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Problem 89 Hard Difficulty

ssm A person whose weight is $5.20 \times 10^{2} \mathrm{N}$ is being pulled up vertically by a rope from the bottom of a cave that is 35.1 $\mathrm{m}$ deep. The maximum tension that the rope can withstand without breaking is 569 $\mathrm{N}$ .What is the shortest time, starting from rest, in which the person can be brought out of the cave?

Answer

8.7 $\mathrm{s}$

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Video Transcript

we begin discretion by calculating the acceleration off this person when the tension is the maximum possible. So we were using that Newton's second law. Then let me set up my reference frame as everything that is pointing up. It's positive, and everything that is pointing down is negative. Newton second bald and tells us that the net force is equal to the mass off the man times his acceleration. The net force is composed in the situation by true forces, the maximum tension force and the weight force. And these is equal to the mass off the man times his acceleration. These is 569 minus 520. These results in 49. Now to calculate the mass off that men we can use his weight No lotus that the wait Is it close to the mass times acceleration of gravity, Then the mass is the weight divided by the acceleration of gravity. So the mass off that man is equals to 520 divided by 9.8 times a and his acceleration is given by 49 divided by 520 divided by 9.8, which is a close to 49 times 9.8 divided by 520 days, is approximately 0.9 to treat meters per second squared. Now that you know his acceleration, we can calculate the time it takes for him to clean that hope up to 35.1 meters. Remember that the distance is the cost of the initial position, plus the initial velocity times that time plus the acceleration times time squared, divided by truth In the situation, we can set up his initial height to be close to zero on his final height to be 35.1, then 35.1. Zico's too. His knees are position is zero. His initial velocity is also equal to zero because he departs from for arrest. Then this is a T squared divided by truth. Therefore, he squared. Is it close to two times 35.1? Divided by eight, these is 70 point true divided by eight. So he is equals to the square it off 70.2, divided by 0.923 and these results in approximately 8.7 seconds