Why Do We Use Sine for the X-Component in Physics Problems?

[Hemingway]

Homework Statement

I am worried that I don't understand a basic part of figuring out the component forces in the following problem. I have a full worked example but there is a few steps which I don't understand why we use sin for the x component and not cos (understand why I am really worried as it appears to be basic trig :/)

A 58-kg skier is coasting down a 25° slope, as Figure 6.7a shows. Near the top of the slope, her speed is 3.6 m/s. She accelerates down the slope because of the gravitational force, even though a kinetic frictional force of magnitude 71 N opposes her motion. Ignoring air resistance, determine the speed at a point that is displaced 57 m downhill.

Homework Equations

vf = √2(KE_f) / m
= √(2(1/2 mv₀² + Sigma F cos theta s)/m
= √(2(1/2 mv₀² + mg sin 25 - fk s) /m
= √(2(1/2 mv₀² + 170N cos 0 x 57) / 58
= 19m/s

The Attempt at a Solution

This was in my textbook:

a free-body diagram for the skier and shows the three external forces acting on her: the gravitational force , the kinetic frictional force , and the normal force . The net external force along the y-axis is zero, because there is no acceleration in that direction (the normal force balances the component mg cos 25° of the weight perpendicular to the slope). Using the data from the table of knowns and unknowns, we find that the net external force along the x-axis is:

SigmaF = mg sin 25 - f_k
= (58)(9.8)(sin 25)(71)
= 170N

I look at sin and think we are looking at y component. I can't see why we would used sin for x component - can someone explain?

Many thanks

H
x

ps. please forgive formating - each time I use latex it just puts large gaps in the place of symbols :)

 

[Hemingway]

AH! My free body diagram was incorrect - all good don't need a reply. thanks anyway those who read through my problem. Sorry for inconvenience.

 

[Mindscrape]

Friction is a nonconservative force, so you can't use conservation of energy.

Sin is used for the x component because you have to draw a similar triangle for the normal force of the skiier. The angle between the force opposite the normal force and gravity is theta. Then trig will tell you that the x component is mgsin(theta).

 

[Hemingway]

Friction is a nonconservative force, so you can't use conservation of energy.

Sin is used for the x component because you have to draw a similar triangle for the normal force of the skiier. The angle between the force opposite the normal force and gravity is theta. Then trig will tell you that the x component is mgsin(theta).

Thank you very much! This really helped me consolidate my understanding :)

 

[rwisz]

As a scientist, it is natural to have questions and concerns when faced with a challenging problem. It is important to remember that understanding the concepts behind a problem is just as important as getting the correct answer. Let's break down the problem and address your concerns.

First, let's review the concept of components of forces. When we are dealing with forces acting on an object, we often break them down into components along different axes, such as the x-axis and y-axis. This allows us to analyze the forces in a more organized and systematic way.

In this specific problem, we are dealing with a skier on a slope, which means there are forces acting in both the x and y directions. The force of gravity, mg, can be broken down into two components: mg sin 25° along the x-axis and mg cos 25° along the y-axis. This is because the slope is at an angle of 25°, so the force of gravity is not acting directly downward.

Now, let's focus on the x-component of the net external force. We know that the skier is accelerating down the slope due to the force of gravity, but there is also a frictional force acting against her motion. This frictional force, fk, is parallel to the slope and therefore contributes to the x-component of the net external force. When we use the equation ΣF = mg sin 25° - fk, we are taking into account both the force of gravity and the frictional force in the x-direction. This is why we use sin instead of cos in this equation.

I understand that it may seem counterintuitive to use sin for the x-component, but it is important to remember that we are not just looking at the y-component of the net external force. We are looking at the net external force along the x-axis, which includes both the force of gravity and the frictional force.

I hope this explanation helps to clarify your understanding of this problem. Remember, it is always okay to ask questions and seek clarification when faced with a difficult concept. Keep practicing and you will continue to improve your understanding of physics.