Incline/Spring/Pulley Friection problem

[Kevin713]

Hey guys. The last couple days, I've been poring over problems like this:

"Block A with a mass of 10kg rests on a 30 degrees incline. The coefficient of kinetic friction is 0.20. The attached spring is parallel to the incline and passes over a massless, frictionless pulley at the top. Block B, with a mass of 8kg, is attached to the dangling end of the string. The acceleration of B is (magnitude and direction)?"

Now, I know how to solve these problems. First you determine the direction of acceleration (80N dangling vs. (100sin30N + 0.20*Normal (for friction))). Once you know this, you simply make the system of equations:

Here, acceleration is up incline for Block A, down for Block B (and assuming g=10N/kg):

mAa = T-100sin30-(0.20*100cos30)

mBa= mBg- T

Then solve for a.

But, in going through countless posts about this online, I've seen people claim that they don't have to determine the direction of acceleration beforehand. They simply assume it (either up the incline or down the incline), then if the acceleration comes out negative at the end, they say it's simply the same magnitude, but in the opposite direction as assumed.

The problem with this is that if, say, the acceleration this problem were down the incline for Block A/up for Block B, then the equations would be (correct me if I'm wrong):

mAa = 100sin30-(0.20*100cos30)-T

mBa = T - mBg

And in calculating this, one realizes that the answer (including magnitude) is different.

Now, I know that all of this confusion can be averted by simply calculating yourself the direction of acceleration first (in fact, the direction is often stated in problems), then making the system of equations. However, as I've seen many people online claim that this isn't necessary, or always assuming that Block A accelerates up the incline, I'm a bit confused. In my numerous calculations (as an aside, I've been thinking so much about them that last night I even dreamed about this one stupid problem), if one randomly assumes a direction to the acceleration, and it comes out negative, then you've assumed incorrectly, and must change the assumed direction and recalculate (which will give you a different answer).

For example, in the problem above, the answer is .69 m/s2 downwards for Block B, but 2.6 m/s2 and -2.6m/s2 are other commonly-had responses.

Sorry for the long post. Any ideas?

 

[tiny-tim]

Welcome to PF!

Hi Kevin713! Welcome to PF!

But, in going through countless posts about this online, I've seen people claim that they don't have to determine the direction of acceleration beforehand. They simply assume it (either up the incline or down the incline), then if the acceleration comes out negative at the end, they say it's simply the same magnitude, but in the opposite direction as assumed.

The problem with this is that if, say, the acceleration this problem were down the incline for Block A/up for Block B, then … the answer (including magnitude) is different.

Now, I know that all of this confusion can be averted by simply calculating yourself the direction of acceleration first (in fact, the direction is often stated in problems), then making the system of equations. However, as I've seen many people online claim that this isn't necessary, or always assuming that Block A accelerates up the incline, I'm a bit confused.

I agree with you …

if friction is involved, then the equations for movement one way will be different from those for the other way …

even in the simple case of no movement, but asking for the hanging weight when the system is about to move, there will be two answers governed by two different equations.

If you made the wrong guess, you would need to start again!​

 

[Kevin713]

Ok, thanks tiny-tim. I guess I was looking for an easier, symmetric way of doing the system of equations for these type of problems, but I guess one has to always be sure and find the direction of the net force/acceleration first...