Newtonian mechanics, incline, acceleration FUN

[neesh]

"Two boxes, m1=1.0 kg witha coefficient of kinetic friction of 0.10 amd m2=2.0 kg with a coefficient of 0.20, are placed on a plane inclined at 30 degrees. (a) What acceleration does each box experience? (b) If a taut string is connecting the boxes with m2 farther down the slope, what is the acceleration of each box?"

So, I have part (a) (I believe!):
x direction: m1: m1gsin30 -Ffr=m1a1 (same for m2, but sub m2 for m1)
y direction: Fn = m1gsin30

For m1, migsin30=(muk)Fn = m1a1, you place in the appropriate values from the question and solve for a1= 4.1 m/s2
You go through the similar steps for m2 and solve for a2= 3.2 m/s2

But I'm a little confused on how exactly to explain part (b):
I think that the accelerations would be the same, but we were taught that whenever boxes are connected by a taut string, the acceleration for the boxes is the same, and it is treated as a system. But, since m1 is in back and has a faster acceleration, wouldn't it go faster so that the cord is no longer taut and the boxes are essentially not connected? Or am I thinking about it wrong and it should be treated as a system?
thanks in advance!

 

[andrevdh]

If you view the two masses as a system the tension in the string cancels out since they are the same magnitude but in opposite directions and therefore constitute an internal force.

 

[kyle8921]

I would like to clarify that Newtonian mechanics is a fundamental theory of classical physics that describes the motion of objects based on the laws of motion and the concept of forces. It is named after Sir Isaac Newton, who developed the theory in the 17th century.

In this scenario, we are dealing with an inclined plane, which is a common example used to explain Newtonian mechanics. The acceleration of an object on an inclined plane is affected by the force of gravity, the normal force, and the force of friction. The coefficient of kinetic friction, as given in the question, is a measure of the resistance to motion between two surfaces in contact.

In part (a), we are asked to calculate the acceleration of each box on the inclined plane. This can be done by using the equations of motion, as correctly shown in the response. The acceleration of each box is affected by the forces acting on it, and the resulting value is dependent on the masses and the coefficients of friction.

Now, in part (b), we are introducing a taut string connecting the two boxes. In this case, the two boxes are considered as a system, and the acceleration of the system will be the same for both boxes. This is because the string is taut, and any force acting on one box will also act on the other box. Therefore, the acceleration of the system will be the same for both boxes, regardless of their individual masses and coefficients of friction.

To answer the question, the acceleration of each box will still be the same as calculated in part (a), but now it will be considered as the acceleration of the system. This may seem counterintuitive, as you mentioned, the box with a larger acceleration may seem to pull the other box along and make the string no longer taut. However, in reality, the tension in the string will adjust to maintain the acceleration of the system, and the string will remain taut.

In conclusion, Newtonian mechanics provides a fundamental understanding of the motion of objects, and it can be applied to various scenarios, including inclined planes and systems connected by strings. By using the equations of motion and considering all the forces acting on the objects, we can accurately calculate their accelerations and predict their motion.