7 a rectangular wooden beam is to be cut from a circular log...

7. A rectangular wooden beam is to be cut from a circular log of diameter d. What are the dimensions b and h in order to have the strongest beam? Assuming bending about the z axis. (b=sh=d) h b Ød

Transcript

00:01 Hello, i have a question which says that the strength of a beam with a rectangular cross -section is directly proportional to the product of its depth so let us say this is the rectangular cross -section and this is the depth and this is let us say width so given that strength is directly proportional to the depth and directly proportional to a square of its width so in this case it is definitely like this so this could be equal to as equal to k h w square where k is the constant of proportionality okay now second part of the question says find the shape of the strongest beam that can be cut from a log with a circular log of diameter so diameter is being given circular lock let us say okay like this this is circular log and diameter is being given so since this is this will be cyclic quadulateral so the diagonal must pass through the center as this is 90 degrees so this is the diameter now i have to find the value of h and w for maximum strength now i have to maximize this or minimize this and i have to use maximum minimum for this but there are two variables h and w so i have to convert it in a single variable so i will be applying pythagoras theorem in this triangle h squared plus w square will be equal to d square therefore this was w square over there so w square will be equal to d squared minus h square d is 9 so 9 square 81 minus h square now i will be plugging in the value of w square here so s will be equal to will be equal to k h into 81 minus h squared so k at 81 h 81 h minus h q now i have to find first the critical point for that d s by d h because there is only one variable now should be equal to zero so k 81 minus 3 h squared should be equal to 0 now k cannot be equal to 0 which means 81 minus 3 h square should be equal to 0 now this could be written as 3x square should be equal to 81 h square should be equal to 27 therefore h should be equal to 27 under root therefore under root of 27 which is 3 root 3 3 3 inches now, see, this is the value of h.

03:25 So, w will be equal to 81 minus h square under root, 81 minus 27 under root, 54 under root, k...

Question

7. A rectangular wooden beam is to be cut from a circular log of diameter d. What are the dimensions b and h in order to have the strongest beam? Assuming bending about the z axis. (b=sh=d) h b Ød

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