Pulley Problem: Find the acceleration of M2 & M3

[Adriano25]

1. Homework Statement
Pulley is mass less accelerating downward at 2.0 m/s².
M1 = 4.0 kg
M2 = 2.0 kg
M3 = 3.0 kg

Find the acceleration of M2 & M3.

Homework Equations

I made up the slope positive pointing up.
Since M2 and M3 have the same acceleration, a2=a3 ⇒ a
Since it's accelerating downwards, Apy = -2.0 m/s²

Mass 1:
ΣF1y = m1a1y
T-m1g = m1a1y
T = m1a1y + m1g (1)

Mass 2:
ΣF2y = m1ay
T-T1 = m2ay (2)

Mass 3:
ΣF3y = m3ay
T1-m3g = m3ay (3)

Pulley:
apy = (a1y+ay) / 2 (4)

The Attempt at a Solution

I added eq. (2) & (3) to get rid of T1
T-m3g = m2ay+m3ay (5)

I plugged eq. (1) into (5)
m1a1y + m1g - m3g = ay(m2+m3)

Solve for a1y:
a1y = (ay(m2+m3) - m1g + m3g) / m1 (6)

Plug (6) into (4)

apy = [ (ay(m2+m3) - m1g + m3g) / m1) + ay ] / 2

Solve for ay:
ay = (m1g-m3g+2m1apy) / m2+m3+m1) = 2.87 m/s²

By the answer key, the acceleration for m2 and m3 is -2.87 m/2². How do I know that is negative and not positive? Is it because m2 + m3 have a higher mass than m1, thus when the pulley is accelerating downwards, m1's acceleration points up and m2 & m3's acceleration point down?

 

[Chestermiller]

I get 2.87 downward. Let "a" be the downward acceleration of m2 and m3 relative to the pulley. So the total downward acceleration of m2 and m3 is (a+2). Using this approach, what is the upward acceleration of m1?

 

[Adriano25]

I get 2.87 downward. Let "a" be the downward acceleration of m2 and m3 relative to the pulley. So the total downward acceleration of m2 and m3 is (a+2). Using this approach, what is the upward acceleration of m1?

The acceleration of m1 would be -1.13 m/s². I think I finally caught my mistake. Since the pulley is accelerating downwards, the accelerations for m1, m2 & m3 are downwards as well. Then, solving the following equations using a negative sign on all accelerations, I get 2.87 downward:

Mass 1:
T = -m1a1y + m1g (1)

Mass 2:
T-T1 = -m2ay (2)

Mass 3:
T1-m3g = -m3ay (3)

Is this the right approach for this problem?

 

[Chestermiller]

The acceleration of m1 would be -1.13 m/s². I think I finally caught my mistake. Since the pulley is accelerating downwards, the accelerations for m1, m2 & m3 are downwards as well. Then, solving the following equations using a negative sign on all accelerations, I get 2.87 downward:

Mass 1:
T = -m1a1y + m1g (1)

Mass 2:
T-T1 = -m2ay (2)

Mass 3:
T1-m3g = -m3ay (3)

Is this the right approach for this problem?

I can't follow your notation, but if you get 2.87, you must have done it right.