pulley acceleration helppp?

Question

If the small block has mass m=10kg, find the mass M of the larger block to give the smaller block an upward acceleration of 3 m/s2
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*sorry my drawing sucks...

Answer 1

Draw a force diagram:
On the small block, there are 2 forces:
1) gravity: Fg = mg = 98 N (it points downward)
2) tension, T (it points upward)

Newton's 2nd law tells us that the sum of external forces on an object is equal to the object's mass times its acceleration. So, for the small block:

F = -mg + T = ma

Go ahead and solve this for T (T is the only unknown quantity in the above equation):

T = ma + mg = m(a + g) = (10)*(3 + 9.8) = 128 N

Now, for the second block, there are 2 forces acting on it:
1) gravity: Fg = Mg (downward)
2) Tension: T = 128 N (upward)

Now, the blocks must accelerate together(or else the string would stretch), so the acceleration of the larger block is the same as the smaller block (only it is in the downward direction):

F = -Mg + T = -Ma (here, a = +3 m/s²)

Solve this for M:
-Mg + Ma = -T
M(a - g) = -T
M = -T / (a-g) = -(128) / (3-9.8) = 18.82 kg

Answer 2

with ropes and pulleys it is irrelevent because of friction. It is just the ratio of the # of lines on the biggest block and the pull line

Answer 3

m2g-m1g = 3(m1+m2)
m2(g-3) = m1(g+3)
m2 = m1(g+3)/g-3)
m2 = 10(9.8+3)/(9.8-3)
m2 = 10(12.8)/(6.8)
m2 = 18.8 g