What Acceleration Will the System Move With?

[wicked_vixen]

Homework Statement

.........__m5____
...|\....../....|
...|..\....../...|
..m1.|...m2....m4...|.m6
...|...\..../.....|
...|...45.\___m3___/30...|- All the boxes (m₁, m₂, m₃, m₄, m₅) are connected by strings.
- m₂ and m₄ are leaning on the wall.
- m₃ and m₅ are standing
- m₁ and m₆ are hanging
- the first incline (the one at the left) has an angle of 135 degrees (conventional) >> thus 45 degrees north of west (in a manner of speaking)
- the second incline has an angle of 30 degrees

m₁=35 kgm₂=15kg
u_s2=0.40
u_k2=0.20

m₃=7.0kg
u_s3=0.70
u_k3=0.60

m₄=5.0kg
u_s4=1.0
u_k4=0.80

m₅=15kg
u_s5=u_k5=0

m₆=10kg

AT WHAT ACCELERATION WILL IT MOVE?

Homework Equations

g = 9.8 m/s²
sin 30 = 0.500
cos 30 = 0.866
sin 45 = cos 45 = 0
F_f = u*F_N
v_f = v_i + at
d = v_it + ¹/₂at²
v_f² = v_i² + 2ad
v = (v_f + v_i) / 2

The Attempt at a Solution

I tried to get all m*u_s and added them all together but it doesn't seem right. And I didn't know where to go next and I'm not sure if I even started right.

 

[kyiydnlm]

draw free-body diagrams first; then you will find the answer by yourself.

 

[tomcue]

I would approach this problem by first identifying all the forces acting on the system. These include the weight of each object (mg), the normal force (FN) from the inclined planes, and the frictional forces (Ff) acting on the boxes that are in contact with the surfaces. The equations for these forces are:

- Weight (mg): where m is the mass of the object and g is the gravitational acceleration (9.8 m/s2)
- Normal force (FN): FN = mg*cos(theta), where theta is the angle of the inclined plane
- Frictional force (Ff): Ff = us*FN, where us is the coefficient of static friction and FN is the normal force

Next, I would draw a free body diagram for each object and write down the equations of motion for each one. This will allow me to determine the net force acting on each object and use Newton's second law (F = ma) to solve for the acceleration of the system.

For example, for m1, the equations of motion would be:

- Fx: T1*cos(45) - Ff1 - T2*cos(135) = m1*a
- Fy: T1*sin(45) + T2*sin(135) - m1*g = 0

Solving these equations simultaneously will give the acceleration of the system.

I would also need to consider the motion of the other objects and how they interact with each other. For example, the tension in the strings (T1 and T2) will be affected by the frictional forces acting on m2 and m4.

In addition, I would need to consider the rotational motion of the system, as m2 and m4 are leaning against the wall and may cause the system to rotate. This would require me to apply the equations of rotational motion (torque and angular acceleration) to solve for the acceleration of the system.

Overall, this is a complex problem that would require careful analysis and application of various equations and principles of physics.