00:01
So let's start with the statement for this question.
00:03
It says that in the figure there is a solid cylinder, right, which is this red color cylinder.
00:13
It has a radius that is equal to 10 centimeters, right? and it has a mass that is equal to 12 kilograms, right? it says that it starts from rest, right? and it rolls without slipping.
00:30
A distance of l that is equal to six meters, right? down the roof that is inclined at an angle that is equal to 30 degrees, right? so this is the distance that this balls cover, right? this is till the l is the distance from the initial position of the wall till it reaches the roof edge of the house, right? and then it falls down.
00:58
Then it says that this roof is inclined and an angle theta that is equal to 30 degrees.
01:05
Now the first part of this question is that we need to find out the angular speed, right? of the cylinder, about its center as it leaves the roof right? and the second is that if its height, roof edges at height, right? so if this height is equal to six, sorry, five meters, a half far horizontally from the roof, it does a cylinder hit the ground? right.
01:37
So we are not talking about the vertical.
01:40
We are not talking about this motion of the ball, but we are talking about the horizontal, right? so we are talking about the horizontal distance.
01:51
This ball will cover.
01:54
Right.
01:55
So let's.
01:55
Start with the solution for this question.
01:58
Now, the solution for this question is very easy.
02:04
So what we're going to do over here, the first step is that we are going to take the ground that is over here in this green color.
02:10
We are taking, why are you going to take it as zero height, right? so the total height at which the ball is initially present, right? let's say that it's given by h's dash, right? and this would be equal to the, height of the house till the roof right which is this point and it's equal to five meter plus the distance the height at which the ball is present so that would be basically six sign theta right now six is the distance that it will cover and theta because it's inclined at an angle right so we have six meter sign theta now we know that theta is basically 30 degrees, right? so age is equal to five meters plus six meter, sine 30, right? so when we calculate this distance, the total height becomes equal to eight meters, right? now that we have the total height of height at which the ball is initially present, right? we will begin with the energy conservation, right? because according to the law of energy conservation, the initial energy is always equal to the final energy.
03:32
Now, since the ball asked by the statement is originally at rest, right? so it means that the initial energy this ball will have, right? that would be the potential energy and that would be m times g times h dash right? that is a total height.
03:50
And the final energy, right? that would be basically till the l point, right? so that would be basically the sum of potential energy, right? plus potential energy until the height it reaches this total age height, right? so we would write age over here, not age dash, right? then the potential energy, right, then the kinetic energy, that would be half times mv square.
04:20
Right.
04:21
And then we have the rotational kinetic energy.
04:23
So that would be half times.
04:26
Moment of inertia times the angular speed square right now what we are going to do that according to the law of energy conservation initial energy must be equal to final energy right so we would have m times g dash equal to m times g plus half mv square plus half i times omega square right now what we're going to do is basically that we are going to plug in the expression for the moment of inertia and the linear speed right so we know that v is equal to omega times r right and i is equal to half m times r square right so we're going to plug these two expressions over here in this equation right and we would have m g h that is equal to m g h right plus we have half m times omega r whole square plus we have half um half m r square right and then the omega square remains the same right so when we open up the parenthesis and then we rearrange this equation a little bit.
05:56
You would end up having an equation that will look like this.
05:59
We have mg parenthesis starts h prime or h -tash negative h, and that is equal to omega -square half mr -square plus 1 divided by 4mr square.
06:15
So what we are going to do over here is that as you can see, this is an equation that will work for this part, right? so what we're going to do is that we are going to plug in the values over here.
06:25
Now, m is 12, right? g is the gravitation acceleration, so it holds the value of 9 .8 meter per second square.
06:33
Right.
06:33
Then we have h that, h dash, right? that was the total height and it's 8.
06:38
Negative, the height that was given, it's 5.
06:42
Right.
06:42
Now, omega square remains the same...