# Conical Pendulum with varying string length

In summary, when considering a conical pendulum with a rotating ball attached to a string or rubber band, the angle and height will change depending on the angular velocity. However, regardless of the type of rubber band used, the height will remain the same as long as the angular velocity is constant. This is due to the constraints for steady state at different radii, as well as an energy-based argument.
Consider a conical pendulum like that shown in the figure. A ball of mass, m, attached to a string of length, L, is rotating in a circle of radius, r, with angular velocity, ω. The faster we spin the ball (i.e., the greater the ω), the greater the angle, θ, will be, and thus, the smaller the height, h will be.

But now imagine that instead of suspending the ball from a string, we use something like a rubber band that can stretch. Then the faster and faster we spin the ball, the more the rubber band stretches. So we expect L will increase, θ will increase, and h will decrease.

Now let's take a set of different rubber bands. The original lengths of all the rubber bands, with the mass, m, suspended from them vertically, are all identical. However, with a given amount of force, each rubber band stretches a different amount. In other words, say we can model a rubber band as if it were a spring. Then different rubber bands have different spring constants. But remember that they all have the same length, L, when the mass, m, is suspended vertically, before anything starts to spin.

So why is it that regardless of which rubber band you use, as long as you keep ω the same, h remains the same? I mean I worked out the equations and found this to be true. However, I found it to be an unexpected result. Can anybody give me some intuitive insight as to why this should be so?

By the way, this is totally independent of exactly what the nature of the force is that the rubber band exerts, as long as the force is a function of the amount the rubber band stretches.

So why is it that regardless of which rubber band you use, as long as you keep ω the same, h remains the same? I mean I worked out the equations and found this to be true. However, I found it to be an unexpected result. Can anybody give me some intuitive insight as to why this should be so?
Consider the constraints for steady state at different r:
- the vertical component of T is constant
- the horizontal component of T is proportional to r
This is exactly what you get, if h is held constant.

vanhees71
There are a lot of unexpected things. For example, imagine that you have a ring in the vertical plane. Take any point A on the ring and connect this point with the lowest point O by a thin smooth pipe OA. Put a small ball into the pipe at the end A. The ball will roll down to the point O inside the pipe . The time of rolling down does not depend on which point A on the ring you take.

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Can anybody give me some intuitive insight as to why this should be so?
I always try to find an Energy Based argument for this sort of question. In the steady state, energy stored due to the modulus of the elastic is not relevant and you can replace it with a string.
The gravitational Potential Energy will be mg(L-h) and the Kinetic Energy will be L m sin(θ)ω2/2.I equated the two and expressed it all in terms of L,θ and ω and ended up with:
2 = 2gL(1-cosθ)/cosθ and the L cancels out on either side and the angle is just a function of the angular velocity.

I think that's ok but the usual caveats apply with my algebra.

Alternatively, you can replace a conical pendulum with two superimposed plane pendulums in phase quadrature and in orthogonal planes. The Energy of each individual pendulum will be exchanging KE and PE during their cycles and the sums of the KEs and PEs will be constant.

## 1. What is a conical pendulum with varying string length?

A conical pendulum with varying string length is a type of pendulum that swings in a circular motion instead of a back and forth motion like a traditional pendulum. The string length of the pendulum changes as it swings, creating a cone-like shape.

## 2. How does the string length affect the motion of a conical pendulum?

The string length of a conical pendulum affects the speed and period of its motion. As the string length increases, the speed of the pendulum decreases, and the period of its motion increases. This is due to the change in the gravitational force acting on the pendulum.

## 3. What factors can affect the motion of a conical pendulum with varying string length?

The motion of a conical pendulum can be affected by several factors, including the length and mass of the string, the mass of the pendulum bob, and the angle at which the pendulum is released. Other factors such as air resistance and friction can also play a role in the motion.

## 4. How is the motion of a conical pendulum with varying string length calculated?

The motion of a conical pendulum can be calculated using the principles of circular motion and Newton's laws of motion. The period of the pendulum can be calculated using the formula T = 2π√(L/g), where T is the period, L is the string length, and g is the acceleration due to gravity.

## 5. What are some real-world applications of a conical pendulum with varying string length?

Conical pendulums with varying string length are commonly used in physics demonstrations and experiments to study the principles of circular motion. They can also be used in timekeeping devices, such as pendulum clocks, and in amusement park rides, such as the swing ride.

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