Finding Tension in a Frictionless Ladder

[Daniiel]

Homework Statement

[PLAIN]http://img72.imageshack.us/img72/6052/jdff.jpg

The Attempt at a Solution

For part ii
i found the centre of mass of the ladder by
80x2/80+25 = 1.52m up the ladder
then
force of the wall minus the tension
Fw - T = 0
Fw = (9.8 x (25x0.51 + 80x0.67)) / 2.83 (height of the ladder against the wall)
= 229.76 N
so that's the force on the wall and the tension opposing it
would that be the answer for the tension in the rope?
or would i have to consider the tension doing more work?
like the total force downwards in the y direction?
Mg + mg = 1029 N
or is that just the force the ground is giving?
and I am also not sure how to incorporate the fact that the rope is abit higher than ground level
thanks for help in advanced

 

[ehild]

Would you please show the forces in the picture?

ehild

 

[Daniiel]

[PLAIN]http://img44.imageshack.us/img44/5007/fhdf.jpg
the angle is 19.5 aswell
Fg- force of ground
Fw force wall
T tension
mg gravitational force
i think there's a Mg force as well of the person
I'm having trouble thinking if the height of the rope matters
like half of me thinks it does
but the other half thinks it doesnt
but if it doesn't matter, would that answer be right?

 

[ehild]

You have to include the torque of the rope, too.

ehild

 

[Daniiel]

ohh ok sorry i forgot
tz = 0
but what about for part two?
does that look alright to you?
would the tension on the rope change if its height increases?
when the weight of hte rope doesn't matter?

 

[ehild]

What is tz=0? The tension of the rope is T, its torque is T*its height, anticlockwise. This term is missing from the torque equation

"Fw = (9.8 x (25x0.51 + 80x0.67)) / 2.83 (height of the ladder against the wall)"

ehild

 

[Daniiel]

oh ok
the previous line looks like
i made the pivot at the bottom of the ladder
-h(Fw) + g(x1m1 + x2m2) + 0 T + 0 Fg = 0
then i rearagned for Fw

 

[ehild]

Why do you multiply the tension of the rope with 0? The arm is 0.5 times cosine of the angle between the ladder and the wall.

ehild

 

[Daniiel]

Oh right, so the pivot 0.5 up the ladder where the string is attached
I was just following an example in my book
it doesn't have a ladder with a string though
it says Tnet,z = 0 (torque)
and times those things by 0
damnn
i thought i had this question down
so is the procedure similar just with the upper length of the ladder?

 

[ehild]

so is the procedure similar just with the upper length of the ladder?

yes...

ehild

 

[Daniiel]

i get 183N
by pluging in the new x co-ordinante for the person
9.8(25*0.51 + 80x0.5)/2.83
im pretty positive I am on the wrong track
the centre of mass would stay the same right?
and we would still consider the height to be 2.83? or do i have to find the height without the 0.5m of the ladder?

 

[ehild]

No, you do not have a new x coordinate. You do not need the centre of mass of the whole system. The centre of mass of the ladder is in the middle. You have forces, points where they act and level arms. Calculate the torque of all forces with respect to the point where the ladder stands. The sum must be zero.

ehild

 

[Daniiel]

ohh ok
so
Fw + T + Fg + (2/3 x 80 x 9.8) + (0.5 x 25 x 9.8) = 0
so would all the forces be times'd by H
and now do we apply that zero thing like they did in my book?

 

[ehild]

Fw + T + Fg + (2/3 x 80 x 9.8) + (0.5 x 25 x 9.8) = 0

This equation is entirely wrong. What is torque?

ehild

 

[Daniiel]

(H)Fw + 1/2mg + 2/3 Mg + hT = 0
is that one right?
sorry for the late reply i polished off all of the other questions
514.5 - 2.83 Fw - 0.47T = 0

does Fw - T= 0?
then could we find one of them and solve
Fw = T

514.5 - 2.83T - 0.47T = 0
3.3T = 514.5
T = 156N = Fw ?

 

[ehild]

(H)Fw should have "-" sign, as Fw would rotate the ladder clockwise while the other three forces would rotate anti-clockwise.

- (H)Fw + 1/2mg + 2/3 Mg + hT = 0,

but where did the 541.5 come from?

Yes,the magnitude of Fw is the same as the magnitude of the tension in the rope, Fw=T. But the torque of Fw is negative, that of the tension is positive.

ehild

 

[Daniiel]

oh right, sorry

i got 541.5 just by adding 1/2mg + 2/3 Mg
which i may have done wrong haha one sec
(53.33+12.5)g= 645.1
ahh i mixed up the 1/2 and 2/3
does that look better?
then the same procedure so like

-2.83Fw + 645 + 0.47T = 0
Fw = T
645 = 2.36T
T = 273.3N

 

[ehild]

It looks all right at last. ehild

 

[Daniiel]

thanks loads =D
sorry about being a hassle at the start i wasnt thinking properly

 

[Daniiel]

hey ehild sorry to be a pain again
i full understood it last night
but I am just thinking over it again
so we made the rope the pivot point
so should it it be different
cause i don't get why its 2/3 and 1/2,
and then why Fw is times H would it be times 3?
i understood the first time because they're = to the x components of the distance from where the laddy touches the ground
how come it isnt, say if L = length of the ladder = 3
L2/3 and L/2
but if we take the pivot at 0.5ms up the ladder
would everything change?
like the distance of everything relative to the pivot?
so like i got the guy is 1.5m from the pivot which is just L/2 then the centre of mass would be the same as the original or would it change?
and would T just be 0 because its 0 from the pivot?
and would there be a new height?
i did it all again and this is what it looked like
1/2Mg + 1/2 mg = 2.35T = 219N
am i just over thinking it?
im just confused about the pivot and why we use 2/3 and 1/2 and then HFw
sorry for being a pain

 

[ehild]

I took the bottom of the ladder as the pivot point. This makes things easy as we do not need to calculate with the normal force from the ground.

The torque is force times level arm (the distance of the "attack line" of force from the pivot. See pic.


I will show you a picture where the pivot is the point where the rope is attached to the ladder in the next post.

ehild

 

[Daniiel]

oh thanks, i was just rewriting my post because it was all jibber jabber
so if we took the pivot at the rope we would ignore T because itll be 0T and have to find Fground right?
just another question
i had my head around it last night, but somone at uni questioned it today
how come for the masses of the person and the ladder, we use the x components from the pivot
like
in my head it makes sense
and i don't want to question it haha, but then i think why isn't it how far he is up the ladder?
is it somthing like, that's where the force is acting relative to where the pivot is?
how come the y component isn't considered for the masses? or just like 2m and 1.5 ms

 

[ehild]

Yes, if the end of the rope is the pivot point then its torque is zero. And we have to take the torque of the normal force into account.

As for your next question, remember, the torque is the turning effect of a force. If you push a door, it will not turn if you push in the direction of the pivot. It is easiest to turn if your force is perpendicular to the door and as far from the pivot as possible.

The picture shows a level. Fa is perpendicular to the level, and its attack point is at distance a from the pivot. The level arm is a, the torque is Ta=a*Fa, and clockwise.

Fc is in-line with the level. Its line of attack crosses the pivot point. The torque is zero.

Fb makes an angle with the level. You get the level arm by drawing a straight line thorough the pivot which makes a 90° angle with the force Fb. The length of the straight line between the pivot point and the line of force is the level arm. Tb=b*Fb, and it is anti-clockwise.

Or you can calculate the torque of Fb in an other way. The in-line component of Fb does not rotate the level, its torque is zero. Only that component of Fb counts which is normal to the level (Fn). So Fb=c*Fn



ehild

 

[Daniiel]

thanks ehild that helped a lot =]

 

[ehild]

I do not know how much do you learned about vectors, but the torque is a vector product of the position vector of the "attack point" of the force with the force vector.

T = \vec r \times \vec F

The magnitude of the torque is T = r F sin(alpha) where alpha is the angle between the force and the position vector r.

ehild

 

[ehild]

I promised a picture with the pivot point at the rope. Here it is.

ehild

 

[Daniiel]

oh thanks for doing that one
i've been messing around with it
and trying to see if i'll get the same T
but i havnt got 273 yet
theta is 70.5 right? 90-19.5
and how come the g's dissapear?
thanks for making that picture though
i completely get how it works

 

[ehild]

oh thanks for doing that one
i've been messing around with it
and trying to see if i'll get the same T
but i havnt got 273 yet
theta is 70.5 right? 90-19.5
and how come the g's dissapear?
thanks for making that picture though
i completely get how it works[/QUOTE

OOpppss.. I forgot the g-s. They are everywhere except the tension and Fw.
... and yes, the angle is 70.5°.


ehild

 

[Daniiel]

oh mad i got a very similar error
like .7 off because i rounded everything
thanks heaps ehild =D