Video Transcript
In a hydraulic press, the ratio
between the radii of the two pistons is five over three. What is the mechanical advantage of
the hydraulic press? (A) Five over three, (B) three over
five, (C) nine over 25, (D) 25 over nine.
In this question, we are told that
a hydraulic press with the ratio between the radii of the two pistons is five over
three. And we are asked to find the
mechanical advantage of the hydraulic press.
Before we can answer this, let’s
first recall some information about hydraulic presses and define what we mean when
we are talking about the mechanical advantage of a hydraulic press. Hydraulic presses use the
properties of fluid to transfer and amplify a force to, basically, smush something
really hard. This amplification of force comes
from using pistons with different areas.
To show how this works, let’s
consider a curved pipe filled with fluid, with its ends at the same height. If we were to exert a force on one
of the ends, that force will move the fluid within the pipe until the force is
applied out of the other end. Let’s now put a piston into each
end to receive the force. Recall that pressure, 𝑃, exerted
over an area, 𝐴, is equal to the force, 𝐹, divided by that area.
So the pressure at each end can be
expressed using this equation. The pressure on the first end, 𝑃
one, is equal to the initial force exerted, 𝐹 one, divided by the area of the
opening, 𝐴 one. And the pressure on the other end,
𝑃 two, is equal to the force it experiences, 𝐹 two, divided by the area of that
opening, 𝐴 two. But, the pressure exerted on the
ends of the pipe will be equal in magnitude. So we can take these two equations
for the pressure and set them equal to each other. The force on the first piston, 𝐹
one, divided by its area, 𝐴 one, is equal to the force on the second piston, 𝐹
two, divided by its area, 𝐴 two.
Now then, we are being asked to
find the mechanical advantage of this press. The mechanical advantage of a press
is the ratio of the force exerted on the piston with the smaller area to the force
exerted on the piston with the larger area. In order to find the ratio of
forces, we just need to use the equation we obtained earlier and put both values of
force on the same side. We can do this by dividing both
sides by 𝐹 one, which cancels the 𝐹 one on the left side. Then, we multiply both sides by 𝐴
two, to cancel it on the right side. This leaves us with a ratio of
force on the right side and a ratio of area on the left.
The ratio of the force on the
larger piston to the force on the smaller piston is equal to the ratio of the area
of the larger piston to the area of the smaller one. This means the mechanical advantage
is also equal to the ratio of the area of the large piston to the area of the small
piston.
Now then, looking back at the
question, we are given the ratio of radii, not the ratio of the area, between the
pistons in this hydraulic press. The large piston, which must have
the larger radius, has a radius of five. And the small piston has a radius
of three. Since this equation mentions radii,
it must mean we’re talking about circles. So to find the area of these
pistons, we’ll have to use the equation for the area of a circle. 𝜋𝑟 squared equals 𝐴, where 𝑟 is
the radius of the circle.
Plugging this into the equation for
mechanical advantage and then substituting in our values for the radii, we see that
the ratio of area is equal to 𝜋 multiplied by five squared divided by 𝜋 multiplied
by three squared. The values of 𝜋 will cancel
out. And we will be left with five
squared divided by three squared. Squaring both of these, we get that
the mechanical advantage of this hydraulic press is 25 divided by nine. So the mechanical advantage of this
press is given by option (D). 25 over nine is the correct
answer.
