Video Transcript
A dancer with a mass of 50
kilograms is standing on the tips of the toes of one of her feet. She exerts a pressure of 490
kilopascals on the tips. What is the area, in square
centimeters, of the part of the foot on which she stands?
Okay, so we’re told that there’s a
dancer standing on the tips of the toes of one of her feet. Let’s suppose that this here is the
dancer. We’re told that her mass is 50
kilograms, and we’ll label this mass as 𝑚. We’re also told that she exerts a
pressure of 490 kilopascals on the tips of her toes. Let’s label this pressure as
𝑃. The question’s asking us to work
out the area of the part of the foot she stands on. So that’s the area of her foot
that’s in contact with the floor. This is the area through which all
of the weight force of the dancer must act. The weight force will act
vertically downward, and we’ll label it as 𝐹.
Now, we can recall that the weight
of an object is equal to the object’s mass multiplied by the acceleration due to
gravity. The acceleration due to gravity is
denoted as a lowercase 𝑔, and on Earth it has a value of 9.8 meters per second
squared. So the weight force 𝐹 of the
dancer, which is the force that goes through the tips of her toes, is equal to the
dancer’s mass 𝑚 multiplied by the acceleration due to gravity 𝑔. Substituting in our values for 𝑚
and 𝑔, we get that this force 𝐹 is equal to 50 kilograms multiplied by 9.8 meters
per second squared. Since the mass is given in units of
kilograms, which is the SI base unit for mass, and the acceleration due to gravity
is in meters per second squared, the SI base unit for acceleration, then the weight
force that we’ll calculate from these values must be in the SI base unit for force,
which is the newton. Evaluating this expression, we find
that 𝐹 is equal to 490 newtons.
We can now add this force value to
our diagram. Remember that what we’re trying to
find is the area through which this force acts. We can recall that whenever a force
𝐹 acts over an area 𝐴, the pressure exerted by that force is given by 𝑃 is equal
to 𝐹 divided by 𝐴. In this case, we know the value of
the force 𝐹 and we know the value of the pressure 𝑃. We’re trying to work out the value
of the area 𝐴. This means that we want to take
this equation and rearrange it to make 𝐴 the subject.
To do that, we first multiply both
sides of the equation by 𝐴. Then, on the right-hand side, the
𝐴 in the numerator cancels with the 𝐴 in the denominator. We then divide both sides of the
equation by 𝑃. Now, on the left-hand side, the 𝑃
in the numerator and the 𝑃 in the denominator cancel out. This leaves us with an equation
that says area 𝐴 is equal to force 𝐹 divided by pressure 𝑃.
Before we substitute our values for
𝐹 and 𝑃 into this equation, we should convert our pressure 𝑃 from units of
kilopascals into units of pascals, which is the SI unit for pressure. The unit prefix kilo- means a
factor of 1000, and so one kilopascal is equal to 1000 pascals. In other words, to convert a
pressure from kilopascals into pascals, we need to multiply the value by 1000. That means that the pressure
exerted by this dancer’s weight force is equal to 490 multiplied by 1000
pascals. This works out as 490000
pascals.
Now that we’ve converted the
pressure into pascals, we’re ready to sub our values for the force 𝐹 and the
pressure 𝑃 into this equation to calculate the value of the area 𝐴. With a force measured in newtons
and a pressure in pascals, we’ll calculate an area in the SI unit for area, which is
meters squared. Substituting in the values for 𝐹
and 𝑃 and then evaluating this expression, we calculate that the area of the
dancer’s foot in contact with the floor is equal to 0.001 meters squared.
However, let’s notice that the
question asks us to give our answer in units of square centimeters. We can recall that one meter is
equal to 100 centimeters. Then, taking the square of both
sides of this, we find that one square meter is equal to 10000 square
centimeters. So, to convert an area from units
of meters squared into units of centimeters squared, we need to multiply the value
by a factor of 10000.
We have then that the area 𝐴 is
equal to 0.001 multiplied by 10000 centimeters squared. This works out as 10 centimeters
squared. So our answer to this question is
that the area of the part of the foot on which the dancer stands is equal to 10
square centimeters.
