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Physics/AP Physics B lab

Question
Hello Mr. Kovalcin.
I was wondering if you could help me with a lab.
The lab involves a metal ramp which is set fixed on a table. We are to place a marble on the ramp, and the marble will then travel downwards the ramp and eventually fall somewhere on the floor. Our objective is to find out how many books to stack on the floor to find the optimal height for a specific target. So our teacher will give us a target where the marble should land, and we are to find the perfect height for that from the floor, and according to that height, we will stack the books.

The information I have:
- mass of the marble: .0282 kg
- height of just the ramp on the table: .08m
- height of the table: .92m
(I'm guessing we will need the total height including the table and the ramp height for mgh?), and this is 1 meter.

Another thing to note also is that part of the ramp is protruding off the table, so the ramp is not completely on the table. This is important I am assuming.

Can you send me a diagram? Or take a cross section image with your cell phone and email it to me.
Based on just the verbal description I would assume that the marble will roll down the ramp. When it reaches the bottom of the ramp the marble will become a projectile and will then land on the target.
For the first part of the problem you will need to use energy conservation to determine the speed of the marble as it reaches the end of the ramp. When the marble is at the top of the ramp all of its energy will be gravitational GPE=m*g*dH where dh is the height of the top of the ramp relative to the bottom end of the ramp. You can get this height dh by measuring the length of the ramp L and the angle theta that the ramp makes with the horizontal and then calculate the height of the ramp dh from:
dh=L*sin(theta)
Therefore, the GPE at the top of the ramp becomes: GPE=m*g*L*sin(theta)
As this marble rolls down the incline all of this GPE will convert into kinetic energy. Since the marble is rolling the total kinetic energy will take two forms: linear kinetic energy, 1/2*m*v^2, plus rotational kinetic energy, 1/2*I*w^2.
Since the moment of inertia of a sphere I is equal to I=2/5*m*R^2 and since the angular velocity w is equal to w=v/R the rotation kinetic energy KEa will be:
KEa=1/2*i*w^2=1/2*[2/5*m*R^2]*v^2/R^2=1/5*m*v^2
The total kinetic energy KE of the ball at the bottom of the ramp will be the same of the linear kinetic energy plus the rotational kinetic energy:
KE=1/2*m*v^2+1/5*m*v^2=7/10*m*v^2
Making the gravitational energy at the top of the incline GPE equal to the total energy at the bottom of the incline KE:
GPE=KE
m*g*L*sin[theta]=7/10*m*v^2
Solving for the velocity v of the marble at the bottom of the incline:
v=sqrt[10/7*g*L*sin(theta)]
Now that you know the velocity with which the marble leaves the ramp you can now do a two dimensional kinematics problem to determine where your marble will land.
In the vertical direction the kinematics variables will be:
do=height of the end of the ramp above the floor, df=the height of the target, Vo=V*sin(theta) the vertical component of the initial freefall velocity, Vf=? unknown, a=-9.8m/s^2 and t=? unknown.
In the horizontal direction the variables are:
do=0m assume the origin in zero in the horizontal, df=? where the target needs to be placed, Vo=V*cos(theta) the horizontal component of the velocity, Vf=Vo since the acceleration in the horizontal is zero, a=0m/s^2 there is no acceleration in the horizontal direction and t=? the same time as in the horizontal direction.
If you place the target h meters above the floor df in the vertical will be h and the time to reach the target vertically can be calculated from df=1/2*a*t^2+Vo*t+do which becomes:
h=1/2*-4.9*t^2+V*sin(theta)*t+do  solving for t.
Then, finally, multiplying the time t by the horizontal velocity, V*cos*theta), to find out how far away from the end of the ramp to place the target.
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Physics

Volunteer

James J. Kovalcin

Expertise

I am teaching or have taught AP physics B and C [calculus based mechanics & electricity and magnetism] as well as Lab Physics for college bound students. I have a BS in Physics from the University of Pittsburgh and a Master of Arts in Teaching from same. I have been teaching physics for 34 years. I am constantly updating my skills and have a particular interest in modern physics topics.