How to find equation of motion for object leaving a curved ramp?

[cantleave]

1. Here is the sketch:

http://i.stack.imgur.com/0s1is.jpg

The sketch is supposed to be side-view of the path of the object.

The following values are known:

- r - radius of the circle that describes the path AB of the object
- a - angle that characterizes the part of a circle that describes the path AB of the point
- m - mass of the point
- V0 - velocity

The dashed line is the object's trajectory after it leaves AB. N is the normal force, T is friction and g is the gravitational acceleration.


2. What I need to find out:

1. Equation of motion for AB
2. Equation of motion for BC
3. velocity at B
4. The distance DC



3. The Attempt at a Solution

I was able to solve this problem partially when AB is a straight line and 'a' represents the angle between AB and AD. So far I could come up with only this:

m*(x)'' = -T-mgsin(?) <- in place of the question mark I would need the angle between AB and AD

m*(y)'' = N-mgcos(?)

N = mgcos(?)

T = μN = μmgcos(?)

(x)'' = -g(μcos(?) + sin(?))

(x)' = -gt(μcos(?) + sin(?)) + c1

x = ((-9t^2)/2)(μcos(?) + sin(?)) + c1 + c2

where μ is the coefficient of friction. x and y are functions of the x and y coordinates with respect to time.

How do I deal with the fact the ramp is no longer a straight line but a curved line? I need to solve this problem, otherwise I cannot apply for taking the exam in mechanics 1. I appreciate any thoughts. Thank you very much for your help.



By the way, I'm so happy I found this place and I also signed up for a free membership at educator.com. This is awesome, I never had access to this much knowledge in form of video-courses.

Thank you physicsforum.com and educator.com!

 

[BiGyElLoWhAt]

I'm sorry, what's T?

[edit]
-.-

Found it.

 

[BiGyElLoWhAt]

How familiar are you with polar coordinates?

 

[cantleave]

How familiar are you with polar coordinates?

I'm familiar with them, I took a course on calculus II. I knew it was something with polar coordinates but I have very little knowledge in dynamics and kinematics. I only solved simple equilibrium problems so far. Thank you for helping me.

 

[BiGyElLoWhAt]

No problem. I would personally use polar coordinates to describe the motion AB.

So what all forces are acting on your system? And how can you describe them in terms of the unit vectors r hat and theta hat?

 

[cantleave]

Forces acting on the object you mean? Gravity, friction, the normal force, and velocity. I know that r hat = sin(theta)cos(phi)x hat + sin(theta)sin(phi)y hat and theta hat = cos(theta)cos(phi)x hat + cos(theta)sin(phi)y hat

I'm sorry but I don't know how to describe those forces in terms of r hat and theta hat. Man, I feel such a noob.

 

[cantleave]

Can I do the following: N hat = -N*r hat and v0 hat = v0*theta hat

What I mean is that I happen to find a similarity between the way we deal with unit vectors for polar coordinates and the fact that the normal force is perpendicular to the velocity.

Am I correct?

 

[BiGyElLoWhAt]

XD no worries.

1. velocity is not a force, so the forces are gravity, friction and normal.
2. That is true for your unit vectors, but in spherical coordinates (3d) we only need 2 dimensions. so rhat in this case would be (keeping your notation) < cos(phi)xhat , sin(phi)yhat > = rhat
what's theta hat in 2d?

 

[BiGyElLoWhAt]

Can I do the following: N hat = -N*r hat and v0 hat = v0*theta hat

What I mean is that I happen to find a similarity between the way we deal with unit vectors for polar coordinates and the fact that the normal force is perpendicular to the velocity.

Am I correct?

That's a very good observation, and exactly why I suggested using polar coordinates. I'm not sure that's exactly right though. Double check what you have. What's the magnitude of the LHS, what's the magnitude of the RHS?

(They're pretty close)

 

[cantleave]

If you're referring to the minus sign (which has nothing to do with magnitude xD, but is my best guess) I put that there because normally r hat would point in the opposite direction of N.

I'm lost, I really am.

 

[BiGyElLoWhAt]

No lol.

Maybe this isn't what you meant, but what I'm reading is $\hat{N} = -N\hat{r}$ and $\hat{v_0} = v_0\hat{\theta}$ Also, I wouldn't use v_0 hat, just because v_0 is a constant. so $v_0$ and $\hat{v}$.

What's the magnitude of a unit vector? and what's the magnitude of $N\hat{r}$?

Also rember, unit vectors' sole purpose in life is to denote direction.

 

[verty]

Parts 2,3,4 are all related; if you can answer 3, 2 and 4 become much easier. Ok, how to solve 3? AB is just like a pendulum, right?

Part 1 is significantly more difficult, usually a small-angle approximation is used but in this question that can't be assumed. I think it may be best to just ignore part 1.

 

[BiGyElLoWhAt]

Part 1 is significantly more difficult, usually a small-angle approximation is used but in this question that can't be assumed. I think it may be best to just ignore part 1.

Part 1 IS significantly more difficult, but I don't think "ignore it" is good advice.

 

[verty]

Part 1 IS significantly more difficult, but I don't think "ignore it" is good advice.

This is the Introductory Physics Homework section.

 

[cantleave]

okay...the magnitude of a unit vector is 1 and the magnitude of N$\hat{r}$ is ?(idk but $\hat{r}$ should have the magnitude, sense and direction of $\hat{N}$ otherwise they are not equal)

It's getting late in this part of the globe. I appreciate your help, I'll rejoin the discussion tomorrow, I hope that's okay.

How do I get this LATEX thing work btw?

 

[BiGyElLoWhAt]

True, but I've seen more challenging problems worked through in this subforum before. Maybe this should be moved, but I don't see the point in helping OP solve part of the problem. Just my opinion.

 

[verty]

My opinion is that part 1 is the type of question that one can't really help to solve without explaining how to do it. I mean there is not really a good hint to give that doesn't lead to more questions and the eventual demonstration of how to solve it. And that is against the rules, the whole purpose is that the person asking the question must do the work.

 

[BiGyElLoWhAt]

the magnitude of the unit vector $\hat{N}$ is just what you said, 1, whereas the magnitude of the RHS $(N)(\hat{r})$ seeing as how r hat is a unit vector, is N times 1 , i.e. N.

What you want to do in terms of unit vectors is look for ways to equate the directions wrt each other.

That's fine, catch yourself some shut eye.

 

[cantleave]

I didn't know that solving a particular problem for the OP was against the rules. In this case I'm going to review these concepts BiGyElLoWhAt was trying to get me understand and come back. I am all in for this method, asking questions till the OP manages to solve the problem, but never thought you guys have the patience and time to do it.

 

[BiGyElLoWhAt]

My opinion is that part 1 is the type of question that one can't really help to solve without explaining how to do it. I mean there is not really a good hint to give that doesn't lead to more questions and the eventual demonstration of how to solve it. And that is against the rules, the whole purpose is that the person asking the question must do the work.

I see your point, and I guess if it gets to that point OP will just have some googling to do.

Speaking from personal experience: even if you do end up looking up most of the stuff elsewhere, sometimes it's nice to have someone who knows what they're doing to give you some direction as to how to solve the problem.

 

[BiGyElLoWhAt]

I actually enjoy doing mechanics, and miss mechanics class... (we only had 2 at my university)

but yes: we can't solve problems for you.

 

[verty]

I didn't know that solving a particular problem for the OP was against the rules. In this case I'm going to review these concepts BiGyElLoWhAt was trying to get me understand and come back. I am all in for this method, asking questions till the OP manages to solve the problem, but never thought you guys have the patience and time to do it.

You must understand. A dialog that consists of a person like yourself progressing, getting stuck, having directed help given to them to get them unstuck, etc, that is fine of course. But the type of dialog that goes like this: "use so and so, but how, integrate so-and-so to get so-and-so, I don't understand, do this... (demonstration), I sort of understand but why did you do step X?", there is a rule against this type of help when one does all the work.

Anyway, you haven't solved parts 2,3 and 4 (the easier parts) yet.

 

[cantleave]

Ok, here are the things I don't get so far:
1.) < cos(phi)xhat , sin(phi)yhat > = rhat - My lecturer used "<",">" to denote scalar multiplication of two vectors so I don't get this one.

2.) $\hat{N}$ is not a unit vector, it's the normal force whose magnitude is N.

3.)

whereas the magnitude of the RHS (N)($\hat{r}$) seeing as how $\hat{r}$ is a unit vector, is N times 1 , i.e. N.

OK, this one actually makes sense.

4.)

but in spherical coordinates (3d) we only need 2 dimensions.

You mean spherical coordinates in 2d?

Could you please clarify these things before moving on? Thank you

Also, the other unit vector of polar coordinate system is: $\hat{φ}$ = − sin φ $\hat{x}$ + cos φ $\hat{y}$

 

[verty]

$\hat{N}$ is the unit normal, $\vec{N}$ is the normal vector.

The rest is what I said we won't be doing, picking apart the answers that are trying to help you. You have yet to solve part 3 which I identified as being the easiest part to solve. It's basic pendulum math, any physics book that covers the pendulum will tell you how to solve that. Once you can answer part 3, parts 2 and 4 become easier, it is a parabolic trajectory and you will have enough information to solve it, to find the distance and the formula.

I see no progress here, no attempts.

 

[cantleave]

I was searching for similar solved problems on the net all this week. I'm just wasting my time here. In my class we didn't use that hat sign ever, we either use the arrow to denote vectors or don't use anything in which case the letter denotes the magnitude.

If I could solve this problem using hints I get from various people I wouldn't have come here in the first place. You tell me a few words like "pendulum" and and god knows what and from that I should be able to solve it. I should have told you guys that I'm a beginner, nobody forced you to come here and post on this thread, if you don't have the patience that's ok but please don't tell me i am not putting enough effort into it because that is very offensive.

I'll need to watch a 70+ hours video course on AP physics B and hope that it covers this topic.

 

[Tanya Sharma]

Hello cantleave

Please don't leave . This forum is an excellent place to learn new things.

I am no expert but will try if I can be of any help . And hope that BiGyElLoWhAt and other experts join us.

Leave notations about unit vectors for a while .

When the object makes an angle θ with the vertical
1) What are the forces acting in tangential direction ?
2) What are the forces acting in radial direction ?

Express 1) and 2) in terms of N,T,M,θ .

 

[cantleave]

1.) T (I was told that the velocity vector is not a force)
2.) N

I'm failing to find a relationship between these and the angle θ, except that: N*T=0 (scalar multiplication)

I found the following formula in my lecture notes. It is an equation of motion for an object rolling down on a circle.

v = $\sqrt{(v_{0})^{2}+2gR(1-cosΘ)}$

Can this help me?

 

[Tanya Sharma]

1.) T (I was told that the velocity vector is not a force)
2.) N

No . Don't you think there will be a component of weight mg along these directions ?

1.)

I found the following formula in my lecture notes. It is an equation of motion for an object rolling down on a circle.

v = $\sqrt{(v_{0})^{2}+2gR(1-cosΘ)}$

Can this help me?

No . We are dealing with a different situation.

 

[cantleave]

Yeah, right, I need to consider the projections of M on the radial and tangential directions.

Mcos($\frac{pi}{2}$-Θ) + Mcos(Θ) = M

But how do these relate to T and N?

 

[Tanya Sharma]

Yeah, right, I need to consider the projections of M on the radial and tangential directions.

Mcos($\frac{pi}{2}$-Θ) + Mcos(Θ) = M

You need to consider Mg i.e force , not M (mass ) .

Which of the two i.e Mgcosθ and Mgsinθ is along the tangential direction ?

 

[cantleave]

Mgsinθ is along the tangential direction.

 

[Tanya Sharma]

OK.

So we can write the net force acting on the object at any angle θ as

$$\vec{F} = (mgcosθ-N)\hat{r} + (-mgsinθ-T)\hat{θ} $$

Here $\hat{r}$ is the unit vector in radial direction with positive being outwards and $\hat{θ}$ is the unit vector in a direction perpendicular to $\hat{r}$ in the direction of increasing θ .

Note that θ is measured from the vertical with counterclockwise increasing (positive) .

Does this make sense ?

 

[cantleave]

Yes, this does make sense! Thank you. I suppose though it get's more complicated.

But wait a minute, did I put the friction force in the right sense? Shouldn't it be pointing toward the opposite direction of the tendency of movement? Or do we assume that the object moves only counterclockwise?

 

[verty]

Yes, this does make sense! Thank you. I suppose though it get's more complicated.

So what would be the resultant force at time t on the object at it travels along AB? Assume there is no friction.

 

[cantleave]

If there is no friction

$$ \vec{F} = (mgcosθ-N)\hat{r} - mgsinθ\hat{θ} $$

Or is this what you were referring to?

I can't get this latex work