Find the tension in each cable

[CollegeStudent]

Homework Statement

A lamp has a mass of 18.5 kg and is suspended by two cables attached to the ceiling. The cable on the right makes an angle of 42.2° with the ceiling and the cable on the left makes an angle of 54.4° with the ceiling. Find the tension in each cable? Why is the sum of the two tensions greater than the weight of the lamp fixture?

Homework Equations

weight = mass * gravity

The Attempt at a Solution

So for the tension...(after drawing a sketch of what it would look like)...i ended up with the weight being 181.3kg. There's 2 cables so 181.3/2 = 90.65N. Since each cable needs to support 90.65N, it looks like 2 right triangles, with 90.65N being the opposite side, and us wanting to know the hypotenuse.

so i did sin(42.2°) 90.65/t and sin(54.4°) 90.65/t (where t is tension)

and got 135.0N and 111.5N respectively.

The sum of these 2 tensions is 246.5

Now I just can't understand why exactly they are greater than the weight of the lamp. I don't think it has anything to do with gravity because we already took that into account when figuring out the weight with respect to gravity.

Any hints?

 

[dang3r]

Hint: Think about how the tension in each cable is distributed with respect to your coordinate system. We know the direction gravity acts in, but what about the tensions? How is it distributed in the X and Y directions? Drawing out a FBD will help!

 

[CollegeStudent]

hint: Think about how the tension in each cable is distributed with respect to your coordinate system. We know the direction gravity acts in, but what about the tensions? How is it distributed in the x and y directions? Drawing out a fbd will help!

I'm not sure how other users draw pictures and put them on here...i just used paint. If there's a better way please tell me...nonetheless here's my FBD

Weight = 18.5 * 9.8 = 181.3

I have the angle that would make me use the sin function...if I'm correct.

So as calculated above...the tension in the right cable will be 135.0N and the left cable 111.5N

So you're saying that I should find the components?

Okay ...well I know the y component will be the 90.65 correct?

So for the x components it will be cos(54.4) * 111.5N and cos (42.2) * 135N

Okay so that comes out to

The left cable's x component will be 64.90N and the right cable's will be 99.97N



Oh wait ...are you hinting at the fact that we don't know the dimensions of the picture in the horizontal direction?

 

[dang3r]

I think you are over-complicating things a little bit. The question is 'Why is the sum of the two tensions greater than the weight of the lamp fixture?'

You understand that for the lamp to have a net force of 0 in the Y and X directions, the sum of all forces in those directions is 0. Realize that this means the sum of the Y components of tension and gravity is 0. The key here is the Y components. The questions is asking why the MAGNITUDE of these two tensions is greater than the lamp. Based on the diagram, we know that each tension has a non-zero X component as well right? And, the magnitude of each tension is |T| = √(Tx^2+Ty^2) according to the Pythagorean theorem. Connect the idea that the Y components of tension are equal to gravity in magnitude, but also note that the each tension's magnitude has an X component as well.

Also, imagine what the answer would be if the two tensions were perfectly vertical and had an angle of 90° to the ceiling. Would the sum of the tensions be greater than or equal to the lamp's weight in that case?

 

[CollegeStudent]

You understand that for the lamp to have a net force of 0 in the Y and X directions, the sum of all forces in those directions is 0. Realize that this means the sum of the Y components of tension and gravity is 0. The key here is the Y components. The questions is asking why the MAGNITUDE of these two tensions is greater than the lamp. Based on the diagram, we know that each tension has a non-zero X component as well right? And, the magnitude of each tension is |T| = √(Tx^2+Ty^2) according to the Pythagorean theorem. Connect the idea that the Y components of tension are equal to gravity in magnitude, but also note that the each tension's magnitude has an X component as well.

Am I correct in saying that the tension in the vertical direction are what causes equilibrium with the weight of the picture...and that the fact that there is an X component as well, adds tension?

Because as your second question (quoted below) asks...if it was just straight up and down at 90° , the tension would be equal with the force of gravity on the picture.

So with the added X-component (while equal and opposite to have a net force of 0) just add extra tension?

I'm sorry I'm just having a hard time wrapping my head around this

Also, imagine what the answer would be if the two tensions were perfectly vertical and had an angle of 90° to the ceiling. Would the sum of the tensions be greater than or equal to the lamp's weight in that case?

Well in this case it would be equal to the weight of the lamp because there's no x components.

 

[dang3r]

Am I correct in saying that the tension in the vertical direction are what causes equilibrium with the weight of the picture...and that the fact that there is an X component as well, adds tension?

You are correct. The overall tension takes into account both the X and y component of tension, so you are right in saying the non-zero X component adds tension.
The equation for tension follow:

|T|=(Tx^2+Ty^2)^0.5 describes this where |T| = magnitude of tension and Tx and Ty are its components.

Using this equation and finding|T1| and |T2| should answer your question.

Well in this case it would be equal to the weight of the lamp because there's no x components.

You are correct, so does having a non-zero X component affect your question? Look at the equation above.

 

[CollegeStudent]

You are correct. The overall tension takes into account both the X and y component of tension, so you are right in saying the non-zero X component adds tension.
The equation for tension follow:

|T|=(Tx^2+Ty^2)^0.5 describes this where |T| = magnitude of tension and Tx and Ty are its components.

Using this equation and finding|T1| and |T2| should answer your question.



You are correct, so does having a non-zero X component affect your question? Look at the equation above.

Thanks dang3r you were very helpful!