Moving Ball on sloping surface issue

In summary, Chadlee88 is trying to calculate the velocity of a ball that is moving due to the tilt of a flat square plane. He is struggling to find an equation to do this, and is also having trouble with understanding the physics behind it.
  • #1
Chadlee88
41
0

Homework Statement


Hey, I'm currently working a software project where I have a ball on a flat square plane and that plane can tilt up and down and left and right and at a combined angle in both direction
for example diagonally. I haven't done physics since high school (4yrs ago) and I've been looking at a few kinematic equations but can't seem to put it all together to be able to find the correct formula to use which takes into account the angles of tilt. So i know how much the value of tilt is tilting left and right, And up and down. but what I'm looking for is an equation to calculate the velocity of the ball as it moves due to the tilt of the square plane.


Homework Equations



ax = Acceleration in x direction
ay = Acceleration in y direction
vf = final velocity
vi = initial velocity
xyAngle = angle created by tilting square plane left or right (Tilt bounds: -20 to 20 degrees)
xzAngle = angle created by tilting plane up or down (Tilt bounds: -20 to 20 degrees)
g = 9.8 (gravity)
t = time
a = acceleration which is due to gravity (ball doesn't move unless the board tilts)

Formulas:
ay = (5/7g sin squared (theta)
ax = (5/7g cos (theta) sin(theta)
vf = vi + at

The Attempt at a Solution



See what i was thinking was the ball has an x velocity component (moving left and right on the plane), a y component(which changes as ball moves up or down when board is tilted) and a z component(which changes as the ball moves up and down the board). The ball would therefore also have an x position, y position and z position. This is is my thinking anyways, i may be wrong here.

But what I'm really struggling is how i encompass these 3 components into calculating the velocity equation for the ball which also takes into account the two angles both up and down tilt and left and right tilt and a combination of the two. Like i said previous I've already programmed the maze to give me these 2 values but i really don't have a clue how to put this all together :( i really would appreciate if someone could help me with this cos I've spent a long time getting nowhere.
 
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  • #2
hi, chadlee88, try vector if you do not want too much trouble, especially when you are programming.


[tex]

\left[\begin{array}{c}M_{r}\end{array}\right] =

\left[\begin{array}{ccc}1&0&0\\0&cos(\theta_{x})&-sin(\theta_{x})\\0&sin(\theta_{x})&cos(\theta_{x})\end{array}\right]

\left[\begin{array}{ccc}cos(\theta_{y})&0&-sin(\theta_{y})\\0&1&0\\sin(\theta_{y})&0&cos(\theta_{y})\end{array}\right]

\left[\begin{array}{ccc}cos(\theta_{z})&-sin(\theta_{z})&0\\sin(\theta_{z})&cos(\theta_{z})&0\\0&0&1\end{array}\right]

[/tex]

[tex]

\vec{r}_{normal} =

\left[\begin{array}{c}M_{r}\end{array}\right]

\left[\begin{array}{c}0&0&1\end{array}\right]

[/tex]

[tex]

\vec{F}_{gravity} =
\left[\begin{array}{c}0&0&mg\end{array}\right]

[/tex]

[tex]

f = \left|\vec{F}_{gravity}\bullet\vec{r}_{normal}\right|

[/tex]

[tex]

\vec{F}_{support} = f\vec{r}_{normal}

[/tex]

[tex]

\vec{F}_{equivalent} = \vec{F}_{support} + \vec{F}_{gravity}

[/tex]

[tex]

\vec{a} = \frac{\vec{F}_{equivalent}}{mass}

[/tex]
 
  • #3
Ok...thanx for that...but i have no idea what all those formulas you jst posted mean.
like F support, r normal, F, Mr equivalent etc. Could you please explain what all that is about cos yeh...i don't get any of it. Like i said, i haven't done physics for 4 years. If i don't get the forumulas then i wont' be able to apply them correctly. And also, as i was saying with the angles of tilt of the square platform, i don't specify an angle as such in the x direction, y direction and z direction. what it does is this: starts off as a level square platform then when i press the left arrow key the board tilts to the left by say "K" degrees, if i then press the up key the board tilts by say "P" degrees. what is actually happening is that the board is being redrawn every few milisecs so as i press the arrow keys it firstly re-draws the board at an angle of "P" degrees then draws at an angle of "K" degrees. Because this redrawing is done so fast it appears as though the board is tilting at the same time the user is pressing the arrow keys. so i only have 2 angle values:

-20 < K < +20 (for left and right, K comes from the above passage)
-20 < P < + 20 (for up and down)

I mention this cos i saw ur matrix thing had theta x, theta y, theta z

thanx :) But yeh still neeed help!
 
Last edited:
  • #4
r_normal is unit normal vector of your flat square.
F_support is force on the ball by the flat square.
F_equavilent is equavilent force applied on the ball.
M_r is rotational matrix.
theta_x, theta_y, theta_z are rotation angle about x, y, and z axis respectively.

look, support force on the ball by the square has a direction along r_normal, and gravity dot products r_normal will be gravity division along r_normal; they will be same in magnitude, but opposite direction because the ball cannot pass through the square.

so comes the equations above,
equation 4 is to find magnitude of support force. because support force direction is same as r_normal, support force vector can be found from equation 5.

there are two forces applied on the ball, gravity and support, the equation 6 is just adding two force vectors together for equivalent force on the ball.

you can use Mr directly without knowing why, or google rotation matrix.

for the programming matters, they are just a problem of how you deal with inputs.
 

1. How does the slope of the surface affect the movement of the ball?

The slope of the surface plays a significant role in the movement of the ball. The steeper the slope, the faster the ball will roll down due to the force of gravity. On the other hand, a gentler slope will result in slower movement of the ball as the force of gravity is less.

2. What other factors besides slope can affect the movement of the ball?

Besides slope, other factors that can affect the movement of the ball include the mass and shape of the ball, the texture and material of the surface, and the presence of any external forces such as friction or wind.

3. How does the shape of the ball affect its movement on a sloping surface?

The shape of the ball can affect its movement on a sloping surface in two ways. Firstly, a spherical ball will roll smoothly on a sloping surface, while a non-spherical ball may experience more resistance and have a less predictable path. Secondly, the weight distribution of the ball can also affect its movement, with a heavier side causing the ball to veer towards that side.

4. How does friction impact the movement of the ball on a sloping surface?

Friction can have a significant impact on the movement of the ball on a sloping surface. It can either slow down the ball's movement or change its direction. The rougher the surface, the more friction there will be, resulting in a slower and less predictable movement of the ball.

5. Can the speed of the ball increase on a downward sloping surface?

Yes, the speed of the ball can increase on a downward sloping surface due to the force of gravity. As the ball moves down the slope, it gains potential energy, which is then converted into kinetic energy, resulting in an increase in speed. However, this increase in speed will eventually reach a maximum point due to factors such as air resistance and the incline of the slope.

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