0:00
Pi there.
00:01
So for this problem we have spheres of radius which is given and that radius is equal to 34 .5 centimeters and it has a mass amped that is equal to 1 .8 kilograms and it starts from the reds so we know that the initial velocity speed is equal to zero meters per second and it rolls without its sleep and this surface that is inclined 30 degrees so the angle is given and that is 30 degrees and it has a longitude this longitude in here is we're gonna call that alt and that is the long altitude l in here, and that l has the value of 10 meters.
01:15
So for the first part of this problem, part a of this problem, we need to calculate the translational and the rotational kinetic energy, so we need to find the final speed and the angular speed or well, the angular and the rotational speeds and at the bottom of that.
01:45
So in order to find that, what we need to use is the conservation of mechanical energy.
01:56
Now, we will have in that that the initial mechanical energy is equal to the final mechanical energy.
02:04
So we will have that the initial energy is potential, and it is given by this height that we call age.
02:14
So we will have only potential energy for the initial case.
02:19
So we'll have the mass times the acceleration due to gravity times the height, h.
02:24
And this is equal to the final mechanical energy that is at this point, and if we said that the potential energy at this point is zero, then we will only have when the sphere is at this point, we will only have kinetic energy.
02:49
And we know that in this case, we will have a combination of translational kinetic energy and rotational kinetic energy.
02:57
So the translational kinetic energy is one -half of the mass times the speed square.
03:03
In this case, that is the final speed, plus one -half of the mass of this...
03:11
Well, no, sorry, one half of the moment of inertia of that sphere times the angular speed squared.
03:24
Now, in here we know that the moment of inertia of sphere is 2 over 5 its mass times its radius squared.
03:42
Also, we are going to use the condition of rolling without sleeping, that sets that the angular speed can be written as the translational speed over the radius.
03:56
So with that said, we substitute these two definitions in here.
04:02
So we will attain the following.
04:04
We'll attain the mass, times the acceleration due to gravity, times the height, is equal to one half of the mass, the speed squared, plus one half, of 2 over 5 the mass times the radius squared times the angular speed square, but we put that in here, so that is the translational speed over the radius and all of these to the squared.
04:27
In here we can see that we can simplify this because this is to the square, so we can eliminate these radius with these radios, and we can eliminate the mass in all of this, because it is present in all of these terms, so we will have the acceleration due to gravity times the height, one half of the speed squared, plus 2 over 10 the speed squared.
04:59
So in here we will have that that is equal to 7 over 10 the speed squared.
05:12
And since we want to obtain the translational speed, we solve for that.
05:18
So we will obtain that the translational speed is 10 times the acceleration due to gravity times the height, overt 7.
05:29
And we take the square root of this, of course, because in here we have the speed to the square.
05:35
Now we just need to simply substitute all of the values in here.
05:39
Remember that we can obtain the initial height...