What Is the Minimum Frequency for a Mass in Vertical Circular Motion?

[ninetyfour]

Homework Statement

A 0.2kg mass attached to a rope 1.6m long is being spun in a vertical plane. Find the minimum frequency.

Homework Equations

F = 4(pi^2)(r)(f^2)(m)
Ft min = Fc - mg

- minimum frequency occurs at minimum tension

The Attempt at a Solution

I had this question on a test and spend like 45 minutes on it. I have no idea what to do, and I am curious as to what I should have done. I tried inserting numbers and re arranging formulas and substituting here and there and nothing seemed to work.

HELP? D:

 

[ehild]

F = 4(pi^2)(r)(f^2)(m)
Ft min = Fc - mg

- minimum frequency occurs at minimum tension

What kind of force is F in your first equation?
How much is the minimum tension?

ehild

 

[ninetyfour]

F = 4(pi^2)(r)(f^2)(m)

That F is centripital force. It can also be written as:

F = m(v^2) / r

 

[dulrich]

Thinking physically, if the frequency is not high enough what happens? The mass will not make it to the top of the circle, right? It will execute free-fall motion (i.e., a parabola) until the string catches it again. So the key is to consider the top of the circle.

What are the forces acting? Weight and tension. These combine together to provide the centripetal force maintaining the circular motion. The smallest the tension can be is zero. So we want to find the frequency in which the weight alone is able to provide the sufficient centripetal force.

I think you can probably take it from here...

 

[rwisz]

I understand your frustration with this question. It can be challenging to solve problems like this, but with the right approach and understanding of the concepts, you can find the solution. Here are some steps that you can follow to solve this problem:

1. Draw a diagram: Start by drawing a diagram of the situation. This will help you visualize the problem and understand the forces acting on the mass.

2. Identify the known and unknown variables: In this problem, the known variables are the mass (m = 0.2kg), the length of the rope (r = 1.6m), and the acceleration due to gravity (g = 9.8 m/s^2). The unknown variable is the frequency (f).

3. Determine the forces acting on the mass: In this case, the only force acting on the mass is the tension in the rope (T). This force is directed towards the center of the circular motion and is responsible for keeping the mass in its circular path.

4. Apply Newton's second law: Newton's second law states that the net force acting on an object is equal to its mass times its acceleration (F = ma). In this case, the net force is the tension (T) and the acceleration is the centripetal acceleration (ac). Therefore, we can write the equation as T = mac.

5. Substitute values and solve for frequency: Substituting the known values into the equation from step 4, we get T = (0.2kg)(4(pi^2)(1.6m)(f^2)). Simplifying this equation, we get T = 1.024(pi^2)(f^2).

6. Find the minimum tension: To find the minimum frequency, we need to find the minimum tension. This occurs when the mass is at the top of its circular path, where the centripetal force is equal to the force of gravity (Fc = mg). Therefore, we can write the equation as Ft min = Fc - mg. Substituting the values, we get Ft min = T - (0.2kg)(9.8m/s^2).

7. Substitute values and solve for frequency: Substituting the value of T (from step 5) into the equation from step 6, we get Ft min = 1.024(pi^2)(f^2) - 1.96N. Solving