Boulder on a hill (Vectors and components)

In summary, when a boulder of weight "w" rests on a hillside at a constant angle "a" above the horizontal, the component of its weight in the direction parallel to the surface of the hill is w cos(90-a) and the component in the direction perpendicular to the surface of the hill is w sin(a). This can be found by using the dot product and unit vectors. It is important to note that rotating the diagram to make the slope the x-axis is not necessary, as the same result can be found by simply using the given angle.
  • #1
asqwt
7
0

Homework Statement


A boulder of weight "w" rests on a hillside that rises at a constant angle "a" above the horizontal. The boulder's weight is a force on the boulder that has a direction vertically downward.

In terms of "w" and "a", what is the component of the weight of the boulder in the direction parallel to the surface of the hill?

What is the component of the weight in the direction perpendicular to the surface of the hill?


http://www.cramster.com/answers-jan-11/physics/physics-vectors-boulder-weight-rests-hillside-rises_1096914.aspx

Homework Equations





The Attempt at a Solution



i have no idea...
 
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  • #2
welcome to pf!

hi asqwt! welcome to pf! :wink:
asqwt said:
In terms of "w" and "a", what is the component of the weight of the boulder in the direction parallel to the surface of the hill?

What is the component of the weight in the direction perpendicular to the surface of the hill?

i have no idea...

oh come on, you must know something about components …

what do you know about the component of a vector (or a force) in a particular direction? :smile:
 
  • #3
i know a vector is made up of an x component, and a y component. but that is when the vector is in a diagonal direction.
 
  • #4
hi asqwt! :smile:
asqwt said:
i know a vector is made up of an x component, and a y component. but that is when the vector is in a diagonal direction.

this is in a diagonal direction …

just turn the papaer round a little so that the slope is horizontal :wink:

(no seriously … it does work!)

ie use cos and sin just the way you usually would
 
  • #5
you want me to rotate the diagram till the vector of the rock... (which is pointing down) is horizontal? so rotate right 90 degrees?

am i solving for the length of the slope? sin a = w/length of slope.
 
  • #6
asqwt said:
you want me to rotate the diagram till the vector of the rock... (which is pointing down) is horizontal? so rotate right 90 degrees?

no, until the slope is horizontal …

that'll then be a situation you're familiar with :wink:
am i solving for the length of the slope? sin a = w/length of slope.

let's see …
asqwt said:
In terms of "w" and "a", what is the component of the weight of the boulder in the direction parallel to the surface of the hill?

What is the component of the weight in the direction perpendicular to the surface of the hill?

the length of the slope has nothing to do with it

(what made you think it had?)

the component is always the original magnitude time cos of the angle …

so for one direction it'll be cos(a), and for the other it'll be cos(90°-a) = sin(a) …

which is which? :biggrin:
 
  • #7
the component is always the original magnitude time cos of the angle …

so for one direction it'll be cos(a), and for the other it'll be cos(90°-a) = sin(a) …

how did you get those? i didnt read that in my book :(
 
  • #8
if you've done dot-products, it's the standard dot-product with a unit vector …

you know a.b = |a| |b| cosθ (where θ is the angle between a and b)

well, if b is the unit vector in, say, the x direction (we usually write that as i), then:

a.i = |a| |i| cosθ = |a| cosθ

(since by definition of a unit vector, |i| = 1)

anyway, it does work in the situations you're familiar with, doesn't it? :smile:
 
  • #9
woh i never did dot products. sorry I am making this hard for you when it is supposedly simple = \ i feel like a retard.

but can i get a few things straightened out?

so is the VECTOR of this problem is the weight of the rock going downward?

you want me to rotate my drawing so that the diagrams "hill" is parallell with the gruond?

then i lost you. :( can walk me through the process step by step by any chance? i have no idea why the answer is w sin(a)
 
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  • #10
asqwt said:
so is the VECTOR of this problem is the weight of the rock going downward?

yes

but before we go any further, can i check what you do know …

if i tell you a vector 16 is pointing at 20° above the horizontal, and ask you for the horizontal and vertical components, what are they? :smile:
 
  • #11
assuming you mean the magnitude is 16. vertical is 16 (sin 20) and horizontal is 16 (cos 20)
 
  • #12
yup! :biggrin:

ok, it's the same for any (perpendicular) axes …

choose one to be the x axis, and call the angle for the vector to that axis θ …

then the x component is the vector times cosθ, and the y component is the vector times cos(90°-θ), = sinθ

in this case, the vector is the weight, w, downwards …

so the component along the slope (going down) is … ? :smile:
 
  • #13
im scared i might be wrong... but..

so if i rotate it to make the slope my x-axis

is the x component w cos (90-theta)? since I am not using the angle given?
 
  • #14
(just got up :zzz: …)
asqwt said:
im scared i might be wrong... but..

so if i rotate it to make the slope my x-axis

is the x component w cos (90-theta)? since I am not using the angle given?

that's correct! :smile:

and soon you'll get used to just looking for the angle, and not having to rotate the page :wink:

(btw, i personally think "not-cos" rather than "sin" … i always look for the "correct" angle, and it either is or isn't, so it's either cos or not-cos :biggrin;)
 

1. What is a vector?

A vector is a mathematical object that has both magnitude and direction. It is often represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

2. What are the components of a vector?

The components of a vector are the parts that make up its magnitude and direction. These are typically represented by the horizontal and vertical axes, with the horizontal component being the magnitude in the x-direction and the vertical component being the magnitude in the y-direction.

3. How do you find the components of a vector?

To find the components of a vector, you can use trigonometric functions such as sine and cosine. The horizontal component can be found by multiplying the magnitude of the vector by the cosine of the angle it makes with the x-axis. The vertical component can be found by multiplying the magnitude of the vector by the sine of the angle it makes with the y-axis.

4. How does a boulder on a hill relate to vectors and components?

A boulder on a hill can be thought of as a vector, with its magnitude being the weight of the boulder and its direction being the direction in which it is rolling. The components of this vector would be the force of gravity pulling the boulder down the hill (vertical component) and the resistance of the hill pushing back against the boulder (horizontal component).

5. How can understanding vectors and components be useful?

Understanding vectors and components can be useful in many scientific and engineering fields. For example, in physics, vectors are used to represent forces and motion. In engineering, vectors are used to calculate forces and design structures. In navigation, vectors are used to determine direction and distance. Overall, understanding vectors and components can help us better understand and manipulate the physical world.

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