What am I doing wrong in this problem dealing with Angular Speed

[ixerr]

Homework Statement

For some reason, I keep getting this problem wrong, could someone please look over it and tell me what I am doing incorrectly? Here it is:
A student ties a 410.0 g rock to a 1.25 m-long string and swings it around her head in a horizontal circle. At what angular speed (in rev/min) does the string tilt down at a 13.9° angle, assuming that the local acceleration due to gravity is -9.80 m/s^2?

Homework Equations

Force centripetal = (mass*velocity) / radius

The Attempt at a Solution

To support the 0.410 kg rock at an angle of 13.9° angle below horizontal,
the vertical component of the tension in the string must equal the weight of the rock
weight of rock = 0.410 * 9.8 = 4.018 N
Vertical component of tension = T * sin 13.9°
T * sin 13.9° = 4.018 N
T = 16.726 N

The horizontal component of the tension is the Force centripetal
The horizontal component of the tension = T * cos 13.9° = 16.726 N

Force centripetal = (mass * velocity^2) / radius
16.236 = 0.41 * v^2 / 1.25

v = 7.036 m/s
1 radian = 1.1 m
Angular velocity = 6.396 rad/s

 

[cepheid]

Homework Statement

For some reason, I keep getting this problem wrong, could someone please look over it and tell me what I am doing incorrectly? Here it is:
A student ties a 410.0 g rock to a 1.25 m-long string and swings it around her head in a horizontal circle. At what angular speed (in rev/min) does the string tilt down at a 13.9° angle, assuming that the local acceleration due to gravity is -9.80 m/s^2?

Homework Equations

Force centripetal = (mass*velocity) / radius

The Attempt at a Solution

To support the 0.410 kg rock at an angle of 13.9° angle below horizontal,
the vertical component of the tension in the string must equal the weight of the rock
weight of rock = 0.410 * 9.8 = 4.018 N
Vertical component of tension = T * sin 13.9°
T * sin 13.9° = 4.018 N
T = 16.726 N

The horizontal component of the tension is the Force centripetal
The horizontal component of the tension = T * cos 13.9° = 16.726 N

Force centripetal = (mass * velocity^2) / radius
16.236 = 0.41 * v^2 / 1.25

v = 7.036 m/s
1 radian = 1.1 m
Angular velocity = 6.396 rad/s

The radius of the circle being swept out is not 1.25 m. That's one problem with your solution.

 

[ixerr]

Ah I'm confused, why is it not? And how would I get it then..?

 

[cepheid]

Ah I'm confused, why is it not? And how would I get it then..?

Think about looking at the circle from the side. If the string were horizontal and tracing out a circle, then the radius of the circle would be equal to the length of the string. But since the string is actually rotated a few degrees below horizontal, the radius of the circle being traced out is LESS than the length of the string, and trigonometry can easily tell you how much less.

 

[ixerr]

Alright, that makes way more sense. So I would find the true radius, and just plug it into the formula I was already using, right?

 

[Bread18]

Check your vertical and horizontal components of forces, draw a diagram.

 

[ixerr]

cepheid, I got the radius, which is 1.288 m ,but you said it's supposed to be less..
I did it by drawing it out and through this method:

cos13.9 = 1.25/h
hcos13.9 = 1.25
h = 1.25/cos13.9= 1.288

 

[Bread18]

cepheid, I got the radius, which is 1.288 m ,but you said it's supposed to be less..
I did it by drawing it out and through this method:

cos13.9 = 1.25/h
hcos13.9 = 1.25
h = 1.25/cos13.9= 1.288

cos13.9 = h/1.25 that's where you're going wrong

 

[ixerr]

I thought cosine was adjacent over hypotenuse?
Anyway, I'll go with it. So by doing cos 13.9= h/1.25,
I get h= 1.21m
Therefore,
16.236 = (.41 * v^2) / 1.21
v= 6.93 m/s
angular velocity = 6.30 rad/s
angular speed in rev/min = 61.165

Does this look correct..?

 

[cepheid]

cos13.9 = h/1.25 that's where you're going wrong

Yes exactly, Bread18 is right. You had the reciprocal of cosine instead of the cosine.

 

[ixerr]

Okay, I'm still doing something wrong, though. The answer I just put in was incorrect.
What else am I doing wrong? :/

 

[cepheid]

I thought cosine was adjacent over hypotenuse?

Yes, it is adjacent over hypotenuse. Which means it needs to be the way that Bread18 and I said it should be, NOT the way you initially had it. Check out this diagram:

http://img809.imageshack.us/img809/5593/ballonstring.png
[/URL]

 

[ixerr]

OHH I had the numbers flipped haha, thanks for the diagram, it clears things up. I suck at physics -___-
I am still stuck on what I'm doing wrong now, however.
Thanks for the help so far you guys!

 

[cepheid]

OHH I had the numbers flipped haha, thanks for the diagram, it clears things up. I suck at physics -___-
I am still stuck on what I'm doing wrong now, however.
Thanks for the help so far you guys!

Can you post the steps you used to convert from rad/s to rpm?

 

[Bread18]

v= 6.93 m/s
angular velocity = 6.30 rad/s

How did you convert velocity to angular velocity too?

 

[ixerr]

Okay, so to get from velocity to angular velocity, I took the velocity (7.036) and divided it by 1.1 which gave me 6.396 rad/s.

To get from 6.396 rad/s to revolutions/minute, I did it through this method:

Pi radians=180 Degrees

(6.3/Pi)x180Degrees

=360.96 degrees per second
1 revolution =360 degrees
So, I take 360.96 and divide by 360 to get:
=1.002676141 revolutions per second
I multiply that by 60 and get:
=61.1605 revolutions per minute

Is this right..? I actually put in 61.165 instead of 61.16 for my answer, so that might have been an issue when I was putting in my answer, if I did everything else correctly?

 

[Bread18]

Okay, so to get from velocity to angular velocity, I took the velocity (7.036) and divided it by 1.1 which gave me 6.396 rad/s.

Why did you divide it by 1.1?

 

[ixerr]

Because 1 radian = 1.1 m
I think...

 

[ixerr]

Actually I don't even know.

 

[cepheid]

Okay, so to get from velocity to angular velocity, I took the velocity (7.036) and divided it by 1.1 which gave me 6.396 rad/s.

To get from 6.396 rad/s to revolutions/minute, I did it through this method:

Pi radians=180 Degrees

(6.3/Pi)x180Degrees

=360.96 degrees per second
1 revolution =360 degrees
So, I take 360.96 and divide by 360 to get:
=1.002676141 revolutions per second
I multiply that by 60 and get:
=61.1605 revolutions per minute

Is this right..? I actually put in 61.165 instead of 61.16 for my answer, so that might have been an issue when I was putting in my answer, if I did everything else correctly?

Three points:

1. I think you meant to type 60.1605, but either way it doesn't matter because it is the wrong answer. You introduce a huge error by using 6.3 instead of 6.396. I don't know why you did that.

2. You did WAY too many steps above. You could have avoided the whole "converting to degrees" step by just noting that 1 revolution = 2pi radians.

3. As Bread18 has rightly pointed out, all of this is moot, because 6.396 rad/s is the wrong angular velocity in the first place. What is the relationship between angular velocity (ω) and linear velocity (v)?

EDIT: 1 radian is NOT 1.1 m. This doesn't even make any sense as a statement, because radians measure angles, not lengths. Furthermore, since the radian system defines an angle as the ratio of the arc length to the radius (both of which are lengths), the radian is therefore a dimensionless unit.

 

[ixerr]

The relationship with angular velocity and linear velocity is

v=rw
right?
Sooo... I use that to get w?

 

[cepheid]

The relationship with angular velocity and linear velocity is

v=rw
right?
Sooo... I use that to get w?

Yes, that is correct.

 

[ixerr]

Alright, so by using that equation, I get:
7.036 m/s = (1.21 m) (w)
w = 5.799 rad/s

And by converting it, I get 55.37 rev/min

Is this correct?