- Good morning. Today I am going to
perform my one and only magic trick.
- Really Mr. P.?
♫ Flipping Physics ♫
- Absolutely, but first, we need to do some physics.
- Sure, that makes sense.
[Mr. P.] We start by making sure the table is level,
we do this so we can determine the angle of this incline.
The length of the board is 70.5 centimeters
and the top end of the board is 15.9 centimeters
above the table. Bo, please use this information
to determine the incline angle.
- Sure, sine of theta equals opposite over hypotenuse
and opposite the incline angle is 15.9 centimeters
and the hypotenuse is 70.5 centimeters
taking the inverse sine of the whole equation
gives us the incline angle equals the inverse sine
of the quantity 15.9 divided by 70.5,
which is 13.0342 degrees.
- Now that we have the incline angle,
Bobby could you please tell me all the forces
that are currently acting on this block?
- The force normal is up and perpendicular to the incline,
and the force of static friction is parallel to and up
the incline, and the force of gravity is straight down.
- [mr.p] Bobby, thank you for showing that you should always
draw the free body diagram before breaking forces into components
and then redraw the free body diagram with the components.
Now, use the components of the force of gravity.
- Okay, rather than the force of gravity,
we have the force of gravity perpendicular,
which is perpendicular to the incline and down,
and the force of gravity parallel,
which is parallel to and down the incline.
- Let's now use Newton's second law to determine
the magnitudes of all these forces.
Oh, actually we need the mass of the block first,
which is 121 grams, which we can multiply by one kilogram
over 1000 grams to get 0.121 kilograms,
the mass of the block.
Bobby, Newton's second law please.
- The net force in the y, no wait,
the net force in the perpendicular direction equals
positive force normal minus force of gravity perpendicular
which equals mass times acceleration in the perpendicualr
direction, and the acceleration of the block is
0 in all directions because it isn't moving, so
the force normal equals the force of gravity perpendicular
and we have an equation for the force
of gravity perpendicular, it equals mass times
the acceleration of gravity times the cosine of
theta which then is 0.121 times 9.81 times the
cosine of 13.0342, which is 1.15643,
or 1.16 newtons with three sig figs.
- Notice how the force normal does not equal the force
of gravity which means it does not equal mass
times the acceleration due to gravity.
I know, many times before the force normal was equal
to mass times the acceleration due to gravity,
however that is clearly not always true.
In order to find the force normal,
you have to draw a free body diagram and sum the forces.
Billy, the parallel direction please.
- The net force in the parallel direction equals
the force of static friction, which is positive
because it is up the incline, and the force
of the gravity parallel which is negative because
it is down the incline, which then equals the mass of
the block times its acceleration in the parallel direction.
Bobby already pointed out the acceleration
in all directions of the block is zero,
so the force of static friction equals the force of
gravity parallel, which is equal to the mass of the object
times the acceleration due to gravity,
times the sine of the incline angle,
which is 0.121 times 9.81 times the sine of 13.0342,
which is 0.267709 or 0.268 newtons
with three significant figures.
- At this point, we have determined the magnitudes
for all the forces which are currently acting on th e
block, and everything we've done so far
is typical of the way we solve physics problems.
What I am about to do, is to solve for numbers
for the magic trick.
Again, this portion of what we're doing is for the magic trick only,
and you would only do this part if you are going to
do this magic trick for your friends and family at home,
not when solving a physics problem, got it?
- Sure.
- Yes.
- Okay, I'll take notes then.
- Very nice, Bo.
The mass in the parallel direction equals
mass times the sine of the incline angle
or 121 grams times the sine of 13.0342,
which is 27.2894 or with three sig figs 27.3 grams.
The mass in the perpendicular direction
equals mass times the cosine of the incline angle
or 121 grams times the cosine of 13.0342,
which is 117.883, with three sig figs, 118 grams.
Now, I need a few more things for the magic trick.
You can see I have added a string in the perpendicular
direction and a string in the parallel direction
and each string goes over a pulley
and is attached to a 5 gram mass holder.
Now the reason I figured out our mass' parallel
and perpendicular in grams is because
these masses, which I can add to the 5 gram
mass holders come in gram increments.
Okay, let's start with the parallel direction,
we have 5 grams right there,
I need to get to 27.3 grams or 27
because I only have gram increments,
so 27 grams is what we're going for
in the parallel direction. 5 grams,
if we add 20 to that, we now have 25 grams
and if we add 2 more grams, we now have 27 grams
in the parallel direction. Now let's do the perpendicular
direction, you can see we need 118 grams
in the perpendicular direction,
we already have 5 grams in the 5 gram mass holder
adding 100 grams gets us to 105 grams
in the perpendicular direction,
adding 10 more gets us to 115 grams
in the perpendicular direction, 2 more is 117 grams,
and 1 more gram gets us to 118 grams
in the perpendicular direction.
Now, it might not seem like I have done much
until I do this.
(applause and chatter)
- Ladies and gentle people, I present to you
a floating block. Okay, to be completely honest
with you it's not magic, it's physics,
so who can explain to me what I did?
I took the board away and yet the block remains
exactly where it was before I took the board away,
so why is that? Yes, Bobby?
- You replaced the forces the board was applying
on the block, with tension forces in the string.
- Yeah, if you multiply the mass parallel
and perpendicular in kilograms by the acceleration
due to gravity, you get the magnitudes of the forces
we solved for before in the parallel and perpendicular
directions.
- Right, so the force normal and
the force of static friction are each replaced by tension
forces, and voilà, the board wasn't applying
any forces on the block anymore,
so you could remove it.
[Mr. P.] Exactly, the floating block ends up being a great
visual for the directions of the forces acting on the block,
which is in equilibrium. The force normal from the board
which is now a force of tension, was perpendicular
to the incline and up, the force of static friction,
which is now a second force of tension,
was parallel to and up the incline,
and the force of gravity is straight down.
Thank you very much for learning with me today,
I enjoyed learning with you.