My flu-like symptoms are fading at last (I don't know if it was flu or just a bad cold, but the effect was the same). The weather here in England has been cold, at or below freezing since before Christmas, and well-below at night and we have had several inches of snow and the wind is blowing straight from the Russian steppes - it's cold bbbrrrrrrrr! Last night's temperature fell to 9 degrees below, here and 28 below in Scotland. More snow forecast for this afternoon.

I have cleared a footpath through the snow to my workshop and have dragged an old garden patio heater into it. Unforunately the gas bottle is empty so I am going to get a replacement one today, if I can drive the car to the store without wrecking it - the icy roads round here are lethal! Once the heater's working I shall be able to get back to work and finish this darned wheel!

LATER - got the gas and the heater works!

JC

Be careful with gas heaters, they can be lethal too. (I'm sure you know that)

ReplyDeleteBest of luck, glad you're feeling better.

Yeah- glad you're feeling better. Let me join in here and applaud your- flu battle, shovelling snow, driving the car, fixing a heater etc, etc.

ReplyDeleteAs usual, a report on everything but the wheel(?)

The usual excuses for not doing anything.

ReplyDeleteThere's nothing usual about the current weather here in England! And quit sniping guys, I've told you it'll work and what I'll do if it doesn't. Have a little patience.

ReplyDeleteJC

The only ones not doing anything useful here are those anons making stupid and lame comments. I am 100% certain that John will do exactly what he said he will do - all in good time. Keep up the good work John!

ReplyDeleteHow can you be 100% confident??? - and you're calling us stupid!

ReplyDeleteAwh those lazy anons!

ReplyDeleteA point mass m is attached to a rod of negligible mass and length l and suspended from a frictionless pivot so that it can swing freely in a vertical plane. It is then lifted to the horizontal and released. What is the average downwards force exerted by the rod on the pivot?

ReplyDeleteSorry, in sufficient information, Alex - I don't know. Actually I probably wouldn't try to answer any question like that because the comments area on the blog would become unwieldy.

ReplyDeleteJC

One and a half times the weight of the mass...

ReplyDeleteWhy's that, Alex?

ReplyDeleteJC

Let theta be the angle between the rod and the vertical.

ReplyDeleteThe tension in the rod due to the weight of the mass is then

mg cos(theta) [1]

The kinetic energy of the mass as it falls must equal the loss in potential

energy

mgh = mv^2/2

where h is the distance the mass has fallen = l cos(theta)

so mv^2 = 2mgl cos(theta)

The tension in the rod needed to keep the mass moving in a circular path is

mv^2/l

= 2mg cos(theta) [2]

so the total tension in the rod is [1] + [2]

= 3mg cos(theta)

The rod exerts a downwards force on the pivot given by

ReplyDeleteF = 3mg cos(theta) cos(theta)

Integrating F with respect to theta gives

3mg(theta/2 + sin(2 theta)/4 + C)

so the definite integral of F from theta = -pi/2 to +pi/2

= 3mg([pi/4 + sin(pi)/4 + C]-[-pi/4 + sin(-pi)/4 + C])

= 3mg([pi/4 + 1/4 + C]-[-pi/4 +1/4 + C])

= 3mg([pi/4]-[-pi/4])

= 3mg(pi/2)

and so the average force exerted by the mass from -pi/2 to pi/2 is

3mg(pi/2)/(pi/2-(-pi/2))

= 3mg(pi/2)/pi

= 3mg/2

That is, neglecting friction, a pendulum swung down from the horizontal exerts an average downwards force of one and a half times its own weight.

Hhhmmmm - that's way beyond me, but thank you for the explanation. Does it help?

ReplyDeleteJC

Suppose you have a bar of length 80cm, pivoted in the middle. At the far end you attach a mass of 1kg. 10cm from this end you pivot a pendulum of length 10cm and mass 1kg.

ReplyDeleteThe moment at the far end of the bar is 1kg * 40cm.

The moment at this end of the bar is 1kg * 30cm

When you let go the bar will tip down at the far end until the bar is vertical.

Attach the bob of the pendulum to this end of the rod with a piece of cotton thread so that the pendulum and the bar are both horizontal. If everything is working the bar should stay horizontal.

Now cut the thread so that the pendulum swings down.

The moment at the far end of the bar is 1kg * 40cm.

The average moment at this end is 1.5 * 1kg * 30cm

What happens?

To use your own words, when you cut the thread the bar will tip down at the far end until the bar is vertical, again. I see where you're going with this but interesting as this is Alex, I would prefer it if you could continue this discussion on a forum such as Besslerwheel.com - or by private email. A blog isn't the ideal medium for this. Thanks.

ReplyDeleteJC