Each spring has an unstretched length of 2 m and a stiffness of k = 350 N/m Part

Question

Each spring has an unstretched length of 2 m and a stiffness of k = 350 N/m

Part A

Determine the stretch in OA spring required to hold the 25-kg crate in the equilibrium position shown.

Express your answer to two significant figures and include the appropriate units.

Part B

Determine the stretch in OB spring required to hold the 25-kg crate in the equilibrium position shown.

Express your answer to two significant figures and include the appropriate units.

Please provide a FBD i am trying to understand the question and how to go about as much as possible. (will rate)

Problem 3.46 Each spring has an unstretched length of 2 m and a stiffness of stiffness of k- 350 N/m Figure 1) 350 N/ m (Fiqure 1) Figure 1 12 m 6 m

Explanation / Answer

>> First writing all the co-ordinates,

A = (0,-2,0) = -2 j

B = (-2,0,0) = - 2 i

C = (6,4,12) = 6 i + 4 j + 12 k

O = (0,0,0)

>> Now, let Forces in cables are:- Toa, Tob and Toc

>> Considering Toa

>> As,it is acting along OA

OA = - 2 j

Magnitude = 2

=> unit Vector along AB = - j

=> Toa = - Toa j      ...(1)...

>> Considering Tob

>> As,it is acting along OB

OB = - 2 i

Magnitude = 2

=> unit Vector along OB = - i

=> Tob = - Tob i    ...(2)...

>> Considering Toc

>> As,it is acting along OC

OC= (6 i + 4 j + 12 k) = 6 i + 4 j + 12 k

Magnitude = [ 62 + 42 +122]1/2 = 14

=> unit Vector along OC = (6 i + 4 j + 12 k)/14 = 0.429 i + 0.0.286 j + 0.857 k     

=> Toc = Toc(0.429 i + 0.0.286 j + 0.857 k)      ...(3)...

>> Now, at point one more force is acting, weight of crate

W = - mg k

=> W = - mg k = - 25*9.81 k = - 245.25 k

>> At A, under equilibrium

=> Toa + Tob + Toc + W = 0

=> From (1), (2) and (3),

Toa(- j) + Tob(- i) + Toc(0.429 i + 0.0.286 j + 0.857 k) - 245.25 k = 0

>> Comparing Coefficients on both sides,

=> - Tob + 0.429*Toc = 0

=> - Toa + 0.286*Toc = 0

=> 0.857*Toc = 245.25

=> Solving these,

Toa = 81.845 N

Tob = 122.768 N

Toc = 286.17 N

>> Now, across Spring in OA,

Force, Toa = 81.845 N

and, stiffness, k = 350 N/m

So, Required Stretch in Spring OA = 81.845/350 = 0.234 m

>> Now, across Spring in OB,

Force, Tob = 122.768 N

and, stiffness, k = 350 N/m

So, Required Stretch in Spring OB = 122.768/350 = 0.351 m

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