Pulley question assigning conventions

[putongren]

In the diagram, the pulleys and the strings are massless. The strings do not stretch. The masses of the suspended blocks are m1 and m2, as shown. The magnitudes of the accelerations of the blocks are a1 and a2, respectively. I'm having trouble assigning the conventions for the variables. I assigned the motion downwards as positive.

1. Homework Statement
T₁ = The string connecting m₂, and the string connecting the two pulleys.
T₂ = Left string connecting the left pulley.
a₁ = acceleration of mass 1
a₂ = acceleration of mass 2
m₂
m₁

Homework Equations

Result of FBD:

T₁ + m₂g = m₂a₂
T₂ + T₁ - m₁g = - m₁a₁

The Attempt at a Solution

[/B]
So... I can't isolate a₁ in terms of a₂.

 

[gneill]

View attachment 95262

In the diagram, the pulleys and the strings are massless. The strings do not stretch. The masses of the suspended blocks are m1 and m2, as shown. The magnitudes of the accelerations of the blocks are a1 and a2, respectively. I'm having trouble assigning the conventions for the variables. I assigned the motion downwards as positive.

1. Homework Statement
T₁ = The string connecting m₂, and the string connecting the two pulleys.
T₂ = Left string connecting the left pulley.

If the pulleys and strings are massless, and the string is continuous, can there really be different tensions in different parts of the string?

a₁ = acceleration of mass 1
a₂ = acceleration of mass 2
m₂
m₁

Homework Equations

Result of FBD:

T₁ + m₂g = m₂a₂
T₂ + T₁ - m₁g = - m₁a₁

The Attempt at a Solution

[/B]
So... I can't isolate a₁ in terms of a₂.

Make a sketch where you draw mass m1 in its initial position and then the situation where it has moved down by some distance d. How much more string, in total, is in the sections of string supporting the lowered pulley? Where must that string have come from? So by how much must mass m2 have risen?

 

[putongren]

If m1 is lowered by d, then m2 rose by d?

 

[SDewan]

This is a constrained motion, wherein, the motion of the first pulley is linked to the second block. Try to understand that if the first pulley goes down by a distance d, then the second block will have to go up by a distance of 2d. If you are able to catch this, your question will be solved.
Hint: Draw INDIVIDUAL free body diagrams. That will make you clear.

 

[gneill]

Here's a diagram:

The pulley on the left descends a distance d. The added length to each of the string segments that support it are shown in red. The total added length has to come from somewhere...

 

[putongren]

Thank you for showing me how we get 2d. Do we do a FBD for each mass, pulley, or both? I don't understand how figuring out the distance to be 2d would lead to the solution.

 

[gneill]

Thank you for showing me how we get 2d. Do we do a FBD for each mass, pulley, or both? I don't understand how figuring out the distance to be 2d would lead to the solution.

In your first post you indicated that you were having a hard time relating the two accelerations. You now have a relationship for the two displacements. The same relationship will hold for velocities and accelerations since the motions are constrained by this fixed relationship.

 

[putongren]

How does the mathematical formalism work?

 

[gneill]

It's a matter of differentiating displacement to get the velocity relationship, and differentiating again to get the acceleration relationship. The constant 2 is carried along. So for example, if x₁ and x₂ are the two displacements related by x₂ = 2x₁, then

$\frac{dx_2}{dt} = 2\frac{dx_2}{dt}~~~$ or, $~~~v_2 = 2 v_1$