Vectors Homework: Displacement & Direction at 80°

[Rob123456789]

Homework Statement

You go to an amusement park with your friend Betty, who wants to ride the 90-m-diameter Ferris wheel. She starts the ride at the lowest point of a wheel that, as you face it, rotates counterclockwise.

What is the magnitude of her displacement vector when the wheel has rotated by an angle of 80 ∘?
and
What is the direction of her displacement vector when the wheel has rotated by an angle of80 ∘, measured counterclockwise from the horizontal?

Homework Equations

I have no idea how to do this.

The Attempt at a Solution

I tried doing 45tan(80), but its gives me a wrong answer.

 

[Lucas SV]

You are comparing two points in a circle of radius 45 m (take the centre of the wheel to be the origin). Write their Cartesian coordinates and take the difference: that will be your displacement vector.

 

[TSny]

My hint would be to draw a sketch of the situation. In the sketch, show the Ferris wheel, the displacement vector, and the 80^o angle.

You can use the sketch as an aid to carrying out Lucas' approach. Or, you might be able to get the magnitude of the displacement with a little trig applied to the sketch.

 

[Rob123456789]

Yes, I am trying to do it with trig, but it seems to give me a wrong answer?

 

[Rob123456789]

Can you guys show me how to do it with trig? I feel like I'm just stuck.

 

[Lucas SV]

Consider the triangle with the following vertices: the centre of the wheel, Betty's initial point and Betty's final point. You know one of the angles and you know two of the side lengths. Note that it is not a right angle triangle.

If you make a sketch of this triangle and use some trig knowledge you should be able to find the magnitude.

 

[Rob123456789]

I keep getting √(52.8)^2+(44)^2 = 69 , but it's telling me I am wrong ? I added both of the vectors's components...

 

[Lucas SV]

I added both of the vectors's components...

which vectors?

 

[Rob123456789]

The (0,45) and (7.8,44) calculated from trig

 

[TSny]

The (0,45) and (7.8,44) calculated from trig

Can you describe what the first vector (0, 45) represents? Is it the vector drawn from the initial location of Betty to the center of the Ferris wheel?
Likewise, what does the vector (7.8,44) represent?

 

[Lucas SV]

The (0,45) and (7.8,44) calculated from trig

Adding these components should give the right displacement vector. I think you messed up on adding the components, so you got the wrong magnitude.

 

[TSny]

Check to see if you might have the x and y components interchanged in (7.8, 44) and check the signs of the components.

 

[hmmm27]

Toss Cartesia.

A line joining two points on a circle is called a chord. The length of a chord can be determined by joining the two endpoints to the center of the circle - thus forming an isosceles triangle - and applying some trigonometry.

 

[Michael Ayisi]

You'd get an Issoceles triangle with sides of 45 m with an angle of 80 degrees between them. Using the cosine rule, you should get an answer of 57.85 m. I hope this helps.

 

[Ray Vickson]

Yes, I am trying to do it with trig, but it seems to give me a wrong answer?

I would suggest you avoid as much trig as you can; you will need some, of course, but do NOT make this an exercise in trigonometric manipulations and trig function properties, or a problem in Euclidean geometry. The problem is about finding a vector, and is (I bet) designed to help you think about vectors and how to use them. There are well-structured methods for doing that, and these become straightforward with a bit of practice. This exercise is about getting some of that practice.

Here you have two vectors: (i) a vector pointing vertically upward, from the ground to the center of the carousel; and (ii) a vector pointing from the center of the carousel to the position of your friend. Their vector sum gives you the vector pointing from the ground to your friend, and that is what you want to find.