# Work done on a conical pendulum

• brotherbobby
In summary, the diagram shows the problem with equations for the vertical and horizontal directions. The first equation gives the tension in terms of the variables provided. The question is how to calculate the work done by the tension force and what angle does it have with the velocity at a given instant. The tension is perpendicular to the velocity at all times, meaning the work done by the tension force is 0.
brotherbobby
Homework Statement
The diagram below shows a conical pendulum of mass ##m##, length ##L## and vertical height ##h## rotating with uniform speed in a plane carving out a circle of radius ##r##. The tension in the (massless) wire is ##T##. Calculate the work done by the tension on the pendulum bob after one complete revolution.
Relevant Equations
1. Horizontal and vertical components of the tension force : ##T_V = T \cos \theta## and ##T_H = T \sin \theta##. (Please note the angle ##\theta## is the semi vertical angle - it is drawn or defined with respect the actual vector and the vertical, not the horizontal).

2. Newton's law for the acceleration of a body in an inertial frame : ##\vec a = \frac{\vec F}{m}##.

3. Centripetal force on a body moving with a speed ##\vec v## in rotational motion : ##\vec F = -\frac{mv^2}{r} \hat r##. (Please note that this value does not depend as to whether the rotational motion is uniform or non-uniform. In case of non-uniform motion, both ##\vec v## and ##\vec F## would be functions of time).
The diagram for the problem is shown alongside. In the vertical (##\hat z##) direction we have ##T \cos \theta = mg##.

In the plane of the pendulum, if we take the pendulum bob at the left extreme end as shown in the diagram, we have ##T \sin \theta = \frac{mv^2}{r}## (the ##\hat x## axis of the pendulum is rotating along with it).

The first equation gives us the tension in the string in terms of the variables provided : ##T = mg \sec \theta = mg \frac{L}{h}##.

How to calculate the work done by the tension force ##-## this is where am stuck.

In particular, what is the angle that the tension has with the velocity ##\vec v## at a given instant? If I could find that angle, say some ##\alpha##, I can connect it to the semi-vertical angle ##\theta## as some ##\alpha (\theta)## and evaluate the work done by the tension by the line integral : ##W_T = \int_0^{2\pi} \vec T(\theta) \cdot d\vec l##, where ##d\vec l = \vec v dt## and the (uniform) speed can be connected to the tension, the radius of the pendulum and its mass from dividing the two equations above : ##v^2 = gr \tan \theta##.

##\color{blue}{Any\; help\; would\; be\; welcome}##.

brotherbobby said:
what is the angle that the tension has with the velocity →vv→\vec v at a given instant
In the picture the ##\vec T## is clearly in the plane of the paper, whereas ##\vec v## is perpendicular to it, towards the viewer !

BvU said:
In the picture the ##\vec T## is clearly in the plane of the paper, whereas ##\vec v## is perpendicular to it, towards the viewer !

Yes indeed, sorry I should have realized it. What you mean is that even though the tension ##T(\theta)## is not along the ##\hat z## direction, it is nonetheless perpendicular to ##\vec v## at this instant?

And since this instant is the same as the next, if we keep rotating our plane of paper so that ##T(\theta)## always lies on it, the velocity vector ##\vec v## will be always perpendicular to the tension for the whole motion?

If that is true, am I right in assuming that the work done by the tension force ##T(\theta)## is 0?

Yes. There is no motion in the direction of ##\vec T##.

## 1. What is a conical pendulum?

A conical pendulum is a type of pendulum where the bob (weight) moves in a circular motion rather than a back and forth motion like a traditional pendulum. The bob is attached to a string or rod that is suspended from a fixed point, creating a conical shape.

## 2. How is work done on a conical pendulum?

Work is done on a conical pendulum when the bob moves from one point to another. This is because the bob is constantly changing direction and therefore, its velocity. The force of gravity acts on the bob, causing it to accelerate towards the center of the circular motion. This acceleration requires work to be done, which is calculated by the force of gravity multiplied by the distance the bob moves.

## 3. What factors affect the work done on a conical pendulum?

The work done on a conical pendulum is affected by the mass of the bob, the length of the string or rod, and the speed at which the bob is moving. A heavier bob, longer string, and faster speed will result in more work being done.

## 4. How is the work done on a conical pendulum related to its potential and kinetic energy?

The work done on a conical pendulum is related to its potential and kinetic energy through the law of conservation of energy. As the bob moves from its highest point to its lowest point, it gains kinetic energy and loses potential energy. The work done on the pendulum is equal to the change in potential energy, which is then converted into kinetic energy.

## 5. What is the practical application of a conical pendulum?

Conical pendulums have a variety of practical applications, including measuring the strength of gravity, testing the accuracy of clocks and watches, and studying the motion of celestial bodies. They are also commonly used in amusement park rides and as a visual aid in physics demonstrations.

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