2 trucks pulling a third truck by the means of a pulley

[Jkyou]

Homework Statement

Two trucks tow a third one by means of inextensible ropes and a pulley attached to them (fig. 1). The accelerations of the two trucks are a1 and a2. What is the acceleration of the third truck that is being towed?
(fig.1) is in this link (I'm sorry, I don't know how to post images on websites):
http://www.physics.ox.ac.uk/olympiad/Downloads/PastPapers/BPhO_AS_2007_QP.pdf

m=mass of truck
a1,a2 respective accelerations
a=the shared acceleration (whole system)
T= Tension

Homework Equations

F = ma
Free body diagrams

The Attempt at a Solution

Although the problem didn't specify, I'm assuming the masses of the trucks are equal
Since the total acceleration of the two trucks must be the same,

2ma = ma1-ma2
Thus, a= (a1-a2)/2
Furthermore, ma =T - ma2 for the second truck, so

T = ma + ma2 -> T = m(a1+a2)/2

The force of the tension on the pulley and 3rd truck = 2T

And total force on the pulley, which pulls the third truck = ma

Thus ma = 2T -> a = a1+a2, but the answer key says it's a = (a1+a2)/2

 

[TSny]

Hello. Welcome to PF!

The accelerations of trucks 1 and 2 are not necessarily the same. This is a kinematics problem. You do not need to work with forces.

For problems like this, it is often helpful to use the fact that the total length of a rope must remain constant. Consider introducing distances from a fixed line ("datum") as shown below. Can you relate the distances x₁, x₂, and x_p to the length of the rope connecting trucks 1 and 2?

 

[Jkyou]

So that'll be length = x1 + x2 -2xp ?

 

[Jkyou]

Hello. Welcome to PF!

The accelerations of trucks 1 and 2 are not necessarily the same. This is a kinematics problem. You do not need to work with forces.

For problems like this, it is often helpful to use the fact that the total length of a rope must remain constant. Consider introducing distances from a fixed line ("datum") as shown below. Can you relate the distances x₁, x₂, and x_p to the length of the rope connecting trucks 1 and 2?

Will it be then length of rope = x1 + x2 - 2xp = constant

And perhaps take the double derivative of that?

 

[TSny]

Will it be then length of rope = x1 + x2 - 2xp = constant

And perhaps take the double derivative of that?

Yes!

 

[Jkyou]

Yes!

Ohh I see! That's a really elegant way to do that. Do you think it's possible with forces, just for the fun of it? Or will it be difficult to do that?

 

[TSny]

Ohh I see! That's a really elegant way to do that. Do you think it's possible with forces, just for the fun of it? Or will it be difficult to do that?

The kinematic constraint is what determines the relation between the accelerations, not the forces. But, you can then use the relation between the accelerations and F = ma to relate various forces in the problem.

 

[Jkyou]

The kinematic constraint is what determines the relation between the accelerations, not the forces. But, you can then use the relation between the accelerations and F = ma to relate various forces in the problem.

I see, so the accelerations are dependent on the constraint and not always on the forces situations such as this one. So for example if I were to determine the tension in the string, since 2T = ma where a must be the acceleration of the pulley, I can determine T = m(a1+a2)/4 and etc. Am I on the right track?

 

[TSny]

I see, so the accelerations are dependent on the constraint and not always on the forces situations such as this one. So for example if I were to determine the tension in the string, since 2T = ma where a must be the acceleration of the pulley, I can determine T = m(a1+a2)/4 and etc. Am I on the right track?

Yes. The details would depend on what information you are given and what you are trying to determine.

 

[Jkyou]

Yes. The details would depend on what information you are given and what you are trying to determine.

Ahh I see, thank you so much! I am glad I came here for help ^^

 

[TSny]

OK. Glad I could help.

 

[Seneka]

Yes!

How do you take the double derivate of that and come to the answer?

 

[TSny]

How do you take the double derivate of that and come to the answer?

Welcome to PF!

Please write out the specific equation that you wish to differentiate.

 

[Seneka]

Welcome to PF!

Please write out the specific equation that you wish to differentiate.

The one used to answer the question above...so x1+x2-2xp.I get that the length of the rope is x1+x2-2xp but I don't understand how you differentiated it and how that led to the answer.

 

[TSny]

I get that the length of the rope is x1+x2-2xp but I don't understand how you differentiated it and how that led to the answer.

So the equation is L = x₁ + x₂ - 2x_p, where L is the length of the rope. What do you get if you take one time derivative of each side?

 

[Seneka]

So the equation is L = x₁ + x₂ - 2x_p, where L is the length of the rope. What do you get if you take one time derivative of each side?

I don't know.

 

[TSny]

What does the time derivative of x₁ represent?

 

[Seneka]

What does the time derivative of x₁ represent?

Velocity?

 

[TSny]

Velocity?

Yes. So, you could write the time derivative of x₁ as v₁.

 

[Seneka]

Where are you going with this?

So, x1+x2-2xp=l
v1+v2-2vp=vl?
a1+a2-2ap=al?
a1+a2-al=2ap?
(a1+a2-al)/2=ap?

 

[TSny]

Where are you going with this?

Hopefully to some insight

So, x1+x2-2xp=l
v1+v2-2vp=vl?

What does vl represent? The ropes are given to be "inextensible".
What does this imply about the right hand side of your equation x1+x2-2xp=l?

 

[Seneka]

Hopefully to some insight

I appreciate your patience.

What does vl represent? The ropes are given to be "inextensible".
What does this imply about the right hand side of your equation x1+x2-2xp=l?

vl=0 there is no change is l
you could rearrange to get vp=(v1+v2)/2
I guess if you do the second derivative you get ap=(a1+a2)/2
Am I right?

 

[TSny]

vl=0 there is no change is l
you could rearrange to get vp=(v1+v2)/2
I guess if you do the second derivative you get ap=(a1+a2)/2
Am I right?

Yes. Good work.

 

[Seneka]

Yes. Good work.

OMD! I finally got there. Thanks again for your patience and quick reply! Is there a name for this technique using this 'datum' thing or is it just logic?

 

[TSny]

I don't know of any name for the technique.