Finding Tension and Acceleration on Frictionless Surfaces

[m.piet]

Homework Statement

For the following system 1) Find the tension in the string
2) find the acceleration of each of the masses

The diagram of the system looks like this

...O-----|200g|-----O
...|.\___|____|___/.|
__|__....__|__ ( . ) are placeholders to make the diagram look accurate
|100g|...|200g|

Basically a mass of 200g on a frictionless surface attached to 2 other masses (100g and 200g) hanging over the edge of the surface with a string on frictionless pulleys. String has no mass and can not expand. Gravity is 9.8 m/s^2.

The attempt at a solution
First off I converted all my masses to Kg
I have numbered each mass from left to right from 1-3 and have drawn free body diagrams for them. These are the equations I got for each

1)
Fnet = ma
Fnet = T-Fg
Fg= mg
ma = T-Fg
(0.1)a = T-0.98
So: T = 0.1a + 0.98
a = (T- 0.98) / 0.1

2)
Fnet = ma
Fnet = T1 - T2 (T1 and T2 are the tensions to the 1st and 3rd masses, respectively)
T1 - T2 = ma
T1 - T2 = 0.2a

3)
Fnet = ma
Fnet = T-Fg
Fg = mg
ma= T-Fg
0.2a = T-1.96
So: T = 0.2a + 1.96
a = (T-1.96) / 0.2


I am stuck at whether the mass that is on the surface will even matter to the acceleration and tension of the system because the surface is frictionless so would that mean I could solve as if there is no mass there? Or would I have to isolate the system into 2 systems, 1,2 and 2,3, then solve for each tension and acceleration and go from there?

 

[PhanthomJay]

You're doing OK except for a couple of things. You should first note that tensions in ideal strings wrapped around ideal pulleys are the same on both sides of the pulley. So what you call T in your FBD of mass 1, you should call T1; and what you call T in your FBD of mass 3 you should call T2. Secondly, be consistent with your plus and minus signs; since you have assumed that the lighter hanging mass is moving upward, then mass 3 must be moving downward, and mass 2 must be moving to the right. Then just solve the 3 equations for the 3 unknowns T1, T2, and a.

 

[thomate1]

As a scientist, your approach to this problem is correct. The mass on the surface will not affect the tension and acceleration of the system since it is on a frictionless surface. You can solve for the tensions and accelerations of each mass separately, without considering the mass on the surface. However, isolating the system into two systems (1,2 and 2,3) and solving for each tension and acceleration will also give you the correct answer. Ultimately, it depends on your preferred approach and what information you have available. Both methods will yield the same results.