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In today's lab we will consider the following questions. What happens to objects in free
fall? And how do mass and air resistance affect acceleration in free fall? We will
explore these questions by measuring the acceleration
of different objects in free fall and seeing how they compare to each other and g.
To begin exploring these concepts, let's look at a simple example. If we drop a sheet
of paper and a crumpled up piece of paper
above a table, which will hit the table first? What
differences between these objects will affect their acceleration due to gravity as
they fall?
Let's drop them and see what happens. The ball hit's first. But why? Let's take
another look, first focusing on the motion of
the ball. In the ball's case, we will neglect air resistance. This is only an approximation.
The only force we draw is the force of gravity pointing towards the earth. The ball then
moves with a constant acceleration. This acceleration is independent of the mass of the ball.
Now let's look at how the position, velocity and acceleration of the ball change over time.
Under constant acceleration, the position is quadratic and the velocity is linear. We
also saw a case of constant acceleration with motion on an incline.
Now let's see how the motion of the sheet differs from that of the ball. For the sheet,
we have to include air resistance. This air resistance points in the opposite direction
of the motion of the object and is roughly proportional to the velocity of the object
at low speeds. The sheet does not move with a constant acceleration. When the magnitudes
of the air resistance and the weight are equal, the acceleration is then zero. This
is called reaching terminal velocity.
Let's look at how the position, velocity and acceleration of the sheet change over
time. At t =0 we start with no air resistance
as the sheet is dropped from rest. The acceleration is just the acceleration due to gravity
g. As the magnitude of the velocity increases, the air resistance increases. This causes
the magnitude of the acceleration to decrease. We reach terminal velocity when the air
resistance and the weight cancel each other out, giving
us zero acceleration. The velocity is then constant while the position is linear.
You will use the spark gap timer to measure the acceleration of the brass and plastic
cylinders in free fall. You will then attach the plastic and brass disks to these
cylinders to measure the effects of air resistance.
The spark gap timer works by burning marks in the paper tape at a frequency of 60 Hertz,
or 60 every second. After you've connected the power cord, hold down the white button
during measurement to burn marks on the paper tape. Make sure the tape is aligned
in the spark gap timer, and hold the button down
during your entire measurement. Note the burn marks on the tape, especially the darkest
one on the right from when the measurement 2:40was started.
To measure the acceleration of the cylinders, you will have to attach the paper tape to
the cylinder. Fold the tape to create a thicker portion for the cylinder to grip,
insert in the cylinder and tighten. To measure the
effect of increased air resistance on the acceleration,
you will attach a disk to each cylinder. Open the cylinder to insert the parachute
and tighten. When you are ready to take data, center
the tape in the spark gap timer. Make sure the
piece of tape is long enough to reach the floor. Check that the foam is below the
timer and release the cylinder. Be sure to hold
down the white button while recording your measurement.
Now let's take a look at the tape from that run. The pink circle shows the start
of the measurement. The dots in the white circles
are all the marks from the spark timer. As the cylinder accelerates, the spacing
delta x between the marks increases. Measure the
distance x from each mark to the first mark in centimeters and enter these values
as a column in Excel. You will then use Excel
to calculate the velocity and acceleration. Column A has the position measurements
from the tape in centimeters. We will write an
equation to convert these measurements to meters. Enter your equation starting with
an equals sign in the first box of the column where you want the equation to apply.
In this case, we want to divide the first column
by 100. The equation we type in box B2 is A2
divided by 100. Press enter and drag the corner of B2 to select the whole column. This
will apply the equation to all the values in
column A to populate column B. Follow the same steps
to write equations to calculate the distance between subsequent measurements and the velocity.
You also need to make a column for the time. The spacing between the entries should
be the period of the spark timer.
Once you have calculated the velocity and time, apply a linear fit to your velocity
graph to find the acceleration. Compare the accelerations for the 4 cases, both cylinders
with and without parachutes. What can you say about the effects of mass and air
resistance on acceleration in free fall?