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Old May 3rd 2017, 03:25 AM   #31
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Hi Chip,

This now leads me to counter weights. Would you know a formula in which I can change the following values and play around with different weights and time:

Let's say I want to lift 8500 pounds 100 feet into the air. Let's say the permanent counter weight is 8400 pounds and that I add 2000 pounds to the counter weight side lifting the 8500 pound weight 100 feet. I would like to know how long it would take for the 8500 pound weight to rise 100 feet. Thank you again for all of your help, I am most grateful.
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Old May 3rd 2017, 08:00 AM   #32
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For two masses on a pulley system, M_1 and M_2, using $\displaystyle \sum \vec F = m \vec a$ you have:

$\displaystyle (M_2 - M_1)g = (M_1 + M_2)a $

This gives you a value for the acceleration of the masses. The time it takes to move a certain distance 'd' given initial velocity = 0 comes from:

$\displaystyle d = \frac 1 2 a t^2$

Solve for t.
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Old May 18th 2017, 04:49 AM   #33
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Hi Chip,

Not sure how to ask this, but here it goes. Suppose you have a pulley with a rope. On one side you have a four pound weight and on the other side a three pound weight. Obviously the four pound weight would pull the 3 pound weight up. But what if the 4 pound weight was in the shape of a ball which had a hole drilled into with a rod going through it so you could attach the rope to it and instead of dropping the ball staight down you rolled the ball down say a 15 degree slope. Do you lose any of the energy, in other words would the 3 pound weight rise more slowly rolling the 4 pound weight down a slope as opposed to dropping it straight down? (Pretty sure you would). And if so, by how much? Is there a formula to figure out different weights and slopes?

Cheers,

Kenji
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Old May 18th 2017, 06:39 AM   #34
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If the system is frictionless there is no loss of energy, BUT the masses will definitely accelerate slower. The reason is two-fold:

First, the component of the force of gravity that is pulling on the 4-pound ball in the the direction of the ramp is smaller than if the 4-pound weight is allowed to fall straight down. In fact, at some angle that force will precisely equal the force of gravity acting on the 3-pound weight, and the system will no accelerate at all. And at lower ramp angles the system will run backward - the 3-pound weight will cause the 4-pound ball to roll uphill. The "magic angle" for this is arcsin(m/M) = 48.6 degrees. This comes from $\displaystyle \sum F = ma$:

$\displaystyle Mg \sin \theta - mg = (M+m) a $

Here M is the mass of the large ball, m is the mass of the smaller weight, and 'a' is the acceleration of the system. Positive values of 'a' means the ball is rolling down the ramp and the 3 pound weight is rising. If the left hand side = 0 then 'a' = 0 and the system is in equilibrium, and if the left hand side is negative then 'a' is negative.

Second, in order for the 4-pound ball to roll (as opposed to slide down the ramp), some of the potential energy that is converted to kinetic energy is used to spin the ball (i.e it goes into rotational KE) and some is used to get the masses to moving linearaly:

$\displaystyle \Delta KE + \Delta PE = 0$

$\displaystyle Mg h \sin \theta - mgh = \frac 1 2 Mv^2 + \frac 1 2 m v^2 + \frac 1 2 I \omega^2$

Given that $\displaystyle \omega = v/R$:

$\displaystyle (M \sin \theta)gh = \frac 1 2 v^2(M+m+\frac I {R^2})$

Here 'h' is the vertical displacement upward of the 3-pound weight. From this you can calculate velocity as a function of displacement of the weights. This can be simplified given that for a rolling ball $\displaystyle \omega = v/R$, and for a solid sphere $\displaystyle I = \frac 2 5 MR^2$:

$\displaystyle (M\sin \theta + m)gh = \frac 1 2 v^2(\frac {7M} 5 + m)$
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