Problem with tension on two strings at angles

[lilmissbossy]

mass m suspended by a pair of vertical ropes attached to the ceiling
what are the rope tensions if they comprise a V shape each at an angle theta with the ceiling?

i have played with this problem for a while and i can't seem to find what the relevant equations are really only first time dealing with physics i am a biology student...

 

[rock.freak667]

Split the two tensions into vertical and horizontal components. At equilbrium, the sum of the vertical forces=0 and similarly, the sum of the horizontal forces=0

 

[dynamicsolo]

You should start with a picture of the mass and the ropes (which are not vertical, but canted), showing the forces acting on the mass. There will be the mass' weight force and the tensions from each rope. Is the angle theta measured from the vertical or from the horizontal?

 

[lilmissbossy]

I have drawn a picture which kinda looks like this

_________________
\o o/
\ /
\ /
\___/
[_m_]
didnt know how to inuput a paint drawning. o represents theta

if the ropes were hanging vertically i have the equation T1= 1/2mg and T2=1/2mg as they both equally support the mass.
I am thinking of using sine theta to work out the answer but i am stuck at that point, I am not sure what to use.

 

[lilmissbossy]

ok that picture didnt work at all
sorry

 

[dynamicsolo]

if the ropes were hanging vertically i have the equation T1= 1/2mg and T2=1/2mg as they both equally support the mass.

That would be correct; this also provides a clue to how to deal with the tilt of the ropes.

I'm assuming from the description (and the way you marked your diagram) that the ropes make an angle theta to the horizontal. So the tensions are still equal, but now the weight W = mg acting downward is counterbalanced by the vertical components of the two tensions, rather than the full tensions T = T₁ = T₂. If the tension in one of the tilted ropes is now T' , what is the component of its tension that acts vertically?

 

[lilmissbossy]

Ok so i guess instead of using 1/2 i factor in the angle it hangs on.
maybe something like T1= sin(theta)mg...

again i am not very good at physics at all

 

[tiny-tim]

 tags preserve spacing[/b]

[quote="lilmissbossy, post: 1846206"]I have drawn a picture which kinda looks like this

_________________
       \o        o/
        \         / 
         \       /
          \___/
          [_m_]
didnt know how to inuput a paint drawning. o represents theta[/QUOTE]

Hi lilmissbossy! :smile:

[SIZE="1"](Have a theta: θ :smile:)

Use code tags, 'cos they preserve spacing :wink::

[CODE]
_________________
       \θ        θ/
        \         / 
         \       /
          \___/
          [_m_]

hmm … still needs adjusting … I wonder why?

    _________________
       \θ       θ/
        \       / 
         \     /
          \___/
          [_m_]

 

[dynamicsolo]

Ok so i guess instead of using 1/2 i factor in the angle it hangs on.
maybe something like T1= sin(theta)mg...

You're getting close. If \theta represents the angle each rope makes to the horizontal ceiling, then the vertical component of the tension T' in each rope will indeed be T' sin(\theta).

Now there are two such ropes connected to the suspended mass pulling upward, and the mass' weight mg is pulling downward. Since the mass is not accelerating (indeed, not moving), the sum of the forces acting on it in the vertical direction are

T' sin(\theta) + T' sin(\theta) - mg = ma_y = 0

T' is the only unknown, so you can now solve for that, which is the magnitude of the tension in each rope. (How does it compare to the tension in the case of vertical ropes -- is it larger or smaller? Why would that be?)

 

[tiny-tim]

components

Ok so i guess instead of using 1/2 i factor in the angle it hangs on.
maybe something like T1= sin(theta)mg...

again i am not very good at physics at all

Hi lilmissbossy!

i take it you're not happy with components? You don't know when to use them, or whether to use sin or cos?

    _________________
       \θ    | θ/
        \    Î / 
         \   |/
          \__/___
          [_m_]

Look at the diagram … I've drawn the two components for the tension force in the right-hand string.

The vertical line has a sort-of arrow pointing up, and the short horizontal line should have an arrow pointing right (but doesn't ).

Together, they make up the tension in the string , and they are the right lengths!

So the upward component is Tsinθ (opp/hyp), and the horizontal one is Tcosθ (adj/hyp).

You can always do it this way, just to check, if you're not sure you have the correct θ.​

 

[lilmissbossy]

Ok i understand it now, thank a heap guys. i wish i new about this place a lot earlier...
Cheers

 

[lilmissbossy]

i know this again

i have done two different types of equations both giving the same answer except one is a -ve number. here are the equations

Tsin(θ)+Tsin(θ)-mg=ma=0
rearranged
T= sin(θ)+sin(θ)-mg
T=sin(53)+sin(53)-(55kg)(9.8ms-1)
T= -537.402

then i used the cos equation to check if there was any difference and this is what i got

Tsin(θ)+Tcos(θ)-mg
T=sin(θ)+Tcos(θ)-mg
T=sin(53)+cos(53)-(55kg)(9.8ms-1)
T= -537.6

i am not sure if i am rearranging the equations right or if i have the right method>>>

 

[lilmissbossy]

i also get for the tension on the vertical ropes (no angles) at

T1=1/2mg
T1=1/2(55kg)(9.8ms-1)
T1=269.5

which is basically what i get when i divide the previous answers by 2

should they be the same?

 

[dynamicsolo]

i have done two different types of equations both giving the same answer except one is a -ve number. here are the equations

Tsin(?)+Tsin(?)-mg=ma=0
rearranged
T= sin(?)+sin(?)-mg

This is not rearranged correctly: you cannot simply separate the sine factors from the T's. This should be

2 · T · sin(?) = mg

T = mg / [ 2·sin(?) ]

Now you won't get a negative force magnitude; magnitudes of forces can't be negative.

then i used the cos equation to check if there was any difference and this is what i got

Tsin(?)+Tcos(?)-mg
T=sin(?)+Tcos(?)-mg
T=sin(53)+cos(53)-(55kg)(9.8ms-1)
T= -537.6

Where would a cosine factor come from? From the angle theta being measured from the horizontal, Tcos(?) would be the horizontal component of the rope tension; that component is not involved in supporting the weight.


In the case of the ropes being vertical, the angle theta is 90º (not "no angle"). The tension in each rope is then T1 = T2 = T = mg / [2 sin 90º] = mg / (2 · 1) = mg/2 , as you said earlier.

 

[lilmissbossy]

i got the cosine from tinytim, but i realize now that its not a factor.
ok i see where i rearranged wrong and i now have answers that show a difference in the tension at 90* and at 53*. with the 53* having more tension on the strings. i guess when you think about it, it makes sense that the angled string has a greater tension as it is strected further from its initial point... thank you again this is all very confusing

 

[dynamicsolo]

i guess when you think about it, it makes sense that the angled string has a greater tension as it is strected further from its initial point...

I'm not sure what you mean by the string being "stretched further": the tension in the rope has nothing to do with the distance between the points where it is fastened. The answer to your problem would be the same whether the ropes were half or twice the lengths that are used in the problem (as long as the mass was still suspended and not resting on the floor).

The reason the tensions for the angled ropes must be greater is that the forces along the ropes are not directed fully in the vertical direction, so some portion of those forces have no influence in countering the weight force acting on the mass. Only the component of the tensions that is directed exactly opposite to the weight force will act to balance it; the component of the tensions that acts perpendicularly (horizontally here) will have no effect against the weight.

Consider this situation. You have some object with a string tied to it that you want to lift. If you pull straight up on it, some amount of force will succeed in raising the object. How hard would you have to pull sideways on the string in order to make the object rise?

 

[tiny-tim]

then i used the cos equation to check if there was any difference and this is what i got

Tsin(θ)+Tcos(θ)-mg
T=sin(θ)+Tcos(θ)-mg
T=sin(53)+cos(53)-(55kg)(9.8ms-1)
T= -537.6

i am not sure if i am rearranging the equations right or if i have the right method>>>

Hi lilmissbossy!

You have the wrong outlook, even before you get to the rearranging …

There's no such thing as a "cos equation" or a "sine equation".

There is a vertical equation and a horizontal equation (and of course, an equation for any other direction … but you only need two ).

The horizontal equation in this case is not Tsin(θ)+Tcos(θ) = mg …

it's Tcos(θ) - Tcos(θ) = 0 …

in other words, the horizontal components of the two tensions are equal and opposite, as expected.

Hint: whenever you do one of these problems, start each equation with the words "Taking components in the … direction". ​

oh … and when the string is at angle, it's pulling sideways as well as up, so there's more pull … that's all!

 

[lilmissbossy]

thank you so much i get really confused with all this physics, biology is my degree so i find this all very stressful

 

[mal4mac]

i know this again

i have done two different types of equations both giving the same answer except one is a -ve number. here are the equations

Tsin(θ)+Tsin(θ)-mg=ma=0
rearranged
T= sin(θ)+sin(θ)-mg
T=sin(53)+sin(53)-(55kg)(9.8ms-1)
T= -537.402

then i used the cos equation to check if there was any difference and this is what i got

Tsin(θ)+Tcos(θ)-mg
T=sin(θ)+Tcos(θ)-mg
T=sin(53)+cos(53)-(55kg)(9.8ms-1)
T= -537.6

i am not sure if i am rearranging the equations right or if i have the right method>>>

Confused? Don't panic! This can easily be sorted out, with so many heads on the case. Are you confused by the physics or the mathematics, or both?

-----------Physics----------------

The basic physics of the situation is:

Tsin(θ)+Tsin(θ) = mg

Are you happy with that?

-------------Mathematics-----------
This is basic algebra, just remember not to get thrown by the sin and cos!

You might find simplifying the equations useful. For instance you can say:

a = sin(θ) and b = cos(θ)

Then Tsin(θ)+Tsin(θ) = mg becomes Ta + Ta = mg

Are you happy with that?

Now the steps in the rearrangement might be clearer to you:

Ta + Ta = mg
2Ta = mg
T = mg/2a

Is that clear?

Now you can stick in the sin:

T = mg/2sin(θ)

Happy?

 

[lilmissbossy]

yes thank you i haven't really studied any math since yr 12 in 2002 or ohysics at all for that matter... so i am finding both equally confusing. that layout shows clearly what my problem was. i understand how to set that question out now let's just hope i can remember that for my exam!
Thank you

 

[mal4mac]

This book is quite good for a gentle refresh of school maths (and more):

https://www.amazon.com/gp/product/039304002X/?tag=pfamazon01-20