How Do You Determine Reactions at a Fixed Support in Statics?

[javaistheman]

I'm currently reviewing for a final tomorrow, I need to score a B on the final to stay in school. I'm beginning to panic as I've reviewed all day and have gotten nowhere.

1. Homework Statement

The tension in the cable AB is 800 lb. Determine the reactions at the fixed support C.

The Attempt at a Solution

So I'm basically trying to get a fundamental grasp of these problems and just can't seem to do it. I just need to know the logic behind each step. I have a solution key that is lacking the steps, and the steps are what I need.

Firstly, they write out the equation: 800*[(2i-4j-k)/sqrt21]. I'm assuming that these position numbers came from the difference in distance of B from A. But why is it being divided by the square root of 21? And what piece of information does this equation actually give you?

Next they write out rCA = 4i + 5k. I assume here they are getting the distance component of the CA beam.

The next step, they suddenly have all three component of C figured out. How?

Then in the final step, they have the moment figured out for each component of C. Now, I know to get to here from the previous step you'd multiply the magnitudes of the components by the perpendicular distance. Would it be the perpendicular distance from C to A? I assume this because the force is coming out of point A.

If anybody could give me clarity on these things, I'd be grateful. I don't know where else to turn.

 

[SteamKing]

I'm currently reviewing for a final tomorrow, I need to score a B on the final to stay in school. I'm beginning to panic as I've reviewed all day and have gotten nowhere.

1. Homework Statement

The tension in the cable AB is 800 lb. Determine the reactions at the fixed support C.

The Attempt at a Solution

So I'm basically trying to get a fundamental grasp of these problems and just can't seem to do it. I just need to know the logic behind each step. I have a solution key that is lacking the steps, and the steps are what I need.

Firstly, they write out the equation: 800*[(2i-4j-k)/sqrt21]. I'm assuming that these position numbers came from the difference in distance of B from A. But why is it being divided by the square root of 21? And what piece of information does this equation actually give you?

√21 = length of the vector (2i - 4j - k). You divide the components of this vector to convert it to a unit vector, which is then scaled by the magnitude of the force (800 lbs.) to turn it into a force vector.

Next they write out rCA = 4i + 5k. I assume here they are getting the distance component of the CA beam.

r_CA = 4i + 5k is actually the vector drawn from point C to point A. It's the moment arm for the force vector applied a point C

The next step, they suddenly have all three component of C figured out. How?

What components are you talking about here? The reaction at C?

Then in the final step, they have the moment figured out for each component of C. Now, I know to get to here from the previous step you'd multiply the magnitudes of the components by the perpendicular distance. Would it be the perpendicular distance from C to A? I assume this because the force is coming out of point A.

If anybody could give me clarity on these things, I'd be grateful. I don't know where else to turn.

It's not clear what calculations have been done because you don't provide them.

Since the beam CA is in static equilibrium, ΣF = 0 and the ΣM = 0. The only moment applied to the beam is due to the force vector at point C, and M = r_CA × F,

where × denotes the cross product of the two vectors, r_CA and F.

Because C is fixed, there will be a force and a moment produced here as reactions due to the force applied at A.

Good Luck!

 

[haruspex]

write out the equation: 800*[(2i-4j-k)/sqrt21]

It's an expression, not an equation.
As you observe, the vector (2i-4j-k) is the relative position of B from A. How long is that vector?

write out rCA = 4i + 5k. I assume here they are getting the distance component of the CA beam.

It's the relative position vector from C to A. Distances are scalars.

The next step, they suddenly have all three component of C figured out.

Not sure what you mean by 'C' there. The components of C as a position are (0, 4, 0). Do you mean the torque vector about C of the tension in the rope?
If you have a vector representing the position of A relative to C, and a vector representing a force through A, what would you do to find the torque the force exerts about C?

 

[javaistheman]

It's an expression, not an equation.
As you observe, the vector (2i-4j-k) is the relative position of B from A. How long is that vector?

It's the relative position vector from C to A. Distances are scalars.

Not sure what you mean by 'C' there. The components of C as a position are (0, 4, 0). Do you mean the torque vector about C of the tension in the rope?
If you have a vector representing the position of A relative to C, and a vector representing a force through A, what would you do to find the torque the force exerts about C?

Alright so I now see that the 21 came from squaring each of the 3 units vectors and adding them.

When I was talking about the components of C, I meant Cx, Cy, and Cz. Which I guess is the force at point C in each direction. So basically they multiplied the 800 lb force by the scalars to get each component. Then I now see that they used the rCA numbers and did a dot product with the C components to find the moments.

But what is the logic behind finding the difference in distance for B? I understand the steps now but I'd like to know the actual reasoning and theory behind them.

 

[haruspex]

they ... did a dot product with the C components to find the moments.

I hope they did not.

is the expression with 800 being multiplied by the difference in distance giving you the force at point A?

You're not including the sqrt(21) divisor in that. If you divide a vector by its magnitude, what is the result usually called?

 

[javaistheman]

I hope they did not.

Really? I thought it went like this:

I divided out those scalars to get this: 0.436i - 0.873j - 0.218k.
Then I multiplied each of those by 800 to get the components of C: Cx= 348.8, Cy= 698.4, Cz= 174.4
Then I used the dot product formula to find the moments: Since rCA = 4i+5k, Mcx for example would be = RyFz - RzFy = (0)(174.4) - (5)(698.4) = -3490 ft*lb.

That's not the right process?

 

[haruspex]

Really? I thought it went like this:

I divided out those scalars to get this: 0.436i - 0.873j - 0.218k.
Then I multiplied each of those by 800 to get the components of C: Cx= 348.8, Cy= 698.4, Cz= 174.4
Then I used the dot product formula to find the moments: Since rCA = 4i+5k, Mcx for example would be = RyFz - RzFy = (0)(174.4) - (5)(698.4) = -3490 ft*lb.

That's not the right process?

The right process but the wrong terminology. What you describe is a cross product.