The three objects in the drawing are connected by strings that pass over massles

Question

The three objects in the drawing are connected by strings that pass over massless and friction-free pulleys. The objects move, and the friction between the middle object and the surface of the table is so small it can be ignored. (Assume

m1 = 12.3 kg

and

m2 = 26.0 kg.)

How fast does the middle block accelerate?   m/s2

We want to use a Space Taxi as a model for the middle object. How much force would you need each thruster to apply in order to accurately represent each of the individual forces felt by the block?
(Using multiple thrusters simultaneously).
Thruster 1:   N
Thruster 2:   N
Thruster 3:   N
Thruster 4:   N


Surely we can accomplish this same Taxi motion with fewer thrusters. What's the most economical way to do this?
(Use as few thrusters as possible).
Thruster 1:   N
Thruster 2:   N
Thruster 3:   N
Thruster 4:   N

Explanation / Answer

a)
acceleration of whole system will be same
T1 and T2 are internal forces

a = net force along the string / total mass
= (m2*g - m1*g) / (m1+m2+80)
= (26*9.8 - 12.3*9.8) / (12.3 + 26 + 80)
= 1.13 m/s^2

b)
force of thruster 1 = T1
T1 = m1*g + m1*a
= 12.3*9.8 + 12.3*1.13
= 134 N

force of thruster 2 = N
N = 80*g
= 80*9.8
= 784 N

force of thruster 3 = T2
T3 = m2*g - m2*a
= 26*9.8 - 26*1.13
= 225 N

force of thruster 2 = M*g
M*g = 80*g
= 80*9.8
= 784 N

c)
thruster 2 and thruster 4 force are equal and opposite
Net force on thruster 3 = 225-134=91 N
other 3 forces will be 0

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