Homework Help: Mechanics pendulum question

1. Dec 4, 2005

ElDavidas

Got another question for everyone.

I've been looking over some exam past papers for mechanics and I'm stuck on a problem.

"Suppose that a planar pendulum has a weightless rod of length l and a pendulum bob of mass m. The only external force acting on the pendulum is gravity of magnitude mg.

There is a number c such that, if the pendulum bob passes through the downward position with a speed of magnitude > c, it will eventually pass through the upward vertical position, and if it passes through the downward vertical position with a speed of magnitude < c, it will never reach the upward vertical position. Determine the number c. Neglect friction forces in this problem."

There is a diagram that goes with this problem but it's gonna be difficult to draw. Basically comprises of a horizontal y axis (pointing right), a vertical x axis (pointing downwards) and a vector drawn with an angle delta between the vector and the x axis.

I'm fairly certain you have to use conservation of energy but don't really know where to begin. I understand the concept of conservation of energy but don't know how to apply it to problems.

Thanks

Dave

2. Dec 4, 2005

LeonhardEuler

Alright, you know the formula for kinetic energy is $E_K=\frac{1}{2}mv^2$ and the formula for gravitational potential energy near the surface of the earth is $E_P=mgh$. m is the mass of the object, v is its velocity, h is the hieght and g is acceleration due to gravity. You know that $E_K+E_P=E=constant$. So what you need to do is find the kinetic and potential energies at the initial and final stages of this process. Keep in mind that it does not matter where you choose h to be zero, since only changes in hieght matter. A convenient place would be at the bottom of the pendulum. Think about what the kinetic enrgy should be at the top. Remember: we want to find the minimum
energy we need to put in for the mass to reach the top.

3. Dec 5, 2005

ElDavidas

Ok, I think I follow you.

Been looking over my mechanics notes and they say:
T2 - T1 = V2 - V1

where Ti, Vi represent kinetic and potential energies respectively at times t1 and t2.

So if I make the height 0 at the bottom of the pendulum, this would mean the height of the top of the pendulum is 2l (l is the length of the rod).

Using the formula for potential energy, this implies

V2 - V1 = 2mgl - mg(0) = 2mgl ?

Not quite sure about what to do with the kinetic energies. If the velocity is > c (and therefore the pendulum reaches the top) then can I say:

T2 - T1 = 1/2mc^2 - 1/2mc^2

and if the velocity is < c then

T2 - T1 = 1/2mc^2 ?

Don't think the kinetic energy is right though.

4. Dec 5, 2005

arildno

ElDavidas:
Remember that the minimum velocity c that achieves this, is that all kinetic energy at the bottom position becomes converted into potential energy at the top position.

5. Dec 6, 2005

ElDavidas

Hmmm, ok. So does this mean you let

1/2mc^2 = 2mgl

and solve for c?