How long does it take mass m to fall distance h?

[vandersmissen]

Homework Statement

[PLAIN]http://rawrspace.com/Capture.JPG

So when you draw out a force diagram, we find these two equations
mg-T=ma
rewriting in terms of T
T=mg-ma
and
T-MgSin\vartheta-f=Ma
rewriting in terms of T
T=Ma+MgSin\vartheta+f

I set the two equations equal to each other so I could solve for the acceleration since it is as rest when it starts (I figured that means that initial velocity is 0 )
setting them equal I get:
mg-ma=Ma+MgSin\vartheta+f
Then I moved them so that I could start factoring out terms:
g(m-MSin\vartheta)-f=a(M+m)
finally, solving for a:
[g(m-MSin\vartheta)-f]/(M+m)=a

I also know that y-y_0=V_0t-(1/2)at^2
Since y final is 0 and y_0 is h, I get this equation:
-h=-(1/2)at^2
So this means that
(2h/a)^(1/2)=t

Pluging in a that I solved for above, I get

{(2h)/[g(m-MSin\vartheta)-f]/(M+m)}^(1/2)=t

That was my answer of the time it takes for the mass m to fall a distance h. Does that seem correct to you, or am I overlooking something ?

 

[kuruman]

It looks right to me except your answer has f in it and you are specifically required to write the answer in terms of m, M, h, θ and μ. So how can you get rid of f and introduce μ?

 

[vandersmissen]

Thank you for taking a look. I don't know why I wrote it like that on here, on my paper I have μN for the frictional force instead of just the f for frictional force..

Thank's again for taking a look.

 

[kuruman]

You're still not there because μN won't do either. What is N in terms of the given quantities m, M, h, θ?

 

[vandersmissen]

Well N is the normal force so it is the opposite component the gravitational force which is Mg so it would be -Mgcos\vartheta wouldn't it (assuming I take my g to be positive)? That would still leave me with a g though which isn't in those terms. Is it okay to have g since it is a constant ? If so then I would replace -f with μMgcos\vartheta (it would be positive correct, as -f = -μN = -μ(-Mgcos\vartheta) )

So I would like to write my answer as

{(2h)/[g(m-Msin\vartheta)+μMgcos\vartheta]/(M+m)}^(1/2)=t

 

[kuruman]

Don't forget that, in your original expression, f stands for the magnitude of the frictional force. That is a positive number. So f = μN = Mgcosθ. You already took care of the direction when you wrote down T-Mgsinθ-f=Ma.

 

[vandersmissen]

So It would be μMgcosθ correct since N = Mgcosθ , and f = μN

So that means that it would be -μMgcosθ in the problem instead of -f

t={[(2h)(M+m)]/[g(m-Msinθ)-μMgcosθ]}^1/2

I believe that would be correct as it is now in the correct terms, I moved the M+m to the numerator. Does that seem correct now, If it was -f which took care of the magnitude since it subtracted in the original force equation. Then replacing it with the information I derived above should be correct.

 

[kuruman]

Right.

 

[vandersmissen]

Thank you for helping me understand where I went wrong.