Compute acceleration and tension in the cord

[skysunsand]

Homework Statement

A cord passing over a frictionless, massless pulley has a 4.0 kg object tied to one end and a 12 kg object tied to the other. Compute the tension and acceleration in the cord.


Right now, I'm just trying to calculate the acceleration. It is supposed to be 4.9 m/s^2

Homework Equations

F=ma
T1=T2=0
T1-mg=ma
T2-mg=ma

The Attempt at a Solution

For the 4kg block-

F=ma
T1-mg = ma

T1-39.2 = 4a
Solving in terms of T1 to get
T1= 4a+39.2

For the 12kg block-
F=ma
T2-mg = ma

T2-117.6= 12a
Solving in terms of T2-
T2= 12a+ 117.6

Now, because T1=T2, I can say

12a+ 117.6 = 4a+ 39.2
But then I end up with a=9.8.

The answer is 4.9, which, incidentally, is 9.8/2.

Do I have to divide by 2 because I have two cords? Or is there some other reason I am missing?

 

[Spinnor]

Assuming the following is correct can you solve for T and a?

 

[skysunsand]

Assuming the following is correct can you solve for T and a?

Yeah, you'd set T=4a + 4g , plug that into the other equation for T, to get

12g+(-4a-4g) = 12a
to get
8g=16a
8*9.8 = 16a

a=4.9

Ah. I must have screwed up somewhere in my algebra.

Thank you!

 

[PeterO]

The answer is 4.9, which, incidentally, is 9.8/2.

Do I have to divide by 2 because I have two cords? Or is there some other reason I am missing?

co-incidence almost.

One way to view this: the weight of a 4 kg mass is pulling one way, and the weight of a 12 kg mass is pulling the other way.

If you consider the weight of the 4 kg mass cancells out [balances if you like] 4 of the other 12, we have a net force equal to the weight of an 8kg mass to accelerate this system - which has a total mass of 16 kg.

Now the weight of a 16 kg mass will accelerate a 16kg mass at 9.8 ms^-2, so we would expect that half that weight would achieve only half that acceleration → 4.9 ms^-2

 

[BetoG93]

Another way to solve this problem is to use just two directions for forces. It may sound crazy, but I have seen this procedure in two or more textbooks.
The positive direction would the direction of motion (the direction of falling of the 12-kg block). This way, if m1 is the 12-kg block, and m2 is the 4-kg block, we would have the following system of equations (each equation for a separate free-body diagram):

-T + m1g = m1a
T - m2g = m2a

I got 58.8 N for tension, and 4.9 m/s^2 for acceleration ;)