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Difficulty with this Problem involving a Pully and Cart

Homework Statement
A cart with a mass of 1.75 kg is pulled by a string over a pulley attached to a hanging
mass of 400 g.
a. What is the acceleration of the cart if the pulley is considered massless?
b. The pulley has a radius of 3.5 cm and a mass of 100 g. It can be considered a
solid disc. What is the actual acceleration of the cart?
Homework Equations
Attached
My attempt is also attached. Basically, I think I have (a) down. The only issue is I am getting the wrong answer, and I was hoping one of you could let me know if the answer key is wrong or the way I did it was wrong? The answer is 1.823m/s^2. Also, for the second part, I am really not sure where to go from here. If someone could get any advice that'd be awesome. (Also, the answer to (b) is 1.782m/S^2.
 

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haruspex

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Problem Statement: A cart with a mass of 1.75 kg is pulled by a string over a pulley attached to a hanging
mass of 400 g.
a. What is the acceleration of the cart if the pulley is considered massless?
b. The pulley has a radius of 3.5 cm and a mass of 100 g. It can be considered a
solid disc. What is the actual acceleration of the cart?
Relevant Equations: Attached

My attempt is also attached. Basically, I think I have (a) down. The only issue is I am getting the wrong answer, and I was hoping one of you could let me know if the answer key is wrong or the way I did it was wrong? The answer is 1.823m/s^2. Also, for the second part, I am really not sure where to go from here. If someone could get any advice that'd be awesome. (Also, the answer to (b) is 1.782m/S^2.
Please see post #2 at your prior posting of this problem, https://www.physicsforums.com/threads/general-physics-i-question-moment-of-inertia-problem-difficulty.972618/.
Also, please do not post algebraic working as an image. That is for diagrams and textbook extracts. Take the trouble to type in your work. That is generally much easier to read and allows those responding to quote specific lines for reference.
 
Please see post #2 at your prior posting of this problem, https://www.physicsforums.com/threads/general-physics-i-question-moment-of-inertia-problem-difficulty.972618/.
Also, please do not post algebraic working as an image. That is for diagrams and textbook extracts. Take the trouble to type in your work. That is generally much easier to read and allows those responding to quote specific lines for reference.
Sorry, was told to post again with writing everything. I'll be sure to write out the work next time.
 
Sorry, was told to post again with writing everything. I'll be sure to write out the work next time.
So, I tried setting it up as ma for cart = ma for the crate, but I go the wrong answer. Could you let me know where I am going wrong with that?
 

haruspex

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So, I tried setting it up as ma for cart = ma for the crate, but I go the wrong answer. Could you let me know where I am going wrong with that?
As I posted, draw separate FBDs for the hanging block and the cart. Assign an unknown T as the tension. Write out the ΣF=ma equation for each
 
Alright,
Here i have my fbd1 and fbd2 attached. Any advice on where to continue from here?
 

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I would say the gravity pulls on the mass, and the mass pulls on the string, which pulls on the cart?
 

haruspex

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I would say the gravity pulls on the mass, and the mass pulls on the string, which pulls on the cart?
That does not express the fact that the string stays the same length. What relationship between the two accelerations does express that?
And as Chet notes, your ∑F=ma for the cart is wrong. m2a2 is not an applied force.
 
So geometrically how are the acceleration magnitudes related? (I don’t want to hear about gravity, masses, or forces)
Are the acceleration magnitudes the same? I don't know how not to mention gravity when that is what is pulling on the mass.
 
That does not express the fact that the string stays the same length. What relationship between the two accelerations does express that?
And as Chet notes, your ∑F=ma for the cart is wrong. m2a2 is not an applied force.
Geometrically, they are at 90degree angle? Or atleast the string is.
 

haruspex

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Geometrically, they are at 90degree angle? Or atleast the string is.
That is not a relationship between the magnitudes of the two accelerations and has nothing to do with the total length of string being constant.
 
19,164
3,782
Are the acceleration magnitudes the same? I don't know how not to mention gravity when that is what is pulling on the mass.
Yes, the acceleration magmoudes are the same? If the crate moves downward 1 cm, how mich does the cart move horizontally?
 
That is not a relationship between the magnitudes of the two accelerations and has nothing to do with the total length of string being constant.
I actually think I found a and it makes sense... Now here is my work for (b).
Σt=Ia
T-mg=(6.125*10^-5kgm^2)a
T-3.92N= (6.125*10^-5kgm^2)a
Not sure how to find T though... Did I even set that up correctly? Any advice on where to go from here?
 
Yes, the acceleration magmoudes are the same? If the crate moves downward 1 cm, how mich does the cart move horizontally?
I actually think I found a and it makes sense... Now here is my work for (b).
Σt=Ia
T-mg=(6.125*10^-5kgm^2)a
T-3.92N= (6.125*10^-5kgm^2)a
Not sure how to find T though... Did I even set that up correctly? Any advice on where to go from here?
 

haruspex

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I actually think I found a and it makes sense... Now here is my work for (b).
Σt=Ia
T-mg=(6.125*10^-5kgm^2)a
T-3.92N= (6.125*10^-5kgm^2)a
Not sure how to find T though... Did I even set that up correctly? Any advice on where to go from here?
As I wrote in the other thread, for part b you need three FBDs; one for the cart, one for the hanging mass and one for the pulley.
The tensions in the two sections of string are now different.
Again, the accelerations are related, but for the pulley it is an angular acceleration. To avoid confusion, please use a different label for that; α is customary.
 
As I wrote in the other thread, for part b you need three FBDs; one for the cart, one for the hanging mass and one for the pulley.
The tensions in the two sections of string are now different.
Again, the accelerations are related, but for the pulley it is an angular acceleration. To avoid confusion, please use a different label for that; α is customary.
Great. Okay, so here I have drawn out the fbds for all three.
Is this correct?
 

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jbriggs444

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You realize that you've been asked to type this stuff in, right? "Writing" means "typing". Taking pictures of handwritten text does not count.
 
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Alright, I think I have that done. Please see below.
I want to see 3 separate equations: a force balance on the cart, a force balance on the crate, and a moment balance on the pulley. These equations should have net forces or moments set equal to ma or ##I\alpha## (I don't want to see any ##\Sigma F's##).
 

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