Solving Ball on Slanted Surface: Calculating Radians Travelled Until Fall

[a.mlw.walker]

so i am looking at the issue of a ball, spinning in a circle, and the surface the ball is on, is slanted, towards the centre

- same as spinning a piece of string with a ball on around your head i think - don't worry about that...

I have attached 2 pictures of ball on the surface at one time, tilted at an angle theta.

In the side on image i have labeled the horizontal component Rsin[theta],

and the component = to the force of gravity = Rcos[theta].

thefore R = mg/cos[theta]

therefore horiztonal component Rsin[theta] = mgsin[theta]/cos[theta] or mgtan[theta]

centrepetal force is equal to horizontal component, it is not a new force of its own, so

mv^2/r = mgtan[theta]

[theta] is the angle of the slope.


so the bit i am confused about is whether i can use SUVAT equations here to work out where the ball will end.

for instance if i had taken times, of the ball so i knew the speeds it was traveling at for two separate but consecutive revolutions, (2*PI/t) then i could calculate the deceleration from

dv/dt = a

So if i know these two speeds for two revolutions, and work out the deceleration, can i work out how many radians the ball will now go through until it drops off the slope?

I think that the speed it drops at will always be the same, because the horizontal component, Rsin[theta] less than mg, the ball will begin to fall.

Yeah so basically how do i work out where the ball will be (in radians) when the ball begins to fall from the track.

Thanks

 

[LightHero]

in advance for your help.Yes, you can use the SUVAT equations to work out where the ball will end. To do this, you need to consider the forces acting on the ball, which are gravity and the centripetal force. The centripetal force is equal to mgsin[theta], where [theta] is the angle of the slope. You can then use this to calculate the speed of the ball, using the equation mv^2/r = mgsin[theta]. Once you have the speed, you can then use the equations of motion, such as dv/dt = a, to calculate the acceleration of the ball. Knowing the acceleration, you can then use the equation s = ut + 0.5at^2 to calculate the displacement of the ball, i.e. how far it has travelled after a certain amount of time. This will tell you the radial distance at which the ball will drop off the track.

 

[astrokreng]

for your question! The scenario you have described is a classic example of rotational motion on an inclined plane. In order to accurately calculate the radians travelled by the ball until it falls, we need to consider a few key concepts.

First, we need to understand the forces acting on the ball. As you correctly stated, the horizontal component of the ball's weight is equal to the centripetal force. This means that as the ball moves along the inclined plane, it is constantly accelerating towards the center due to the force of gravity.

Next, we need to consider the forces that are causing the ball to fall off the inclined plane. These forces include the component of gravity parallel to the plane and any friction forces that may be present. These forces will cause the ball to accelerate down the plane until it eventually falls off.

To calculate the radians travelled by the ball until it falls, we can use the equations of rotational motion. These equations relate the angular acceleration (alpha) of an object to its angular velocity (omega) and angular displacement (theta).

We can use the equation alpha = (omega^2)/r to calculate the angular acceleration of the ball as it moves along the inclined plane. Once we know the acceleration, we can use the equations of motion (such as theta = theta_0 + omega_0*t + 0.5*alpha*t^2) to calculate the angular displacement of the ball at any given time.

In order to determine when the ball will fall off the inclined plane, we can use the equation for the vertical motion of the ball (y = y_0 + v_0*t + 0.5*a*t^2). We can set y = 0 to represent the point where the ball falls off the plane, and solve for t. Once we have the time at which the ball falls, we can plug this value into our equations of motion to calculate the angular displacement (in radians) at that time.

I hope this helps to clarify the process for calculating the radians travelled by the ball until it falls off the inclined plane. Keep in mind that this is a simplified explanation and there may be additional factors to consider, such as the shape and mass distribution of the ball. It is always important to carefully consider all relevant factors when conducting scientific calculations.