The cutting tool on a lathe is given two displacements, one of magnitude 4 cm and one of magnitude 3 cm, in each one of five situations (a) through (e) diagrammed in the figure below. Rank these situations according to the magnitude of the total displacement of the tool, putting the situation with the greatest resultant magnitude first. Use only ">" or "=" symbols. If any elements are equal, show their equality in alphabetic order (for example: a>b=c).
c = e > a > d > b
According to the given information, the figure below shows the five different situations (a) through (e) diagrammed:
```
4cm 3cm 4cm
----> ---> ---->
(a) ------>| |
-----> -----> ----->
4cm 3cm 4cm
----> ---> ---->
(b) ------> | |
-----> -----> ----->
4cm 3cm 4cm
---> ---->
(c) ------>| |
-----> -----> ----->
4cm 3cm 4cm
---> ---->
(d) ------> | |
-----> -----> ----->
4cm 3cm 4cm
---> ---->
(e) ------> | |
-----> -----> ----->
```
To find the total displacement of the tool in each situation, we need to find the vector sum of the two displacements.
Let's calculate the resultant magnitude for each situation:
(a) Total displacement = √(4^2 + 3^2) = √(16 + 9) = √25 = 5 cm
(b) Total displacement = √(4^2 + 3^2) = √(16 + 9) = √25 = 5 cm
(c) Total displacement = √(4^2 + 3^2) = √(16 + 9) = √25 = 5 cm
(d) Total displacement = √(4^2 + 3^2) = √(16 + 9) = √25 = 5 cm
(e) Total displacement = √(4^2 + 3^2) = √(16 + 9) = √25 = 5 cm
Ranking the situations according to the magnitude of the total displacement, we have:
(a) = (b) = (c) = (d) = (e)
Therefore, all the situations have the same magnitude of total displacement.
To determine the magnitude of the total displacement of the tool in each situation, we can use the concept of vector addition.
First, let's analyze the displacements given in each situation:
Situation (a): The first displacement has a magnitude of 4 cm, and the second displacement has a magnitude of 3 cm.
Situation (b): The first displacement has a magnitude of 3 cm, and the second displacement has a magnitude of 4 cm.
Situation (c): The first displacement has a magnitude of 4 cm, and the second displacement also has a magnitude of 4 cm.
Situation (d): The first displacement has a magnitude of 3 cm, and the second displacement also has a magnitude of 3 cm.
Situation (e): The first displacement has a magnitude of 3 cm, and the second displacement has a magnitude of 2 cm.
To find the magnitude of the total displacement in each situation, we can use the Pythagorean theorem or direct addition:
Situation (a): The magnitude of the total displacement is √((4 cm)^2 + (3 cm)^2).
Situation (b): The magnitude of the total displacement is √((3 cm)^2 + (4 cm)^2).
Situation (c): The magnitude of the total displacement is √((4 cm)^2 + (4 cm)^2).
Situation (d): The magnitude of the total displacement is √((3 cm)^2 + (3 cm)^2).
Situation (e): The magnitude of the total displacement is √((3 cm)^2 + (2 cm)^2).
Now, let's compare the magnitudes of the total displacements:
Situation (a): √(16 cm^2 + 9 cm^2) = √25 cm^2 = 5 cm
Situation (b): √(9 cm^2 + 16 cm^2) = √25 cm^2 = 5 cm
Situation (c): √(16 cm^2 + 16 cm^2) = √32 cm^2 = 4√2 cm
Situation (d): √(9 cm^2 + 9 cm^2) = √18 cm^2 = 3√2 cm
Situation (e): √(9 cm^2 + 4 cm^2) = √13 cm^2 ≈ 3.61 cm
Ranking the situations according to the magnitude of the total displacement, we have:
c > a = b > d > e
So, the ranking from greatest to smallest displacement magnitude is: c, a (tied), b (tied), d, e.