Friday, October 12, 2018
Unizor - Physics4Teens - Mechanics - Work - Golden Rule of Mechanics
Notes to a video lecture on http://www.unizor.com
Golden Rule of Mechanics
Let's summarize what we have learned about mechanical work in the previous lecture.
1. In a simple case of motion along a straight line with a constant force F acting along a trajectory, the most important parameter that quantifies the result of the action is work defined as W=F·S, that fully characterizes and is fully characterized by speed V of an object:
W = F·S = m·V²/2
In particular, it means that increasing the force by a factor of N and decreasing the distance it acts by the same factor of N would result in the same final speed of an object.
So, we can "win" in distance, but we will "lose" in force and vice versa.
This is the first example of the Golden Rule of Mechanics - there
are many ways to achieve the result, you can reduce your distance, but
you will have to increase the force or you can reduce the force, but you
will have to increase the distance.
In short, as we mentioned above, whatever you win in distance you lose in force and vice versa.
2. In case of a constant force acting at an angle to a straight line trajectory, the difference is only a factor cos(φ), where φ is an angle between a force and a direction of a trajectory. In vector form it represents the scalar product (F·S).
So, the Golden Rule works exactly as above.
3. Recall the formula for work of a force F pushing an object of weight P up along an inclined plane of the length S making angle φ with horizon to the height H:
W = F·S = P·H
We've proven this in the previous lecture and, as you see, it's
independent of the slope of an inclined plane. However, the minimum
effort we have to apply as a force to move an object up the slope is F=P·sin(φ), while the distance equals to S=H/sin(φ).
We can reduce the effort (the force F) by using an incline of a smaller slope, but that would lengthen the distance S we have to push an object.
So, again, we see the Golden Rule in action.
4. Consider lifting some heavy object of the weight P using a lever, applying the force F to the opposite to an object end of the lever.
This is a case of equilibrium in rotational motion and the balance can
be achieved by equating the moments of two forces acting against each
F·Lf = P·Lp
If Sf is the distance the force F acts down and Sp is the distance our object moves up,
Sf/Sp = Lf/Lp and
F·Sf = P·Sp
By using a lever with longer arm Lf for the application of force, we can proportionally reduce the force F
achieving the same result - lifting an object to certain height. An
inverse is true as well - we can shorten the arm and proportionally
increase the force.
In any case, the Golden Rule of Mechanics is observed: "winning"
in force - proportionally "losing" in distance or "winning" in distance"
- proportionally "losing" in force.
All the above examples emphasize the importance of the concept of mechanical work as the quantitative measure and characteristic of the purpose and the result of applying a force. The so-called Golden Rule of Mechanics is just a catchy term that underscores the importance of the concept of work.
Acting with the force to achieve certain goal necessitates performing
certain amount of work that depends on the goal, not on how we achieve