Zachary Tripoli 

MWF 10:40 

Footnote 18 Project 

Applied Project: Roller Derby 

Suppose we have a solid sphere, a hollow sphere, a solid cylinder, and a hollow cylinder rolling down a ramp. To guess which object reaches the bottom first, we need to understand cetain principles of physics such as inertia. 

We will define I as the moment of inertia (about the axis of rotation). The mass will be m, the radius r, and the vertical drop h. The potential energy of each object the, is defined as mgh (where g is the acceleration due to gravity).  

At the bottom of the ramp, the object will have velocity v and angular velocity w such that v = wr. 

After combining these givens, and assuming the energy loss from friction is negligible, the conservation of energy gives: 

mgh =  

We will begin with this formula. Recall that I can be defined as and w =  

Plugging in the new values gives 

mgh =  

     (multiply both sides by m) 

    or     

    where      

Now if y(t) is the vertical distance traveled at time t, then substituting y for h gives 

    and then       

Now we visualize a triangle where our velocity v is the hypotenuse and the vertical side is the change in vertical distance (dy) divided by the change in time (dt). 

      here   

if we substitute v into this equation 

  or 

Now we can solve for time t by multiplying both sides by dt and integrating. 

dt 

T 

     (here we substitute h back in for y) 

The above equation demonstrates that the object with the smallest value for will win the race. 

Now all that remains is to calculate the value of for each object.  

Beginning with the cylinders, the mass of a solid object is defined in the book as 

    where p is the density function (in the case of a completely solid clyinder it is a constant k) 

therefore 

    and because the volume of E, we can rewrite the expression with the volume of a cylinder 

    or          for a solid cylinder. 

For the solid cylinder, the moment of inertia is defined in the book as 

    which can be rewritten subsituing constant k for and using cylindrical coordinates 

              and          so that 

    or      

Here if we plug in our value for k determined from the previous step we get  

    and so 

    for the solid cylinder, and  

For the hollow cylinder, we can regard the object as a lamina, so that from the book the moment of inertia is defined as 

therfore, for the hollow cylinder. 

In order to fine the values of for the hollow and solid spheres, we must first consider a partly hollow ball with inner radius a and outer radius r. To find the value for k, the density constant, we will again use the formula 

    again here the definition of this integral is the volume of E, a partly hollow ball 

    therefore 

Now if we consider the moment of inertia for our hollow ball, we must integrate in terms of our two radii, a and r 

  this will be rewritten using spherical cooardinates 

              and           

plugging this integral into maple gives 

Now if we plug in the k value that was obtained for spheres, the moment of inerta is 

Recall so that 

    for the spheres 

Now as a-->0, the hollowed-out space within the imaginary ball is shrinking, and so the ball itself is becoming fully solid. Conversely, as a-->r, the hollowed-out space within the ball is expanding to approach the outer limit of the sphere, making it completely hollow. 

Thus for the solid ball is         and for the hollow ball it is    

Thus for a solid sphere,  

    therefore we must use LHopital's Rule 

Thus for a hollow sphere,  

Recall that the object witht the smallest vallues of have the fastest time rolling down the ramp. 

solid cylinder  

hollow cylinder  

solid sphere  

hollow sphere  

Therefore the objects will finish rolling down the ramp in the following order: 

solid sphere, solid cylinder, hollow sphere, hollow cylinder.