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22:34:49 11-14-2005 22:34:49 ** When force is parallel to displacement, work is force * distance, so the proportionality with distance follows immediately.
The force comes from the gravitational interaction of the masses of the washers with the Earth's gravitational field. More washers imply more force, and the force is proportional to the number of washers. **......!!!!!!!!...................................
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22:35:18 11-14-2005 22:35:18 Why is this consistent with the idea that the work done on a given cart on an incline is proportional to the vertical distance through which the cart is raised?
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22:36:17 11-14-2005 22:36:17 ** The steeper the incline the more washers are required to pull the cart up the incline. In fact for small inclines and correcting for friction the number of washer is proportional to the slope of the incline. The vertical distance, being the rise through which the cart is raised, is also proportional to the slope of the incline.
So for a given cart the number of washers is proportional to the incline. Since work is proportional to the number of washers we conclude the work is proportional to the vertical distance through which the cart is raised. **......!!!!!!!!...................................
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22:37:06 11-14-2005 22:37:06 How does the work done against friction of the cart-incline-pulley-washer system compare with the work done by gravity on the washers and the work done to raise the cart?
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22:38:01 11-14-2005 22:38:01 ** The work against friction plus the work done to raise the cart are all provided by the gravitational force acting on the descending weight. **
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22:38:15 11-14-2005 22:38:15 How do we establish, using a cart-pulley-washer system on a constant-velocity incline, that the acceleration of cart is proportional to the net force on the cart?
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22:39:13 11-14-2005 22:39:13 ** STUDENT ANSWER: In order to keep the velocity constant the amount of force should be constant but it must overcome the effects of gravity and therefore it must increase at a constant rate . The increase in the ramp slope will correspond to the increase in the force.
INSTRUCTOR RESPONSE: Good start. This is half of it. This indicates how the force on the cart is proportional to acceleration. The other half is that the acceleration of the cart is proportional to the effective slope. Since the force and the acceleration of the cart are both proportional to the effective slope, they are proportional to each other. **......!!!!!!!!...................................
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22:54:51 11-14-2005 22:54:51 How do the experiment with acceleration vs. number of washers and experiment with acceleration vs. ramp slope convince us that for a given cart there is a net force down the ramp which is proportional to the slope in excess of the constant-velocity slope (the slope excess)?
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22:55:51 11-14-2005 22:55:51 ** A graph of acceleration vs. the number of washers in excess of the number required to maintain a constant velocity is consistent with a straight-line graph through the origin. **
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22:57:09 11-14-2005 22:57:09 If for small slopes we find that the net force exerted on a cart is proportional to the slope excess, then why does Newton's Second Law ensure us that for small slopes the acceleration of the cart should in fact be proportional to the slope excess?
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22:57:40 11-14-2005 22:57:40 ** Newton's Second Law tells us that for a fixed mass the acceleration is proportional to the force, so if for small slopes force is proportional to slope excess, then acceleration must also be proportional to slope excess. **
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22:58:12 11-14-2005 22:58:12 How is the slope of the net force vs. acceleration graph for an object reveal the mass of the object? Why should a greater mass result in a greater slope?
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22:58:38 11-14-2005 22:58:38 ** Since F = m a, when we graph F vs. a the slope is m. **
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22:59:22 11-14-2005 22:59:22 How do we obtain our definitions of work and kinetic energy in terms of the equation vf^2 = v0^2 - 2 a `ds and Newton's Second Law?
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23:01:52 11-14-2005 23:01:52 How do the experiment with acceleration vs. number of washers and experiment with acceleration vs. ramp slope convince us that for a given cart there is a net force down the ramp which is proportional to the slope in excess of the constant-velocity slope (the slope excess)?
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23:03:10 11-14-2005 23:03:10 ** A graph of acceleration vs. the number of washers in excess of the number required to maintain a constant velocity is consistent with a straight-line graph through the origin. **
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23:04:01 11-14-2005 23:04:01 If for small slopes we find that the net force exerted on a cart is proportional to the slope excess, then why does Newton's Second Law ensure us that for small slopes the acceleration of the cart should in fact be proportional to the slope excess?
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23:04:22 11-14-2005 23:04:22 ** Newton's Second Law tells us that for a fixed mass the acceleration is proportional to the force, so if for small slopes force is proportional to slope excess, then acceleration must also be proportional to slope excess. **
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23:04:41 11-14-2005 23:04:41 How is the slope of the net force vs. acceleration graph for an object reveal the mass of the object? Why should a greater mass result in a greater slope?
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23:05:23 11-14-2005 23:05:23 ** Since F = m a, when we graph F vs. a the slope is m. **
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23:05:48 11-14-2005 23:05:48 How do we obtain our definitions of work and kinetic energy in terms of the equation vf^2 = v0^2 - 2 a `ds and Newton's Second Law?
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23:06:27 11-14-2005 23:06:27 ** Substituting a = Fnet / m into vf^2 = v0^2 + 2 a `ds we get vf^2 = v0^2 + 2 Fnet / m * `ds, which we easily rearrange to get Fnet `ds = 1/2 m vf^2 - 1/2 m v0^2.
Fnet `ds is defined as work, 1/2 m v^2 is defined as KE, and our equation tells us that the work done by the net force is equal to the change in KE. **......!!!!!!!!...................................
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