How Do You Calculate Mass M1 in a Frictional Inclined Plane Pulley System?

[bulldog23]

Homework Statement

M1 and M2 are two masses connected as shown. The pulley is light and frictionless. Find the mass M1, given that M2 (6.50kg) accelerates downwards at 3.27m/s2, that the angle theta is 20.0o, and that muk is 0.340.

Homework Equations

F=ma

The Attempt at a Solution

I drew my free body diagram for mass M1 and M2, and I also know that the acceleration is the same for both masses. Also, I know that the tension in the string should also be the same for the two masses. From there, I am kinda lost. Any help...

 

[bob1182006]

can you tell us what you got for Newton's second laws on each of the 2 bodies?
from there you can combine and solve for m1.

 

[bulldog23]

for M1 there is going to be a friction force parallel to the plane opposing the motion up the plane. There is a normal force perpendicular to the plane and mg going straight down. Then there is the force from M2 that causes the M1 to move up the ramp.

 

[bob1182006]

Kay but write it algebraically and don't plug in any of the numbers at first and that way you can get a formula for m2 which will depend on the angle/m1 and other things.

 

[bulldog23]

that is the part I am struggling with. This is my first time taking physics.

 

[twotaileddemon]

If you wait a little bit, I'll try my best attempt at the problem

 

[bulldog23]

alright sounds good

 

[bob1182006]

ok well then let's find all of the forces first:

m1:
friction force opposing the motion up the plane
Normal force
mg going straight down (best to split this into it's components)

m2:
ma downward

What other force is missing? From the string connecting the blocks right?

 

[bulldog23]

For M2 there should be a tension force upwards. For M1 there should also be a tension force up the ramp.

 

[bob1182006]

right, can you attempt to write it algebraically? you know all the forces and if say the + axis is going up and to the right you can determine whether that force is aiding the net force or resisting it.

I'll start you off, for m2 you will have something like:

T-mg=ma
mg is - since we assume that up is positive, you could say down is positive but you have to keep using that assumption always, you can't say for m2 down is + but for m1 down is -.

 

[twotaileddemon]

The change in the forces in the x and y direction for block one is
F_x = T - F_f = (m_1)a (Friction force = F_f)
F_y = F_n - (m_1)gcos* = 0 (normal force = F_n)
and for block two is
F_x: (m_2)g - T = (m_2)a
There is no y component on the second block
To help you understand why, rotate m_2 so that the ropes make a straigt line. Then mg is to the right. Realistically it's not, but it helps you understand how the T's are connected

From there, you should be able to substitute equations to find m_1

 

[bulldog23]

tension is aiding the net force, and friction is opposing it. It makes sense what you wrote for M2 but I don't know how to do it for M1.

 

[twotaileddemon]

If you need help (don't look until you try yourself):
T = F_f + (m_1)a = uf_n + (m_1)a
f_n = (m_1)gcos*
T = u(m_1)gcos* + (m_1)a

(m_2)g - (m_2)a = T
(m_2) = T/(g-a)
T = (m_2)(g-a)

u(m_1)gcos* + (m_1)a = (m_2)(g-a)
(m_1) = (m_2)(g-a)/(ugcos* + a)
sorry, had to edit it a bit

 

[bob1182006]

well what forces are helping the block go up? what forces are opposing the motion upwards?

just the sum of those helping are added, opposing are subtracted will = m2a.

 

[twotaileddemon]

I got 13.56 kg as my answer
Which, looking over, is unrealistic since M1 needs to be lighter than M2
I'm still looking over my work, so I won't give up~

 

[bulldog23]

11.97 is not right. Not sure where we went wrong.

 

[bulldog23]

I got 13.56 kg also, but it says that it is incorrect. I used the (m_1) = (m_2)(g-a)/(ugcos*)

 

[twotaileddemon]

I'm still working on it, after I edited my work I got 6.63 now, but that still doesn't seem right. I'm getting closer. so I'll keep trying.

 

[twotaileddemon]

I'm in the middle of editing this.. so it will change, just leaving it hear so I have everything in order~

F_x = T - F_f = (m_1)a (Friction force = F_f)
F_y = F_n - (m_1)gcos* = 0 (normal force = F_n)
and for block two is
F_x: (m_2)g - T = (m_2)a

T = F_f + (m_1)a = uf_n + (m_1)a
f_n = (m_1)gcos*
T = u(m_1)gcos* + (m_1)a

(m_2)g - (m_2)a = T
(m_2) = T/(g-a)
T = (m_2)(g-a)

u(m_1)gcos* + (m_1)a = (m_2)(g-a)
(m_1) = (m_2)(g-a)/(ugcos* + a)

Using that, I had gotten 6.63 kg.. but it should be less still.

 

[bulldog23]

Yeah that still doesn't make sense because M1 must be less than M2. Thanks

 

[twotaileddemon]

I checked, and rechecked various times
My answer seems to be 6.63 kg.. are you sure no numbers were copied down wrong?

 

[bulldog23]

6.63 kg isn't right. I checked the numbers in the problem and they are all correct.

 

[twotaileddemon]

Hm.. this is certainly very strange.
Obviously the answer is less than 6.5
but I checked my equations and cannot find what I did wrong.. I even checked my textbook and made sure all my diagrams were correct oo;

Is this on Webassign or something?

 

[bulldog23]

I know I did the same thing and I cannot figure it out. It's a CAPA problem

 

[twotaileddemon]

Well.. let's try to get through this somehow
Obviously, the first block has a normal force, an mg with a component of mgcos* that opposes it - in the vertical direction - since no acceleration takes place there - F_n - mgcos*= 0
the first block is has tension and a frictional force, which oppose each other, and equal ma. Because it's moving towards the pulley T - F_f = (m_1)a

block 2 has mg holding it down and T going up. It is accelerating. Therefore (m_2)g - T = (m_2)a

Solving for tension on both sides, and subsituting one into the other, I found m_1 as described in my equation on this page (not the last, as they had some errors that I fixed)

I looked over my procedure careful, and I cannot see any mistakes. I am truly at a lost. I'm sorry, but either 1) the problem itself has an error, or 2) I'm missing something, which I cannot see... but everything I says makes sense, right?

 

[bulldog23]

yeah I understand your reasoning but not sure where it goes wrong. There must be a small mistake somewhere

 

[twotaileddemon]

How soon do you need the answer? Maybe someone else might see this and help...
I'd like to stay here, but I'm at my wits end..

 

[bulldog23]

I still have a lot of time. Thanks for trying though, it is much appreciated!

 

[hotvette]

Deleting incorrect/misleading response...

 

[twotaileddemon]

Ah! so you got the same answer? 6.63? then maybe the problem is wrong.. try questioning whoever gave it to you

It's 2:40 am here so I need to get rest, but I'd be happy to help with anything tomorrow if you want. Just look at my profile for contact information if you need it.

 

[bulldog23]

I don't see how it is possible for M1 to weigh more than M2, yet M2 is pulling M1 up the plane?

 

[twotaileddemon]

I don't see how it is possible for M1 to weigh more than M2, yet M2 is pulling M1 up the plane?

Actually, it may be possible for m2 to be accelerating upwards, the M1 down the plane

 

[bulldog23]

but the problem specifically says that M2 is accelerating downwards, so that means that M1 must be going up the plane.

 

[hotvette]

deleting incorrect/misleading response...

 

[twotaileddemon]

Ah... then we certainly do have quite the problem n_n;
anyway, I really have to go now~
I wish you the best of luck~