Newton's Law on Circular Motion

In summary: Of course, the real reason is that it's wrong is that it gives the wrong answer, but you can see that from the units alone: The units of acceleration are m/s², while those of force are kg·m/s². It's not mathematically sound, either, because you're not doing anything with the unknown force.)Okay, thank you very much!So I have:F = ma = m(dv/dt)fk = -(umv^2)/Rdv/dt = -(R/um)(dfk/dt)integrate both sides:Vf = Vi + atVo/3 = Vo + (-(R/um)(fkf)/dt)(t)
  • #1
mc8569
25
0

Homework Statement


Block mass "m" is sliding on frictionless horizontal surface while traveling inside of hoop radius "R". Coefficient of f between block and wall is "u", therefore, speed of block is decreasing. In terms of "R", "v, block's veloc.", "m", and "u", find expressions for:

1. frictional force on the block:

2. block's tangential acceleration (dv/dt).

3. From #2 find time required to reduce speed of block from its original velocity to one-third of its original velocity.


Homework Equations


fk = un
F = ma
ar = v^2/r <-- ar is radial acceleration


The Attempt at a Solution



My process for 1:
fk = un
ar = v^2/R
F = n = ma
n = m(v^2/R)

fk = (umv^2)/R

2:
F = ma
I have no clue on this one.
Would it be from #1, use
v^2 = fR/um
v = sqrt[(fR)/(um)]
And take the derivative of that? And it will be my tangential acceleration?
but the thing is, you have to give the answer in terms of m, R, u, and v only. Can't put in terms of f - friction

3: I assume to be easy with #2? It's just simple kinematics right?

I found a previous inquiry on this problem in the forums but it wasn't very clear, so I wanted to ask it myself.
 
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  • #2
Your answer to question 1 is correct, but fk is negative with respect to the velocity of the block. It is also parallel with the velocity, a tangential force, acting along the tangent of the circle, and causing tangential acceleration. Now, how do you get acceleration from force?

ehild
 
  • #3
You get acceleration from force by using F = ma

So would I:
fk = ma
a = fk / m?
Could you please give me some kind of hint, something that triggers my thinking? Because you're posing the question that I'm essentially asking.

If a = fk / m
i would simply have:
fk = -(umv^2)/R

a = -(uv^2)/R

Is that correct? I feel that is too simple. Especially because the question poses: "Find the block's tangential acceleration dv/dt", kind of hints towards taking some kind of derivative of a velocity equation? Please tell me what you think.
 
  • #4
ahhh please help me someone.. I need help on this by Wednesday =(
 
  • #5
mc8569 said:
a = -(uv^2)/R

Is that correct?
Yes.
I feel that is too simple. Especially because the question poses: "Find the block's tangential acceleration dv/dt", kind of hints towards taking some kind of derivative of a velocity equation? Please tell me what you think.
That hint will come in handy in part 3.
 
  • #6
You have written
fk = μ*mv^2/R.
Since v is slowing down, fk is not a constant.
So d(fk)/dt = (μm/R)*2v*dv/dt.
Here dv/dt is the tangential acceleration.
 
  • #7
Yes! Thank you for answering =DDD

Now, I did what you said, which is also what I originally asked if I should do and I have
dv/dt = -(R/(2umv))(d(fk)/dt)

The issue I have, which I presented when I originally asked if I should do that, is with the fact that the answer must be in terms of m, R, u, and v. I have in terms of fk...

That is where my problems are, because I have nowhere to go that I know of. I can't take the derivative of fk to plug in, because I will simply end up canceling everything out by using the same equation.
Help mee! I'm so hopeless on this problem right now..
 
  • #8
If you're still talking about part 2, you already solved it in post #3.
 
  • #9
Oh.. okay awesome!

But I don't get what you mean by the hint about dv/dt will come into play for part c) because how I am solving for c) is quite simple:

with a = -(uv^2)/R

Vf = Vi + at
(1/3)Vo = Vo + (-uv^2/R)t
I assume that Vo = v and therefore
t = (2R)/(3uv)

Is that right? Thanks sooo much! So helpful
 
  • #10
No, not right. You can't use Vf = Vi + at, since the acceleration is not constant. (That's why it seemed so easy!)

That's where the hint comes in:
a = dv/dt = -(uv^2)/R

Use a bit of calculus.
 
  • #11
Doc I've given it thought but I really have no clue what to do, what direction to go in.

Should I integrate the equation? I don't know what would that even do though.
 
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  • #12
should I bring up dt and bring over v^2 to the left and all its coefficients, and integrate that from Vo to Vo/3?

like such:

dv/dt = -uv^2/R
(-R/u)(1/v^2)dv = dt
integrate that equation?? It doesn't seem to make much sense though. I'm so clueless..
 
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  • #13
mc8569 said:
should I bring up dt and bring over v^2 to the left and all its coefficients, and integrate that from Vo to Vo/3?
Yep. (Where you leave the constants is up to you.)
like such:

dv/dt = -uv^2/R
(-R/u)(1/v^2)dv = dt
integrate that equation??
Yep.
It doesn't seem to make much sense though.
Why not?
 
  • #14
If I integrate it should I be integrating it from Vo to Vo/3? Or do I do something else with the integrated equation for t. I think I will integrate from Vo to Vo/3 that seems logical.

Also, could you give me a brief explanation as to why doing:

fk = -(umv^2)/R
v^2 = -fR/um
2v(dv/dt) = -[df/dt]R/um
isolate dv/dt

is incorrect for solving for acceleration? Both seem right to me mathematically, so I don't understand how I would know which one to do, say if I were doing an exam & had no physics forums help.
 
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  • #15
mc8569 said:
If I integrate it should I be integrating it from Vo to Vo/3? Or do I do something else with the integrated equation for t. I think I will integrate from Vo to Vo/3 that seems logical.
Makes sense to me.

Also, could you give me a brief explanation as to why doing:

fk = -(umv^2)/R
v^2 = -fR/um
2v(dv/dt) = -[df/dt]R/um
isolate dv/dt

is incorrect for solving for acceleration?
But what does this accomplish? All it does is express the acceleration in terms of the derivative of the unknown friction force. Just going around in circles.

Note that the instructions tell you to express your answers in terms of R, v, m, and u. This doesn't get you any closer to that goal.
 

What is Newton's Law on Circular Motion?

Newton's Law on Circular Motion states that an object moving in a circular path at a constant speed experiences a force directed towards the center of the circle. This force is known as centripetal force and is necessary for the object to maintain its circular motion.

What is centripetal force?

Centripetal force is the force that acts towards the center of a circle and is required for an object to maintain its circular motion. In other words, it is the force that keeps an object moving in a circular path instead of flying off in a straight line.

What is the relationship between centripetal force and velocity?

The centripetal force required for an object to maintain its circular motion is directly proportional to the square of its velocity. This means that as the velocity of the object increases, the centripetal force required to keep it moving in a circle also increases.

How does mass affect circular motion?

The mass of an object does not affect its circular motion directly. However, a larger mass may require a higher centripetal force to maintain its circular path, as stated by Newton's Second Law which states that force is equal to mass multiplied by acceleration.

What are some real-world examples of circular motion and centripetal force?

Some common real-world examples of circular motion and centripetal force include the Earth revolving around the Sun, a pendulum swinging back and forth, and a car making a turn on a curved road. Other examples include a roller coaster moving along its track, a planet orbiting its star, and a satellite orbiting Earth.

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