Thinking Like an Engineer: An Active Learning Approach (3rd Edition)
Thinking Like an Engineer: An Active Learning Approach (3rd Edition)
3rd Edition
ISBN: 9780133593211
Author: Elizabeth A. Stephan, David R. Bowman, William J. Park, Benjamin L. Sill, Matthew W. Ohland
Publisher: PEARSON
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Chapter 3, Problem 1MDP
  1. 1. Prove the law of the lever.
Expert Solution & Answer
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To determine

Prove the law of the lever.

Answer to Problem 1MDP

The law of the lever is proved.

Explanation of Solution

Consider the lever consists of rigid bar is shown in Figure 1.

Thinking Like an Engineer: An Active Learning Approach (3rd Edition), Chapter 3, Problem 1MDP , additional homework tip  1

According to the law of lever, in equilibrium condition the torque due to effort force is equal to the torque load.

τeffort=τload (1)

Refer to Figure 1, write the formula for τeffort and τload.

τload=Fede (2)

τload=Fldl (3)

Substitute equation (2) and (3) in equation (1)

Fede=Fldl (4)

Proof:

Consider the uniform bar ABCD suspended by middle point M as shown in Figure 2.

Thinking Like an Engineer: An Active Learning Approach (3rd Edition), Chapter 3, Problem 1MDP , additional homework tip  2

Refer to Figure 2; the bar is divided into two parts as AEFD and EBCF. Consider the middle point of AEFD is K and the middle point of EBCF is L.

The force of AEFD is represented as FAEFD and the force of EBCF is represented as FEBCF. The force FAEFD and FEBCF are proportional to GI and IH respectively.

Write the relation between two forces.

FAEFDFEBCF=GIIH (5)

Consider the forces FAEFD and FEBCF are concentrated at K and L.

Modify equation (4) for FAEFD and FEBCF as follows.

FAEFD×MK=FEBCF×ML

FAEFDFEBCF=MLMK (6)

Substitute MLMK for FAEFDFEBCF in equation (5) to prove the law of the lever.

GIIH=MLMK (7)

Consider l and b is used to name the GH bar and MI bar respectively.

GH=lMI=b

Write the expression for GI.

GI=12GH+MI

Substitute l for GH and b for MI to find GI.

GI=l2+b=l+2b2

Write the expression for IH.

IH=GHGI

Substitute  l for GH and l+2b2 for GI to find IH.

IH=l(l+2b2)=2ll2b2=l2b2

Write the expression for IH.

ML=MI+IH2

Substitute b for MI and l2b2 for IH to find ML.

ML=b+(l2b2)2=b+l2b4=4b+l2b4=l+2b4

Write the expression for GK.

GK=GI2

Substitute l+2b2 for GI to find GK.

GK=(l+2b2)2=l+2b4

Write the expression for MK.

MK=GMGK

Substitute 12GH for GM.

MK=12GHGK

Substitute  l for GH and l+2b4 for GK to find MK.

MK=l2l+2b4=2ll2b4=l2b4

Substitute l+2b2 for GI and l2b2 for IH to find the ratio of GItoIH.

GIIH=(l+2b2)(l2b2)

GIIH=l+2bl2b (8)

Substitute l+2b4 for ML, and l2b4 for MK to find the ratio of MLtoMK.

MLMK=(l+2b4)(l2b4)

MLMK=l+2bl2b (9)

Compare equations (8) and (9).

GIIH=MLMK (10)

Compare equation (7) and (10), which satisfies the law of the lever.

Conclusion:

Thus, the law of the lever is proved.

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