# Determine the speed and tension of keys swinging in a circular path

• Traced
In summary: At the top of the swing, the keys have kinetic energy and, by virtue of their height, gravitational potential energy. As they descend, they lose GPE but get faster. Other than a tiny bit of air resistance, their total energy stays constant.At the top of the arc, how much KE do they have?In descending from the top of the arc to the bottom, how much GPE do they lose?So what is their KE at the bottom?The phrase "minimum speed at the bottom of the circle" is a bit confusing. Technically, that would be zero. So, how about "speed at the bottom of the circle, assuming minimum speed at the
Traced
Homework Statement
Keys with a combined mass of 0.100 kg are attached to a 0.25 m long string and swung in a circle in the vertical plane.
a)Determine the slowest speed that the keys can swing and still maintain a circular path.
b)Determine the magnitude of the tension in the string at the bottom of the circle.
Relevant Equations
Fnet = Fg+ Ft
Fc = Fg + Ft
Fc= mv^2 / r
Fg = mg
a)Determine the slowest speed that the keys can swing and still maintain a circular path.

Fnet = Fg + Ft
Fc = Fg + Ft

When Ft = 0, Fc = Fg
So, Fc = mv^2 / r and Fg = mg

mv^2/r = mg

v = √gr
v = √9.81 * 0.25
v = 1.56 m/s

Therefore, the slowest speed that the keys can swing and still maintain a circular path is 1.56m/s.

b) Determine the magnitude of the tension in the string at the bottom of the circle.

In order to determine the magnitude of the tension in the string at the bottom of the circle I would need to find the minimum speed at the bottom of the circle. How can I find that? I know that I need to include velocity from part a, but not sure what else to do.

The minimum velocity should happen at the top of the circle, and potential energy becomes increased velocity at the bottom of the circle, where you have two forces acting vertically and in the same direction.

Traced
Traced said:
I know that I need to include velocity from part a, but not sure what else to do.
Can you think of a relevant conservation law?

Traced
haruspex said:
Can you think of a relevant conservation law?
I just re-read through all of my lessons and I couldn't find any conservation laws. Is there another way to solve this question? The law of conservation of energy maybe? I just haven't seen that get used in any examples in my textbook for questions like this.

Last edited:
I saw someone online calculate the minimum speed at the bottom of the circle like this:
v=√v^2_top + 2gr
= √3 * v_top
= 2.71 m/s.
I don't really understand this equation, and I don't know what it's called or why it was used.

After this, to solve for the tension they did

Ft = mg + mv^2 / r

Ft = 3.92 N

I understand the tension part, just not how they solved for the minimum speed at the bottom of the circle.

Traced said:
The law of conservation of energy maybe?
That's the one, and this is what @Lnewqban described in post #2.
Traced said:
I saw someone online calculate the minimum speed at the bottom of the circle like this:
##v=\sqrt{v_{top}^2 + 2gr}##

Traced
haruspex said:
That's the one, and this is what @Lnewqban described in post #2.
Okay. Could you show me the steps I need to take to arrive at that equation? Or help me get started please!

Would I be using mv^2/r = mg?

Traced said:
Would I be using mv^2/r = mg?
No, because at the bottom there will be tension.
As you have been told twice already, it's conservation of energy that you need.
At the top of the swing, the keys have kinetic energy and, by virtue of their height, gravitational potential energy. As they descend, they lose GPE but get faster. Other than a tiny bit of air resistance, their total energy stays constant.
At the top of the arc, how much KE do they have?
In descending from the top of the arc to the bottom, how much GPE do they lose?
So what is their KE at the bottom?

The phrase "minimum speed at the bottom of the circle" is a bit confusing. Technically, that would be zero. So, how about "speed at the bottom of the circle, assuming minimum speed at the top". Yeah, I know, same thing.

From there... at the top you've PE (vertical) and KE (horizontal) ; at the bottom, no PE right ? just KE(/RE).

So, where did the PE go ?

## 1. What is the formula for determining the speed of keys swinging in a circular path?

The formula for determining the speed of keys swinging in a circular path is v = √(g * r), where v is the speed, g is the acceleration due to gravity, and r is the radius of the circular path.

## 2. How do you measure the tension of keys swinging in a circular path?

The tension of keys swinging in a circular path can be measured using a tension meter or by calculating the centripetal force acting on the keys, which is equal to the tension in the string.

## 3. What factors affect the speed and tension of keys swinging in a circular path?

The speed and tension of keys swinging in a circular path are affected by the length of the string, the mass of the keys, the radius of the circular path, and the force applied to the keys.

## 4. How does the speed and tension of keys swinging in a circular path change with different string lengths?

The speed of keys swinging in a circular path increases as the string length decreases, while the tension increases as the string length increases. This is because a shorter string allows for a smaller radius, resulting in a faster speed and higher tension.

## 5. Can the speed and tension of keys swinging in a circular path be controlled?

Yes, the speed and tension of keys swinging in a circular path can be controlled by adjusting the length of the string, the mass of the keys, and the radius of the circular path. The force applied to the keys can also be controlled to affect the speed and tension.

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