Verifying the Limits of Newton's Second Law with Blocks

[IHateMayonnaise]

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:

$$a=\frac{m_3g-(m_1+m_2)g\mu_k}{(m_1+m_2+m_3}$$
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:
$$\sum(1,x): T+F_{f,s}-F_{f,k}=(m_1+m_2)a$$
$$\sum(1,y): F_{n,1}-(m_1+m_2)g=0$$

$$\sum(2,x): F_{f,s}=m_2a$$
$$\sum(2,y): F_{n,2}-m_2g=0$$

$$\sum(3,x): 0=0$$
$$\sum(3,y): m_3g-T=m_3a$$

Where the frictional forces take on the familiar value of the coefficient times the normal. Solving, I get:

$$a= \frac{m_3g+\mu_sm_2g-\mu_kg(m_1+m_2)}{m_1+m_1+m_3}$$


For part b) while testing the familiar limits, i found that as $$m_3 \rightarrow \infty$$ then $$a=g$$, which makes sense. I found that if $$m_2=0$$, then we get what we expect. And when I calculate when $$m_2 \rightarrow \infty$$, $$a=0$$. BUT, when I try to calculate as $$m_1\rightarrow \infty$$, $$a=\frac{-\infty}{\infty}$$, 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 $$a$$, the maximum acceleration: $$a=\mu_sg$$, which doesn't make sense either since it should depend on mass.

Thoughts??

 

[kuruman]

Part (a) is OK. It is the same result as if blocks 1 and 2 were glued together to form a single block of mass m₁+m₂.

Part (b) is not OK. The force that accelerates m₂ (net force) is the force of static friction, f_s. Newton's 2nd Law says that it must be equal to m₂ times the acceleration from part (a). As you increase m₃, you increase the acceleration which means that f_s must increase accordingly. But f_s cannot increase forever. At the point where f_s can no longer increase, and ony then, can you say f_s = μ_sN. You have all the ingredients to you can put it together from this point on.

 

[IHateMayonnaise]

Thanks for the reply kuruman.

So my equation for the acceleration is correct then? You say that I cannot include fs = μsN until the point in which the static friction can not keep up with the acceleration of blocks 1 and 3, so then it shouldn't be in my equation for the acceleration?

What about part c)? Under what condition does block 2 stay on top of block 1? I understand that this only happens if fs = μsN, eg if μs=fs/N (or less), but in this equation what is fs?

 

[kuruman]

The force of static friction f_s is whatever is necessary to provide the observed acceleration. Because it is the net force on mass m₂, you can write

$$f_{s}=m_{2}a=m_{2}\frac{m_{3}g-\mu_{k}g(m_{1}+m_{2})}{m_{1}+m_{2}+m_{3}}$$

That is always true unless the top mass starts sliding. At the threshold of when sliding is just about going to happen, the force of static friction has reached its maximum value and can no longer "provide the observed acceleration" past that value. What does the above equation become then? What is the maximum value of the force of static friction?

 

[IHateMayonnaise]

When:

$$\mu_s\geq \frac{F_{f,s}}{F_n}=\frac{am_2}{g}\left(\frac{m_{3}g-\mu_{k}g(m_{1}+m_{2})}{m_{1}+m_{2}+m_{3}}\right)$$

?

Also what about the problem with negative acceleration when m_2 -> infinity?

 

[kuruman]

Start with

$$f_{s}=m_{2}a \leq \mu_{s}F_{n}$$

Replace a and F_n with the appropriate quantities.

Also what about the problem with negative acceleration when m_2 -> infinity?

What about it? Do you really think that if you make mass m₂ large enough, say the size of a battleship, the masses will accelerate in the opposite direction? I didn't think so. What is more likely to happen instead? Remember, the acceleration you have has been derived assuming that the masses accelerate so that the hanging mass drops.