Centripital Force and Springs Combined

[Jennings]

1. Homework Statement
(Attached is an image depicting the situation)
A ball of mass m = 1.50kg is hung from a spring attached to a shaft. The length of the spring with the mass hanging from it is Lo = 50.0cm. The shaft then starts to rotate such that the spring stretches to a length L = 65.0cm. The spring constant of the spring is k=1450N/m.

A) At what angle theta does the rotating spring make with the vertical
B) How fast is the ball moving
C) How much work had to be done on the shaft to get the ball moving at the angle and speed given in parts a and b

Homework Equations

Springs : Fs = -kx
centripetal Force : Fc = mv²/r

The Attempt at a Solution

First thing I did was solve for the length of the spring at equilibrium (assuming the spring is massless). First I solve for the change in distance due to the mass attached.

mg = kx
mg/k = x = (1.50kg)(9.81m/s²)/(1450N/m)
x = 1.015cm

Length at Equilibrium
50.0cm - 1.105cm = 48.99cm

Now that I know the equilibrium length I will find the total force being applied to the spring since the length is known.

Change in length from equilibrium

65.0cm - 48.99cm = 16.01cm = .1601m

The total force acting on this spring is

F = kx = (1450N/m)(.1601m) = 232.1N

I don't even know if I'm on the right track with this. I have a free body diagram that I will post, but again I have no idea if I'm on the right track.

If I've made any mistakes please let me know.

 

[Jennings]

I don't know how to attach the file of my free body diagram

 

[Chestermiller]

So far you are on the right track.

Chet

 

[Jennings]

So far you are on the right track.

Chet

Hello Chestermiller. In order to solve for angle theta Atleast one more component of the force must be known. Weight is straight down and I'm guessing the centripetal force must make up rest of the net force. Isn't centripetal force is purely horizontal?

 

[Chestermiller]

Hello Chestermiller. In order to solve for angle theta Atleast one more component of the force must be known. Weight is straight down and I'm guessing the centripetal force must make up rest of the net force. Isn't centripetal force is purely horizontal?

Please write out your force balances on the mass in the x and y directions.

Chet

 

[Jennings]

Here is my attempt at solving for the centripetal force. I don't think its correct but it's all I know how to do

 

[Chestermiller]

How come your force balance equations don't have the components of the spring force acting on the mass in the horizontal and vertical directions?

Chet

 

[Jennings]

How come your force balance equations don't have the components of the spring force acting on the mass in the horizontal and vertical directions?

Chet

I thought that the spring force is the net force? That's how I was able to find the net force. Please clarify

 

[BvU]

It would be wise to see F_c as a resultant force, i.e. the net force. It is the sum of the two forces that work on the mass: one from gravity, the other from the spring.
The resultant force is what makes the mass follow a circular trajectory.
You calculate its magnitude (you need that for B.), but in A) the exercise asks for an angle...

 

[Chestermiller]

I thought that the spring force is the net force? That's how I was able to find the net force. Please clarify

The spring force is not the net force. The net force is the vector sum of the spring force and the weight. Now please, write out the force balances in the horizontal and vertical directions in terms of the horizontal and vertical components of the spring force, and the weight of the mass.

Chet

 

[Jennings]

It would be wise to see F_c as a resultant force, i.e. the net force. It is the sum of the two forces that work on the mass: one from gravity, the other from the spring.
The resultant force is what makes the mass follow a circular trajectory.
You calculate its magnitude (you need that for B.), but in A) the exercise asks for an angle...

Sounds good. Only there's one problem. I don't know the direction of the spring force, only its magnitude.

What I know :
Magnitude and direction of the weight
Magnitude of the spring force
Don't know:
Direction of spring force
Magnitude and direction of resultant centripetal force.

What can we do?
Is there a trick?

 

[Jennings]

Sounds good. Only there's one problem. I don't know the direction of the spring force, only its magnitude.

What I know :
Magnitude and direction of the weight
Magnitude of the spring force
Don't know:
Direction of spring force
Magnitude and direction of resultant centripetal force.

What can we do?
Is there a trick?

Wait hold on I think I got something

 

[Chestermiller]

No, there's no trick. You express the components of the spring force in terms of sine and cosine of theta. Then you write the force balances. Then you solve for theta.

Chet

 

[Jennings]

Chet[/QUOTE]

No, there's no trick. You express the components of the spring force in terms of sine and cosine of theta. Then you write the force balances. Then you solve for theta.

Chet

Something else needs to be known.
The sum of forces in the y direction must be 0 so that we can have an equation.

(Fs means force of spring)
ΣFy = Fscosθ - mg
ΣFx =Fssinθ

If the sum of forces in the y direction = 0 then we have our equation. Otherwise i don't kneo

 

[Chestermiller]

The sum of forces in the y direction must be 0 so that we can have an equation.

(Fs means force of spring)
ΣFy = Fscosθ - mg
ΣFx =Fssinθ

If the sum of forces in the y direction = 0 then we have our equation. Otherwise i don't kneo

Well, then you have your equation for the y direction. That gives you theta.

What is your force balance equation for the x direction?

Chet

 

[Jennings]

 

[Jennings]

 

[Jennings]

First attachment is solving for the angle which I found to be 86.4 degrees. (It really doesn't look that way in the picture I hope I didn't do it wrong). Second attachment is the sum of forces in direction x which is also the magnitude of the centripetal force

 

[BvU]

Looks good. Now on to b) and c) ?