Classical Mechanics (Taylor) 1.39

[ZomboTheClown]

So this was a textbook problem my professor did in lecture. I felt like I followed along with the logic as she went along, but after a few days and looking back it, I can't seem to recreate it genuinely.

Homework Statement

A ball is thrown with initial speed v₀ up an inclined plane. The plane is inclined at an angle $$\phi$$ above the horizontal, and the ball's initial velocity is at an angle $$\theta$$ above the plane. Choose axes with x measured up the slope and y normal to it. Write down Newton's 2nd law using these axes and find the equations of motion as a function of time. There's also a request to prove that the ball will land a specified distance away from the launch point, but it is simple enough to do once the equations of motion in the specified coordinate system are found.

Homework Equations

System is ideal (frictionless/no air-resistance). The equations just follow from N2.

The Attempt at a Solution

I will just list what my professor did and compare it with my attempts to recreate the solution. So of course, when launched the ball will want to trace out a parabola, but the presence of the inclined plane will not allow it to complete it. I am now actually not quite certain if the professor made her free-body diagram at the instant the ball hits the inclined plane, or just an arbitrary point during flight. Anywho, she and Taylor ends up with the components of weight as w = (-mgsin(phi), -mgcos(phi)) which makes sense easily enough.

However, when I put together my equations using N2, things seem to go awry.

For the x-hat direction, F_x = -mgsin(phi) = ma_x which was pretty easy off the bat.

However, for the y-hat direction I set my equation up as:

N - mgcos(phi) = ma_y while the professor seems to do..
-mgcos(phi) = ma_y which would make sense if N = 0.

I am making the assumption here that the ideal place to make the free-body diagram is at the instant where the ball makes contact with the inclined plane. I thought of a couple ideas of how N could be zero and give me the N2 equations as given and the best I have now is considering the instant:

N - mgcos(phi) = -mgcos(phi), which would give me the N = 0 for
-mgcos(phi) = ma_y, but it seems somewhat awkward. Although, I almost want to say it actually does make sense since this would be the instance/situation/condition in the free body diagram set up that separates this whole problem from just another ball sliding down an incline plane problem. Sorry for the length and a lack of a picture, and thanks in advance to anyone who decides to help out!

 

[diazona]

I am making the assumption here that the ideal place to make the free-body diagram is at the instant where the ball makes contact with the inclined plane.

Seems like a rather strange assumption to me...

 

[ZomboTheClown]

I suppose it is. I just tried out the problem with the probably better free body assumption of an arbitrary position in flight and it works out perfectly. I guess I had some sort of problem with how this approach would somehow incorporate the existence of the incline's angle, but it turns out it is encoded in the choice of how I want to break up the components. Fun fun

 

[diazona]

Yeah, from what you were describing it seemed like your professor was analyzing the motion of the ball in flight. That's what you always do for projectile motion.

The only time it makes sense to draw a free-body diagram for when the ball is touching a surface would be if you're trying to analyze the bounce event itself, e.g. to determine the coefficient of restitution or to find the max. force or impulse exerted on the surface.

 

[ZomboTheClown]

Ah, yep thanks. It's certainly a ride going back to mechanics 1.5 years after freshman year. I just for some reason totally forgot it's up to me in this situation how I want to break up the force components and that I don't need a concrete surface to be in contact with it for me to work with the situation.