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IHateMayonnaise
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Homework Statement
Imagine that there is a slab (call it block 1) on top of which sits a block (call it block 2). The slab connects to a cable that passed through a pulley and attaches to another block (call it block 3), hanging freely. Assuming that the coefficient of kinetic friction is the same for both blocks (and the same with static friction):
a) Find the acceleration of the system.
b) Verify the familiar limits.
c) Under what conditions does block 2 stay on top of block 1?
2. Homework Equations /The attempt at a solution
If blocks 1 and 2 move together, then all three blocks have the same acceleration:
[tex]a=\frac{m_3g-(m_1+m_2)g\mu_k}{(m_1+m_2+m_3}[/tex]
If block 2 begins to slip, then we still take all the accelerations as the same (block 2 is just beginning to slip, so it is still pretty much moving with block 1 (i.e. same acceleration) and it's frictional coefficient will be static)
TO MAKE A LONG STORY SHORT: DID I SET UP NEWTON'S 2ND LAW CORRECTLY?
then Newtons 2nd Law for block 1, block 2 and block 3(respectfully) are:
[tex]
\sum(1,x): T+F_{f,s}-F_{f,k}=(m_1+m_2)a
[/tex]
[tex]
\sum(1,y): F_{n,1}-(m_1+m_2)g=0
[/tex]
[tex]
\sum(2,x): F_{f,s}=m_2a[/tex]
[tex]\sum(2,y): F_{n,2}-m_2g=0[/tex]
[tex]\sum(3,x): 0=0[/tex]
[tex]\sum(3,y): m_3g-T=m_3a[/tex]
Where the frictional forces take on the familiar value of the coefficient times the normal. Solving, I get:
[tex]a=
\frac{m_3g+\mu_sm_2g-\mu_kg(m_1+m_2)}{m_1+m_1+m_3}[/tex]
For part b) while testing the familiar limits, i found that as [tex]m_3 \rightarrow \infty[/tex] then [tex]a=g[/tex], which makes sense. I found that if [tex]m_2=0[/tex], then we get what we expect. And when I calculate when [tex]m_2 \rightarrow \infty[/tex], [tex]a=0[/tex]. BUT, when I try to calculate as [tex]m_1\rightarrow \infty[/tex], [tex]a=\frac{-\infty}{\infty}[/tex], which doesn't make sense. Did I screw up Newtons 2nd law?
For part c) all I did was use Newton's 2nd law for the second body and solved for [tex]a[/tex], the maximum acceleration: [tex]a=\mu_sg[/tex], which doesn't make sense either since it should depend on mass.
Thoughts??