what other assumptions and simplifications we've made in the system.