Finding Acceleration and Stopping Time in a Multi-Block Lifting System

[ChrisBrandsborg]

Homework Statement

A lifting system can be modeled as two blocks of mass m1 and m2 hanging from two pulleys, connected by ropes to a third block of mass M = 100 kg, lying on top of a horizontal surface. There is gravity g = 9.80 m/s2.

a) Ignoring at first friction, find an expression for the acceleration of the blocks, as a function of the masses and gravity.

b) Now, we assume that the system starts at rest, and that there is static friction between the block M and the surface it is lying on. How much diffe- rence in mass can m1 and m2 have, for the system to not start moving, if the coefficient of static friction is μs = 1.00? Provide an expression as well as a numerical value.

c) Next, we assume that the system is moving with an initial speed vi = 2.00 m/s, when suddenly kinetic friction between M and the surface is triggered (the brakes are switched on). The coefficient of kinetic friction is μk = 0.700. How long does it take for the system to stop if m2 = m1 = 100 kg ? Provide an expression as well as a numerical value.

Homework Equations

ΣF = ma
F* = F

(Newton´s laws)

The Attempt at a Solution

I think I did a correctly, but I have some problems solving b.
I put positive direction downwards to the left.

a) Firstly I separated the system into 3 systems:

1) a_m1 = m₁g - T₁
2) a_m2 = T₂ - m₂g
3) This one is the third block, lying on the top of the surface.

a_m3 = T₁ - T₂

then add together:

so that: a (m₁ + m₂ + m₃) = m₁g - T₁ + T₂ -m₂g + T₁ - T₂

a = (m₁ + m₂)*g / (m₁ + m₂ + m₃)

b)
I know that μs = 1.00.
Fs = nμs = mgμs = 100kg * 9.80 * 1.00 = 980N

That means that m1g - m2g = 980N
Then divide by g:

m1 - m2 = 100kg

Is that the answer? That the difference in mass has to be less or equal to 100kg?

 

[ChrisBrandsborg]

b)
I know that μs = 1.00.
F_s = nμ_s = mgμs = 100kg * 9.80 * 1.00 = 980N

That means that m₁g - m₂g = 980N
Then divide by g:

m₁ - m₂ = 100kg

Is that the answer? That the difference in mass has to be less or equal to 100kg?

 

[ChrisBrandsborg]

I tried c as well:

v_i = 2.00 m/s
μ_k = 0.700
How long does it take for the system to stop?

F_k = ma
μ_kmg = ma

a = μ_kg

v = v_i + μ_kgt

t = v_i/μ_kg

t = 2.00/0.700*9.80 = 0.292s

Correct? Or do I have to use the masses as well? m1 = m2 = 100kg?