A conical pendulum is formed by attaching a 500 g ball to a...

Key Concepts

Conical Pendulum

A conical pendulum is a system where a mass is attached to a string and moves in a horizontal circle, causing the string to trace out the surface of a cone. This arrangement leads to a situation where the tension in the string is resolved into vertical and horizontal components, making it a useful example of force equilibrium and circular motion analysis.

Force Equilibrium

In a conical pendulum, the forces acting on the mass are in equilibrium in the vertical direction. The vertical component of the tension in the string balances the gravitational force, ensuring that the mass remains at a constant height while the horizontal component provides the necessary centripetal force.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path. In a conical pendulum, the horizontal component of the tension in the string acts as the centripetal force, directing the mass toward the center of the circle, and is essential for maintaining uniform circular motion.

Tension Decomposition

The tension in the string is split into two components: one that counters the weight of the object (vertical component) and one that accelerates the mass centripetally (horizontal component). Analyzing these components using trigonometric relationships is central to solving problems involving conical pendulums.

Angular Speed and Unit Conversion

Angular speed quantifies how fast an object rotates around a circle and is commonly expressed in radians per second. In many practical problems, however, this value is converted into revolutions per minute (rpm) for easier interpretation. Understanding the conversion between these units is vital for relating theoretical analysis to real-world applications.

Transcript

00:02 We're asked to figure out expressions for the tension and the angular velocity of a conical pendulum, which is a pendulum where you have, it actually kind of swings around in a circle at the base instead of just back and forth.

00:21 So it moves in a circle, so not just in a plane.

00:25 So you have some, this procession around here.

00:29 And so what's happening is that, you know, this thing is moving, which, some angular velocity omega and we have a string here and then we have the length of that string and then the radius of the circle that this bob is moving in so if you look at it from the top basically it would be moving in a circle so just swinging around in a circle now let's see here we have using force balances we know that in the vertical direction here there's no acceleration in that direction so we know that the acceleration is zero and then then the forces are t cosine of theta, where t is the tension in the string, and then t cosine theta is the component in the vertical direction.

01:13 And then we have minus weight, the weight, mg, is the, this is the force balance in the vertical direction.

01:21 Now in the horizontal direction or the radial direction, so we're just going to look at, you know, wherever this is in the radial direction, we know that the acceleration, the centripetal acceleration is minus m r omega square and that we know the net tension or the net force acting in this plane is minus t sine theta so those are going to be in the same direction both in the radial direction now if we want to get um these things in terms of the geometry that was given so the in terms of l and r we know that uh cosine of theta here is h over l a opposite over adjacent and then we know h from the pythagorean theorem is the square root of l squared minus r squared so we can then solve this equation here for t and substitute for cosine of theta so we get the t equals the weight times the length of this arm this string divided by the square root or the length of the string squared minus the radius of the circle squared so one thing you can look at here and when you when you when you derive things symbolically, it's nice because you can actually look at some of limited cases.

02:45 So let's say if r goes to zero, so if r goes to zero, then l squared of l squared is l, and then the t just becomes the weight, which makes sense because that means that, you know, if r was zero, then this thing isn't swinging around anything.

02:59 And we just basically could be spinning, but it's not swinging.

03:05 So even if it's spinning, the tension and the string is still going to be just the weight of the mass at the end of the string.

03:13 So that makes sense.

03:15 And then you can, you know, you can look at like, same thing if, if l gets very big with respect to r.

03:21 So if this becomes negligible compared to this, then you get the very same thing.

03:25 So, you know, this was very, very long and the, you know, the radius was very small, you're not going to get, the tension is going to be approximately the weight of the bob.

03:36 Now, to get the, to get the angular velocity, we can use that we need sign of theta and that's just r over l.

03:47 And so we can cancel out a minus sign here...

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