# Uniform circular motion and acceleration

Tags: acceleration, circular, motion, uniform
 P: 109 Hi spacecadette, ok so let try and help you by starting you off in the right direction, so the question says that we can spiral motion as uniform circular motion combined with a constant upward velocity. So lets do just that, we will first investigate this Uniform circular motion bit. So as discussed in the other post, there is no resultant force in the vertical direction, this is evident from the fact that the vertical velocity is constant. Now what this means is that as far as the UCM goes, is doesn't matter how fast the bird is moving vertically as it is not accelerating. Now if you look back at my diagram above, you will see that I have shown the forces acting on the bird, now actually I have "grouped" forces and whats actually going on physically is slightly different. You may be wondering why we have a force acting at an angle like that, well firstly we don't have to think about it as a single force acting at and angle, well think about it like two forces. Well call the force acting vertically Y and the force acting horizontally X, original huh :D. Our Y force comes from the lift produced by the thermal producing a force vertically. We know our resultant vertical force must be zero, which means Y must equal mg, can you see that? Now on to our X force, hopefully you have done stuff on circular motion so you should be familiar with the following equation: $$F = m\frac{v^2}{r} = mr\omega ^2$$ which describes the force required to cause centripetal motion or uniform circular motion. Now you should also be aware that this force always acts toward the center of the radius of rotation of your UCM, which is why we have a force X acting horizontally. Now from you question it kind of leave us to imagine what physical reason the bird has a force acting towards some center, but if you can imagine we could say that the bird is at an angle so thats its wings are not level with one another (you might like to look up motion on a banked surface to read more about this :D), the physical effect of this would be that the lift force would not act totally upwards but be angle (hence the diagram) cause both a vertical and horizontal force. So where does this leave us, well I have described two forces X and Y, but as should be evident from the last paragraph we actually only have one literal force acting on the bird. Now you should be able to use the data provided in the question to work out the the horizontal component of the force acting on the bird and you should be able deduce the vertical component of the force acting on the bird, (note you may find it a little strange as you dont know the mas of the bird, so you will have the force described in terms of m, don't worry about that as that will all be fixed) which should lead to a system of equations like so: $$F_{vertical} = Y = F_L sin\theta$$ $$F_{horizontal} = X = mr\omega ^2 = F_L cos\theta$$ Now something we can do with this is divide one equation by the other. What this will do is eliminate FL (note that I used FL to describe the force due to lift acting at an angle to the bird) and you should see that it will eliminate the mass of the bird from our equation too, i will use Ya and Xa to denote the accelerations of the bird in their associated directions, we get: $$\frac{Y}{X} = \frac{Y_a}{X_a} = \frac{sin\theta}{ cos\theta} = tan\theta$$ from there you can work out the angle of the acceleration on the bird. Now this all deals with the acceleration of the bird, none of this deals with the velocity direction of the bird, that if you do need that ill post after this one had been solved, hope that all helps, the values 8 and 5 you used in your last post cant be put in any equation like you did to a get an answer, both of those values must be used to find the the centripetal acceleration on the. Also do check you value for the acceleration, I haven't done the calculation myself but there would be know way to find a value to the magnitude of acceleration on the bird without knowing the vertical and horizontal components of the lift force acting on the bird. Have fun with the question spacecadette :D