- #1

brotherbobby

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- 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).

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}##.