Understanding equations and their relationships (multiple equal signs)

[HoboBones]

Homework Statement

Personal question

Relevant Equations

Used as examples: Ideal gas law & net force on object in fluid

Hello all,

I'm doing some major review for a upcoming final and am making sure to fill in some arithmetic gaps. I am trying study using only my equation sheet given for my course.

I am having a hard time understanding the relationships between some of the equations that are given. Specifically, equations and relationships with multiple "equal" signs.

Here are two examples:

Net force on object in fluid: F_net = ma = F_B - W

Ideal gas law: pV = NK_BT = nRT = nN_AK_BT

Using the net force on a object in fluid for example, is it meant to be read from left to right? Are we able to manipulate these equations anyway we want?

F_net = ma = F_B - W

F_net = ma ?
F_net = F_B - W ?
F_B - W = ma ?
etc.

hoping to get some clarification, thank you!

 

[haruspex]

is it meant to be read from left to right?

Equality is transitive. If a=b and b=c then a=c. You can take them in whatever pairs you like.

 

[HoboBones]

Equality is transitive. If a=b and b=c then a=c. You can take them in whatever pairs you like.

Thank you,

Here's a example where I keep getting hung up on.

F_net=ma=F_B-W

The example gives F_net = F_B + 17.5 N - W = 0

where did ma go? Why is the zero placed on the far right?

my interpretation using my given equation sheet is: 0 = F_B + 17.5 N - W, because we know F_net = 0

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[pbuk]

Here's a example where I keep getting hung up on.

You shouldn't get hung up on this: $F_B + 17.5 \mathrm N - w = 0$ is the same as your interpretation $0 = F_B + 17.5 \mathrm N - w$.

 

[haruspex]

where did ma go? Why is the zero placed on the far right?

Same answer to both questions. You are given the system is static, so ma=0.

 

[Orodruin]

You shouldn't get hung up on this: $F_B + 17.5 \mathrm N - w = 0$ is the same as your interpretation $0 = F_B + 17.5 \mathrm N - w$.

Or, to paraphrase @haruspex :

Equality is symmetric. If a=b then b=a. You can take them in any order you like.

 

[Lnewqban]

Here's a example where I keep getting hung up on.

F_net=ma=F_B-W

The example gives F_net = F_B + 17.5 N - W = 0

where did ma go? Why is the zero placed on the far right?

The term ma (mass times acceleration of the body) could have still be shown, if of any interest.

The proper relation should be understood as

Acceleration of a body = Net force acting on the mass / Mass

Meaning that, any body will be accelerated in direct proportion to the net force (vectorial summation of all the forces) acting on it, and in inverse proportion to its mass.

It is not always like that, but in the shown case, the vectorial summation of all the forces (upward direction is assumed to be positive and downward to be negative) happens to be zero.
The mass multiplied by zero is zero; therefore, the ma term can be eliminated form further calculations.

 

[HoboBones]

The term ma (mass times acceleration of the body) could have still be shown, if of any interest.

The proper relation should be understood as

Acceleration of a body = Net force acting on the mass / Mass

Meaning that, any body will be accelerated in direct proportion to the net force (vectorial summation of all the forces) acting on it, and in inverse proportion to its mass.

It is not always like that, but in the shown case, the vectorial summation of all the forces (upward direction is assumed to be positive and downward to be negative) happens to be zero.
The mass multiplied by zero is zero; therefore, the ma term can be eliminated form further calculations.

Thank you, I really appreciate it. Light bulb moment