Uniform circular motion and acceleration

[spacecadette]

Homework Statement

It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 8.00 {\rm m} every 5.00 {\rm s} and rises vertically at a rate of 3.00 {\rm m/s}.

Homework Equations

Find the direction of the bird's acceleration.

The Attempt at a Solution

I found the bird's acceleration to be 12.6m/s^2
But I have no idea how to find it's direction. It's being asked for as an angle.

 

[Galadirith]

Hi spacecadette, well for this one a good diagram would be very useful. First think about all the forces that are acting on the bird. We have its weight acting down we know this one for sure, but it is moving upward at a constant speed, which must mean that the resultant force in the vertical direction is zero, so there must be another force acting upwards, we also know that there is a force acting towards the center of the spiral.

Now considering that this is a bird, we know that they can produce their own lift as we see from the question its using thermals. The question doesn't state so, so we will assume there are no other forces acting on the bird.

now this seems slightly odd, as if only the thermal is causing lift for the bird there will only be one other force acting in a upward direction, plus his wight acting down means we only have two actual forces acting on the bird. But what if we consider the bird is at an angle, which means the lift it generates would act both towards the center and upwards, x and y components of that force respectively. So have a think about all that, see if you can get a nice diagram draw up and have a go, see how far you can get and if you get stuck well see if we can guide you in the right direction :D

 

[spacecadette]

I understand the concept behind it but I don't know how to go about solving it.

 

[Galadirith]

Ok spacecadette, here's a diagram as to what is actually happening.
http://weald10k.fileave.com/PF001.png
This illustrates what forces are actually acting on the bird with mg being the weight and F_L the force due to the thermal. You should be able to get a bit further with this diagram, what you need to be able to do is resolve forces horizontally and vertically, and understand that the resultant horizontal force is equal to the centripetal force (not acceleration :D i hear Newtons second law ;-) ) of the bird.

So have a go at that now, I can't think what else and can tell you without giving you the answer to the question, if you think hard about it you should be able to get it, you may not get it straight away but persevere :-)

 

[chrisk]

Wait a moment. Since the bird is rising at a constant velocity there is no acceleration and hence no force in the upward direction. The only net force is that of centripetal acceleration. Recall that for uniform circular motion the tangential velocity is constant and only the direction of the velocity vector is changing thus producing the centripetal acceleration and this component of acceleration is pointed at the center of the circle. The bird is rising at a constant vertical velocity so find an angular expression for the position of the bird using a coordinate system with the vertical axis coincident with the center of the spiral and the origin on the ground. Then use some calculus to determine the change in angle with respect to time (this is the vertical angular velocity), the take the derivative of this to find the angular acceleration.

 

[Galadirith]

Hi chrisk, em not quite sure what you meant by no Force in the upward direction. You right if you meant that there is no Resultant force in the upward direction, but there has to be a force in the upward direction to act in opposition to the weight to make the resultant force zero or the bird would accelerate towards the earth! :D

Not wishing to refute your explanation chrisk, but I don't think that is what the question is about, it is not asking for any sort of angular acceleration or anything of the sort it is asking for a constant angle of the force acting on the bird relative to the bird. This sort of question although not quite the same as either, uses the ideas from a conical pendulum and motion on a banked surface :D

 

[chrisk]

Hi Galadirith, I did mean a net force of zero in the vertical direction. Thanks for pointing this out. The issue I have is what would be the vertical force if no net acceleration exists? An initial acceleration upward has to occur to get the bird to rise but then the upward force has to reduce to the magnitude of the weight of the bird to produce a constant velocity.

 

[spacecadette]

Thank you for your help everyone. I'm still having trouble with it though.
I need to Find the direction of the bird's acceleration and also Find the angle between the bird's velocity vector and the horizontal.

I tried using Vinitial = Vsin(t)

I plugged in 8 =(3)Sin(5)

I took the arc sin of 8/15 and got 32.23.

It's wrong but I still don't know how to go about it.

 

[Galadirith]

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 let's 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 what's 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 that's 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 don't 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 F_L (note that I used F_L 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 Y_a and X_a 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 can't 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