Fall 2023 4. [11 pts] For the beam and loading shown below,...

Fall 2023 4. [11 pts] For the beam and loading shown below, use the superposition method to show that the upward reaction $R_C = \frac{5WL}{8} - \frac{48EI\delta_o}{L^3}$, assuming the gap closes. You must show all of your work to receive credit.

Transcript

00:01 This problem being asked to use either a sum difference or a half angle formula to find the tangent of 195 degrees.

00:07 Well, remember, because we're not using calculator, we have to think about our angles that are on the unit circle.

00:13 And how we could get 1905 degrees.

00:16 Well, i know that 225, that's one of our nice angles on our unit circle.

00:20 And 30, that's another angle, nice angle on our unit circle.

00:23 If i take 225s and subtract 30, i'll get 195.

00:26 So i can rewrite this as the tangent of 225 minus 30 degrees.

00:31 So in this particular case, i'm going to use our difference formula for tangent.

00:35 Well, what is that? well, it says that the tangent of alpha minus beta is equal to the tangent of alpha minus the tangent of beta, all divided by 1 plus the tangent of alpha times the tangent of beta, where in this case, 225 is alpha and 30 is beta.

00:52 So let's substitute these values into our formula.

00:54 Well, that would mean we have the tangent of 2202.

00:56 25 degrees minus the tangent of 30 degrees, getting divided by 1 plus the tangent of 225 degrees times the tangent of 30 degrees.

01:06 And now we just need to simplify.

01:08 Well, i know that the tangent of 225 degrees.

01:12 Remember, its reference angle would be 45, and the tangent of 45 degrees is 1.

01:17 Well, because 225 is in the third quadrant, tangent is also positive there.

01:21 So the tangent of 225 is 1.

01:24 Then we have the tangent of 30 degrees.

01:26 Well, that's equal to root 3 over 3.

01:29 So now we can substitute these values into the denominator.

01:32 So that means we're going to have 1 plus 1 times root 3 over 3.

01:35 Well, 1 times root 3 over 3 is simply just root 3 over 3.

01:39 So we'll leave this as it.

01:40 Well, now we just need to go ahead and simplify...

Question

Fall 2023 4. [11 pts] For the beam and loading shown below, use the superposition method to show that the upward reaction $R_C = \frac{5WL}{8} - \frac{48EI\delta_o}{L^3}$, assuming the gap closes. You must show all of your work to receive credit.

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