## Physics Question

ArielM
Posts: 52
Joined: Mon Oct 02, 2017 9:15 pm

### Physics Question

Just a general physics question to get further understanding. Is it safe to say that that generally (general convention) sine indicates forces acting in the Y direction (along the Y-axis), and cosine indicates forces acting in the X direction (along the X-axis)?

Just trying to get an understanding of this concept. This concept can also be found in the Physics Lesson 1 video on the slide posted below of the Free Body Diagram.

Sincerely,
-Ariel Morrow
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NS_Tutor_Andrew
Posts: 520
Joined: Mon May 23, 2016 1:47 pm

### Re: Physics Question

Hi ArielM,

That is generally the case, but not always. Ultimately, it depends on the coordinate system that you set up and the angles that you're given. The bedrock definition that you can rely on is that sine = opposite/hypotenuse and cosine = adjacent/hypotenuse (often remembered through the mnemonic "SOHCAHTOA").

I'm attaching a sketch of a problem involving motion on an inclined plane that illustrates this point. You may look at this and say "wait, why is Fgrav(x) = Fgrav*sin(θ)?" After all, we have the intuition that motion in the x direction corresponds to cosine. The issue here is that θ refers to the angle that the plane forms with the ground, not the angle that Fgrav makes with Fn -- which is the angle we need to worry about when resolving the force into its x and y components. Instead, θ corresponds to the angle between Fgrav and the dotted line that we would get by extending Fn down. The x-component of Fgrav corresponds to the opposite line from this angle, so we need to use sin(θ) = opposite/hypotenuse here.

So, with all of that in mind, there are two ways that you can approach this. Option 1 is to use the shorthand of assuming that cosine is used for the x-component and sine is used for the y-component, and then memorizing any counterexamples as they come up. Option 2 is to become familiar enough with the trigonometry that you can figure it out on the fly no matter what problem you encounter. All things being equal, option 2 is probably better, but in a pinch, you can work with option 1 -- there aren't that many different setups that get tested on the MCAT, so you can memorize some common variations and probably be OK. That said, if you can check your work, so to speak, using SOHCAHTOA, that's really helpful.

Another helpful technique can be to apply common sense to a problem that you may be able to visualize intuitively. Imagine changing the angle at which you're applying the force. What will happen at extreme points? To take a simple example, imagine that you're applying a force to move a block on a horizontal plane. If you apply the force almost horizontally, then almost all of that force will "go" to moving the block. If instead, you apply the force almost vertically, that will be very inefficient -- very little of that force will "go" to moving the block horizontally. That's a clue that you need to use cosine, because cos(0) = 1, so at an angle very close to 0, Fcos(θ) will almost equal F, while cos(90) = 0, so at an angle very close to 90, Fcos(θ) will be almost zero.

Hope this helps, & best of luck!
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Andrew D.
Content Manager, Next Step Test Prep.
ArielM
Posts: 52
Joined: Mon Oct 02, 2017 9:15 pm

### Re: Physics Question

Hi Andrew,

So, ultimately, to get the maximum efficiency of pushing a box along a horizontal plane, you would want the force applied to equal (or be close to) the force (θ) of the plane that the object is being pushed along. It would not make sense to push a box along a horizontal plane with an applied force of, say, 1N, if the force is applied in the vertical direction at an angle(θ) of, say angle(θ) = 90°. If the 1N force applied is applied to the box in the vertical direction, the use of cosine would be inefficient because the cosine 90° = 0, and that would mean that the box would not go anywhere- so despite the applied force of 1N, the angle at which the force is applied makes it like nothing is happening to the box.

In contrast, if the 1N force is applied to the box in the horizontal direction, then the use of sine would be efficient because sine 90° = 1; the use of sine is efficient at this angle because it gives the closest/gives the greatest value that corresponds to the applied force of 1N.

Is this a correct understanding?

Sincerely,
-Ariel Morrow
NS_Tutor_Andrew
Posts: 520
Joined: Mon May 23, 2016 1:47 pm

### Re: Physics Question

Hi Ariel --

The important thing to note about this whole discussion is that this is a trick to use if it helps you - if it winds up creating more confusion than insight, then it's not worth using.
ArielM wrote:Hi Andrew,

So, ultimately, to get the maximum efficiency of pushing a box along a horizontal plane, you would want the force applied to equal (or be close to) the force (θ) of the plane that the object is being pushed along. It would not make sense to push a box along a horizontal plane with an applied force of, say, 1N, if the force is applied in the vertical direction at an angle(θ) of, say angle(θ) = 90°. If the 1N force applied is applied to the box in the vertical direction, the use of cosine would be inefficient because the cosine 90° = 0, and that would mean that the box would not go anywhere- so despite the applied force of 1N, the angle at which the force is applied makes it like nothing is happening to the box.
Yes! So this is a hint that the correct formula for work here is W = F*d*cos(θ), because that pushing with an angle of 0 to the horizontal will mean that you're getting the most out of the work.
In contrast, if the 1N force is applied to the box in the horizontal direction, then the use of sine would be efficient because sine 90° = 1; the use of sine is efficient at this angle because it gives the closest/gives the greatest value that corresponds to the applied force of 1N.

Is this a correct understanding?
I'm not sure how this second example fits in. Let's just keep it simple and think about a force being applied in almost the horizontal direction -- say at 1° from the horizontal to a box lying on a horizontal surface. Our physical intuition tells us that this orientation will be very effective for pushing the box -- almost all of our force will go towards moving the box horizontally. Now we consider whether we want to use sine or cosine. Cos(1°) is almost equal to cos(0°), which is 1 -- indicating that F will not really change by multiplying it by cos(θ), which corresponds to our physical intuition. This means that cos is right to use here. What if we use sine? Sin(1°) almost = sin(0°), which is 0. Since we need to multiply F by the trigonometric term, that would mean that F would be almost 0, or almost no force would be going towards pushing the box. This contradicts our physical understanding of what must be happening, so we don't want to use sine here.

Hope this helps!! But that said, if this is more confusing than helpful, than this trick might not be worth the energy .
Andrew D.
Content Manager, Next Step Test Prep.
ArielM
Posts: 52
Joined: Mon Oct 02, 2017 9:15 pm

### Re: Physics Question

Andrew,

Right, right and right. Absolutely- I agree, you are 100% correct. I will only use if it is helpful.

Sincerely,
-Ariel M.