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A2kg mass is dropped to the ground through a conducting loop. The mass starts 3 meters off the ground with zero velocity, and it lands with a velocity 7 meterslseco...

Question

A2kg mass is dropped to the ground through a conducting loop. The mass starts 3 meters off the ground with zero velocity, and it lands with a velocity 7 meterslsecond. You may take g to be 9.8 m/s^2 and ignore all other forces than gravity and electricity + magnetism: How much energy did the conducting loop dissipate (release) as heat and how do you know?

A2kg mass is dropped to the ground through a conducting loop. The mass starts 3 meters off the ground with zero velocity, and it lands with a velocity 7 meterslsecond. You may take g to be 9.8 m/s^2 and ignore all other forces than gravity and electricity + magnetism: How much energy did the conducting loop dissipate (release) as heat and how do you know?



Answers

A 1.0 g piece of copper falls from a height of $1.0 \times 10^{4} \mathrm{m}$ from an airplane to the ground. Because of air resistance it reaches the ground moving at a velocity of $70.0 \mathrm{m} / \mathrm{s}$. Assuming that half of the energy lost by the piece was distributed as thermal energy to the copper, how much did it heat during the fall?

In this problem, if we want to be able to find the kinetic energy and gravitational potential energy and each woman in time, we need functions for those quantities with respect to time. But at this very moment in time, we know how to find these two. In terms of another variable first, we have a kinetic energy With respect to velocity as 1/2 times mass tens velocity squared. And we have the gravitational potential energy with respect to height, mass. Temps gravity times height. So what we can do is replace velocity and height with functions that include time. So let's begin working on the kinetic energy. We know that the object is accelerating due to gravity. So it's velocity will simply be property times time. Therefore we can replace that in the kinetic energy equation that we know and say that kinetic energy with respect to time will be equal to one half times the mass times the expression of velocity that we have found. So that is gravity squared times time squared. Great. Now we have the function with respect to time. Let's do the same for the gravitational potential energy. Now, instead of velocity, we have the variable of height and we can build an equation for height. We can say that hide is equal to the initial height, so 300 m minus one half times the acceleration due to gravity times a times square. We can substitute that back into this equation. And we get that the gravitational potential energy with respect to time, it's equal to the mass times gravity times initial height minus one half times the mass times gravity squared because we have gravity here and we have gravity here. So we have to multiply them and then we have time square, which is what we need. Great. Now let's substitute the numbers in these two functions now that we have the expressions with the numbers. What we need is to know how much time passes between the moment we dropped the ball or the mass of the object at the moment it hits the ground. So let's calculate the total time. You have time total. Mhm. And we have this equation. We know that The hike it travels is going to be equal to 1/2 times the acceleration due to gravity times time total squared. Therefore the total time. It's useful to the square root of two times the maximum height over graphing. So that's simply still times 300 and 600 over 9.8. And that is going to give us roughly eight seconds. Okay, so now we're ready to set up a table so that we can compute the values for each second. Now, since this is a very lengthy calculation and it's just about replacing or substituting the time back into the equation. We can use some calculator. In this case I decided to use Excel, but you can use any a program. You would like to process this data and notice that we have a constant mechanical, total mechanical energy. So this table yes. Our final answer.

For this problem. On the topic of electric fields, we have a body with mass M, which carries a charge to and falls from rest a height of age above the ground near the surface of the earth. The gravitational acceleration is G. An electric field with constant component E. In the vertical direction exists. We want to find an expression for the speed V. Of the body when it reaches the ground. In terms of its mass charge, the height above the earth, the acceleration, gravity, the gravitation acceleration of the earth and the electric field strength. We then want to find the limiting cases of this expression. Now the net force on the falling objects in electric field is in some of the force due to gravity and the force due to the electric field if the falling object carries a positive charge in the force on the object due to the electric field acts in the opposite direction to the force of gravity. Now the net upward force acting on the object, F is equal to F E minus F G, which is equal to the electric force, Q times E minus the gravitational force, M times G. And by Newton's 2nd law, this must equal to M. times acceleration a. Now this corresponds to a downward acceleration A which is equal to g minus Q E over. Em now recall that the speed of an object in free fall, we can find from charismatics and we have V F squared is equal to Vinod squared plus to a delta Y. So in this case we get the final speed of the object V to be the square root of two A. H since its initial velocity is zero. And so if we substitute in this equation, we have A is equal to V squared over to age is equal to G minus Q E over em, which means that we can find a different expression for the velocity V and they've lost Tv is the square root of to age into the g minus Q E over em. So we have an expression for the velocity of this falling object with charge Q. Now next you want to know what happens in the limiting cases. Now we can see from this expression that if the value G minus QE over em is less than zero, then the argument of the square root is negative and this means the value is non real and the body does not fall.

Surgery in Chapter 29 problem 50 Lying here. So we have a square root side links 27 centimeters. It has a resistance. This is a seven point I Holmes. It says it's initially in a beef fueled 0.755 Kessler with its playing perpendicular tow the beef you. But it is a move from the field any times 40 milliseconds and we want to calculate the energy that was disappointed in this process. So the electrical energy is dissipated because there is a current that flows Google Bush actually the resistance. So we know that the the dissipation is given by the power, which is ice where are and the which means the energy dissipated. His power comes out the teeth. So this becomes ice where are guilty. So now we're gonna figure out what I is. Well, the induced D in math is given by the negative change. Then I gotta clocks with his back time and this becomes negative a time of change and be field because ace constant over changing time. So this is our e in them so that we can write the eye as being over art, which is negative, eh? don't the over delta t r So now we can square I and plug it in above to see that he is now a squared. Don't be squared over. Don't the tr never unplug? Listen so a squared is well, let's 0.8 point 27 meters squared so square It's where it is for the fourth. No, don't be is our final was zero minus our initial. So that's negative. 0.755 Tesler Square that I don't The tea was 40 milliseconds. That's 0.4 seconds and times our resistance, which was 7.5 homes. So we can plug this in and we get the energy dissipated. Waas 1.1 times 10 to the negative two rules cool.

What is the gravitational potential? Energy off a person would be a mess Him Any height h above the ground? Well, gravitational potential energy. You is defined as him times a gravitational acceleration G and H tight about the ground. So if we have an example three person off mess 68 kilograms. We know G to be 9.8 reaches per second squared and the height above the ground to me. 82 0.3 meters. This gives us a potential energy. My gravitation of the gravitational potential energy. This person that is 54 when you ate, you know Jules or 54,000 800 jewels.


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