Parametric Equations for a Rotating Object on an Incline

[Hello4749]

Hi, I started off by writing the parametric equations as if the rock was rolling on a flat surface.

What I got by drawing sin/cos graphs and basing my answers off example problems:
x = 1t + 5sin(2πt)
y = 10 - 5cos(2πt)
x is horizontal distance, y is vertical distance, t is in minutes

Are these close to correct or helpful in solving the problem? I'm mainly not sure how to account for the incline or angle of the hill. I know that it'll change the y equation a lot. Should I be doing something like adding the 40 degrees into the parenthesis? Any help would be appreciated, thank you in advance.

 

[Deleted member 4993]

Hi, I started off by writing the parametric equations as if the rock was rolling on a flat surface.

What I got by drawing sin/cos graphs and basing my answers off example problems:
x = 1t + 5sin(2πt)
y = 10 - 5cos(2πt)
x is horizontal distance, y is vertical distance, t is in minutes

Are these close to correct or helpful in solving the problem? I'm mainly not sure how to account for the incline or angle of the hill. I know that it'll change the y equation a lot. Should I be doing something like adding the 40 degrees into the parenthesis? Any help would be appreciated, thank you in advance.

If I were to do this problem, I would start off like your equations - then rotate the x - axis 40^o clockwise and re-derive those equations.

 

[markraz]

check out Rodrigues' rotation formula

 

[Hello4749]

If I were to do this problem, I would start off like your equations - then rotate the x - axis 40^o clockwise and re-derive those equations.

Thank you for the reply! I'm not sure how I would go about rotating the x-axis 40 degrees clockwise.

Would it be something like this?

40 degrees = 2π/9
x = 1t + 5sin(2π/9t)
y = 10 - 5cos(2π/9t)

Would you mind expanding a bit more on how to re-derive the equations? I was thinking that I have to change something about the 1t and 10 in the equations since those would only be accurate if it was rolling on a flat surface, but I'm not sure how to change it to account for the hill.

Thanks again!

 

[Hello4749]

check out Rodrigues' rotation formula

Thank you for the response! I just did a quick search, and it looks like it involves vectors/matrices and probably some other topics that I haven't reached yet. Is there any alternative to using Rodrigues' rotation formula?

Thanks again!

 

[Cubist]

You can do this without matrices and vectors. Initially just think about the movement of the center of the rock (this doesn't wobble as the rock rotates). Come up with two functions for the movement of the centre:-
\(x_c=f_{cx}(t)\)
\(y_c=f_{cy}(t)\)

Consider how far the center moves in one minute. And what direction is this movement in, and where will the center be at t=0 and t=1? Plug these values into your functions to check that they are correct.

After you have these two equations, then the movement of the spot can be added. This is done in a similar way to your equations in post#1 except you'll use the functions \(f_{cx}, f_{cy}\) as the center of the circular movement:-
\(x=f_{cx}(t)+5sin(2\pi t)\)
...except use the content of your functions rather than literally writing "\(f_{cx}(t)\)"