Engineering Mechanics I exercises - 6 problems, Exercises of Mechanics

6 problems on Engineering Mechanics I

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1.050: Checklist Exercises
Due: Wednesday –September 12, 2007 (In Class)
MIT –1.050 (Engineering Mechanics I)
Fall 2007
Instructor: Markus J. BUEHLER
This subject, 1.050 Engineering Mechanics I, builds upon what you have learned in
your physics and mathematics courses, in particular at MIT, 8.01 and 18.01. The following
exercises, which are inspired from problem sets in 8.01 and 18.01, are designed for you to
refresh your memory after the long summer. They are also important for us to get a good
feel on what we can build together –now that we start to explore the world of Engineering
Mechanics. In all these exercises, specify your assumptions, and feel free to add any physics
quantity you may need to provide an engineering answer. Furthermore, specify all the
resources you use for your solution.
Team Building and Team Work We strongly encourage you to form Homework teams
of three students. Each team only submits one solution for correction. We expect true team
work, i.e. one where everybody contributes equally to the result. This is testi…ed by the team
members signing at the end of the team copy a written declaration that ā€œthe undersigned have
equally contributed to the homeworkā€. Ideally, each student will work …rst individually through
the homework set. The team then meets and discusses questions, di¢ culties and solutions, and
eventually, meet with TA or instructor. This …rst assignment is a good occasion for team building.
1. A waiter’s job: A new waiter is given the task of holding a tray of size 5l, with three
drinks of various weights. The drinks are distributed in the tray as displayed in the
…gure. Calculate the location along the tray where the waiter should place his hand in
order to lift and equilibrate all the items.
w
4w
w
l l l2l
Figure by MIT OpenCourseWare.
pf3
pf4
pf5

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1.050: Checklist Exercises

Due: Wednesday ƱSeptember 12, 2007 (In Class)

MIT Ʊ1.050 (Engineering Mechanics I) Fall 2007 Instructor: Markus J. BUEHLER

This subject, 1.050 Engineering Mechanics I, builds upon what you have learned in your physics and mathematics courses, in particular at MIT, 8.01 and 18.01. The following exercises, which are inspired from problem sets in 8.01 and 18.01, are designed for you to refresh your memory after the long summer. They are also important for us to get a good feel on what we can build together Ʊnow that we start to explore the world of Engineering Mechanics. In all these exercises, specify your assumptions, and feel free to add any physics quantity you may need to provide an engineering answer. Furthermore, specify all the resources you use for your solution.

Team Building and Team Work We strongly encourage you to form Homework teams of three students. Each team only submits one solution for correction. We expect true team work, i.e. one where everybody contributes equally to the result. This is testiƖed by the team members signing at the end of the team copy a written declaration that Ƭthe undersigned have equally contributed to the homeworkĆ®. Ideally, each student will work Ɩrst individually through the homework set. The team then meets and discusses questions, diĀ¢ culties and solutions, and eventually, meet with TA or instructor. This Ɩrst assignment is a good occasion for team building.

  1. A waiterƭs job: A new waiter is given the task of holding a tray of size 5 l, with three drinks of various weights. The drinks are distributed in the tray as displayed in the Ɩgure. Calculate the location along the tray where the waiter should place his hand in order to lift and equilibrate all the items.

w

4 w

w

l 2 l l l

Figure by MIT OpenCourseWare.

  1. The Ɣoating egg:

(a) When Alberto placed an egg in a container full of water, he noticed that it sank. Why was that so? Explain the phenomenon. (b) Someone told Alberto a trick to make the egg Ɣoat: add some salt to the water, mix thoroughly, and surely enough, the egg Ɣoated! If the container Alberto used had 1 liter of water, how much salt did he add to make the egg Ɣoat? Make any assumptions you think appropriate, just make sure to justify your procedure. Use the handout from the CRC Handbook of Chemistry and Physics posted in the for data on concentration of salt and mass per unit weight of water.

water salt + water

class website

  1. A drag force exercise: From the top of a skyscraper, one decides to drop a ball with an initial velocity V 0. In a Ɩrst approach, the drag force FD due to air resistance can be approximated in the form, FD = cmV where c is the drag coeĀ¢ cient, m the mass of the object, and V the velocity.

(a) Set up the di§erential equation, and solve for the position of the ball as a function of time, z (t). Use the appropriate boundary conditions. (b) Calculate the terminal velocity for the motion of the ball. Assume that the height of the building and the time to reach the Ôoor are large enough to allow the ball to reach the terminal velocity. (c) Numerical application: Using a drag coe¢ cient c = 0: 1 s^1 , plot the velocity of the ball as a function of time v (t) for three di§erent initial velocities: V 0 = 10; 100 ; 200 m/s. For this numerical application, use MATLAB to construct the plot. Inter- pret your results.

V 0

h

FD = cmV

  1. Building of a channel: You have just been hired as an engineer to design a channel for water Ć”ow. The available material has a width w, and it is bent into three straight sections as observed in the Ɩgure. Your goal as the designer is to maximize the Ć”ow of water on the channel, which is directly proportional to the maximum cross-section area.

(a) Find the optimal dimensions l and for the channel displayed in the Ɩgure. (b) A colleague of yours in the company proposes the use of a semi-circular shape for the same size of cross-section area obtained in part (a). Would this be advan- tageous? Explain your answer in terms of other engineering aspects involved in optimization problems (e.g. material cost, labor).

w āˆ’ 2 l

l l

α α