How to Use Newton's Laws to Solve a Sliding Ramp Problem

[ningaman129]

Homework Statement

Two packages at UPS start sliding down a 20 degree ramp. Package A has a mass of 5.0kg and a coefficient of friction of .20. Package B has a mass of 10kg and a coefficient of friction of .15. How long does it take package A to reach the bottom? The package are positioned next to each other on the ramp, and the distance from package A to the bottom is 2.0m.

Homework Equations

Newton's Second Law equations.
$$\sum$$ (FnetA)x=mAaX

$$\sum$$ (FnetA)y=mAaY

$$\sum$$ (FnetB)x=mBaX

$$\sum$$ (FnetB)y=mBaY

The Attempt at a Solution

I tried solving this by putting the horizontal and vertical components in their respective places, but that didn't get me that far, or at least I don't think so. For both y-components of the net force n=mAgcos(20) and n=mBgsin(20), well at least I think so. really I'm not quite sure if I am going in the right direction because I don't know how I am going to solve for time.

 

[Riogho]

Have you guys done the conservation of energy theorem yet in class?

If so, you can find this by realizing that U1 + K1 + W = U2 + K2

 

[ningaman129]

no I don't think we covered that yet.

 

[cepheid]

Solving for time is no problem because you know that package A was moving under uniform acceleration, and you know what distance it travelled. There is a simple kinematics formula that applies to motion under constant acceleration.

Of course, you need to calculate what that acceleration *is*. I think that the slightly tricky part of this problem is (if I am interpreting the problem correctly) the fact that the two packages are in contact, meaning that the net force on package A in the "down the ramp" direction depends on three things: its weight, friciton, and the contact force from package B pushing on it.

 

[pc2-brazil]

The problem says that the packages are "next to each other", but seems to left implicit that they are in contact. Also, it lefts implicit that package A is closer to the end of the ramp than package B.
If I understood the problem correctly, it is necessary to see whether packages A and B will be always next to each other during the motion (if they weren't, then you could simply treat the problem as if package A was alone). Since B's mass is bigger, its weight is bigger, so it would move faster if it were alone in the ramp; so they will be always next to each other during the motion.
I would solve this problem this way: first, I would draw a sketch of the situation, a free-body diagram representing all the forces acting on package A (four forces: weight, friction, contact force from package B pushing on it --as cepheid said--, and the normal force that the ramp exerts on the package -which has the same direction and opposite orientation to one component of the weight).
Secondly, I would decompose weight, "cancel" what needs to be canceled and see what is the net force. Then, find the acceleration and, since I have distance and acceleration, find the time.