00:01
The first task here is to draw a free body diagram and make sure to show the coordinate axes and reference point on that diagram.
00:10
Excuse me.
00:12
So if this is our box here, i'm going to assume we're on the earth.
00:16
So the force of gravity is directed straight downward and has a magnitude m .g.
00:22
We're on an inclined plane.
00:24
So the normal force is going to be directly out of the incline.
00:28
I'm going to label that capital n.
00:32
And we are also told that there is friction between the object and the incline, and the object is sliding down the incline, so the friction force is going to be directed.
00:45
Up the incline.
00:49
And so those are all of the forces acting on the object, but we need to consider some other things.
00:57
So it said to show our coordinate axes.
01:01
And for an incline plane like this, usually it is helpful to rotate the coordinate axes.
01:09
And you can choose whether you want positive x to be up or down the incline.
01:13
Because the object is moving down the incline, i'm going to make positive x down the incline and positive y out of the incline.
01:22
This means that we will eventually need to decompose the gravitational force into its x and y components.
01:30
So i'll go ahead and add that to my diagram.
01:34
Right now, where the x component is going to be mg times sine of theta, and the y component will be mg times cosine of theta.
01:47
And then lastly is to label our reference.
01:53
I'm assuming what's meant by that is the reference point for zero gravitational potential energy.
02:01
So i will go ahead and just put that reference point here, y equals zero, which means that our object is up here at y equals 1 .25 meters based off of the diagram.
02:19
Okay, now that we've done that, we can move on to part b, where we're to determine the initial speed of the object using newton's rules.
02:28
So first, i want to find the acceleration of the object.
02:36
And to do so, i want to find the sum of the forces in the x direction because we're accelerating in the x direction.
02:42
So the sum of the forces in the x direction will be equal to mass times acceleration in the x direction, which will be equal to in the x direction.
02:53
Direction, we have the friction force in the negative x direction and the x component of the gravitational force in the positive x direction.
03:07
For the force of friction, that is always equal to the coefficient of kinetic friction times the normal force.
03:21
And in this case, because the only forces acting in the y direction are the normal force and the y component of the gravitational force, they must be equal in magnitude because we're not accelerating in the y direction.
03:36
So i'm going to replace the normal force here with the y component of the gravitational force.
03:50
At this point, we want to just solve for the acceleration.
03:53
We can do that by dividing by the mass, and we get that the acceleration then must be equal to negative mu sub kg cosine of theta, plus g sign of data.
04:11
Now, in order to actually figure out what the initial speed is going to be, we want to utilize the kinematic equations.
04:22
So specifically, we don't know how long the amount of time that it takes for our object to come to rest.
04:30
So i'm going to use the equation that v squared is equal to v.
04:36
The initial squared plus 2a.
04:41
D.
04:43
We know that our final speed is zero because the object comes to rest.
04:49
And so we can solve for the initial velocity and it's going to be equal to the square root of negative to a times d.
05:03
And we can of course go ahead and plug in the value that we or sorry, the expression that we found for acceleration, and we would get that the initial velocity is equal to the square root of negative 2.
05:19
D.
05:20
I'm going to also pull out a g because it's a common factor, times negative, mu, sub k, cosine of theta, plus sine of theta.
05:34
And at this point, you can just plug in the values that were given in the problem, where the distance is 1 .15 meters.
05:43
I used 9 .81 meters per second squared for g.
05:48
The coefficient of friction is 0 .32, and the angle theta is 15 degrees.
05:55
And when you plug it all in, you end up with an initial speed or an initial velocity of 1 .065 meters per second.
06:09
So that is part b determining the initial speed using newton's rules.
06:16
We want to try to do the same thing but use energy theorems.
06:21
So i'm going to try and squeeze it in on the left hand side here where we've got a little bit of space.
06:26
But i apologize that i'm probably going to have to write a little bit small to make it all fit.
06:33
Our energy theorem is essentially going to be that the initial energy is going to be equal to.
06:40
To the final energy plus any energy that is lost due to non -conservative forces.
06:47
So initially our object is at a height of 1 .25 meters.
06:53
So we have an initial energy of mgh initial, where h initial, all right here is 1 .25 meters...