Document 6540193

Sample Lab: The ball-drop experiment
Wayne Hacker
c
Copyright Wayne
Hacker 2009. All rights reserved.
c
Hackernotes: Physics 210 Lab, Wayne
Hacker 2009. All rights reserved.
1
Contents
1 Getting Started
2
1.1
Before you read this lab . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Before you dare set foot in the lab . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.4
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Introduction
3
3 Theory
3
4 Lab procedure
4
5 Results and analysis
5
c
Hackernotes: Physics 210 Lab, Wayne
Hacker 2009. All rights reserved.
1
2
Getting Started
1.1
Before you read this lab
• Watch the introductory video introducing the photo-gate timer mechanism.
• Watch the introductory video introducing the ball-drop mechanism.
1.2
Before you dare set foot in the lab
• Read this entire lab assignment and any instructions on the use of any apparatus
needed for the experiment.
1.3
Apparatus
• ball-drop mechanism
• metal ball (do not use a plastic ball or the mechanism won’t work)
• photo-electric timer with drop pad
1.4
Objectives
• To expose you to some of the complications that can arise when trying to measure
the motion of an object in a real lab.
• To give you a chance to apply some of the error analysis that you learned in the
previous assignment to an actual hands-on application that has minimal technical
complications.
• To determine the relationship between the fall time t and the fall distance y (i.e,
y = y(t)).
• To measure the magnitude of the acceleration due to gravity g.
c
Hackernotes: Physics 210 Lab, Wayne
Hacker 2009. All rights reserved.
2
3
Introduction
The objective of this experiment is to determine the equation of motion (displacement
as a function of time) for a freely falling object near the Earth’s surface. In the course
of the investigation you will arrive at a value of g.
You will drop a ball from a number of different heights and measure the time that it
takes to fall. You will then try to determine the relationship between the fall time t and
the fall distance y.
If the ball starts at rest and if air resistance can be neglected, the ball’s motion should
be governed by the equation y = 12 gt2 . For purposes of this lab, we will assume that you
don’t know this equation. You will attempt to fit your data to an equation of the form:
y = ktn . Using some basic properties of logarithms should allow you to determine the
values of k and n. Ideally, you should find that k = g/2 and n = 2.
3
Theory
The general kinematic equation governing an object moving vertically, subject to a constant acceleration, is
1
(1)
∆y = v0 ∆t + a(∆t)2 ,
2
where v0 is the initial velocity, ∆y = y − y0 is the displacement, ∆t = t − t0 is the time,
and a is the constant acceleration. In particular, this equation governs the motion of a
ball falling from rest. Let the ball’s initial position be y0 = 0, and the time of the ball’s
release be t0 = 0. Let the positive y-axis point downward, to avoid negative distances
and velocities; then a = g. Since the ball is initially at rest, v0 = 0. Then equation (1)
becomes
1
(2)
y = g t2 ,
2
Here t is the amount of time that it takes for the ball to fall a distance of y, starting
from rest and subject only to the force of gravity.
Now suppose that you didn’t know this relationship, as was the case for Galileo; but that
you suspected a power-law relationship of the form
y = ktn
(3)
where k was a constant and n a real number (not necessarily an integer). How could you
test this hypothesis, and how could you determine the values of k and n? This is the
goal of this lab.
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Hackernotes: Physics 210 Lab, Wayne
Hacker 2009. All rights reserved.
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If we take the natural log of equation (3), we get
y = k tn
ln( )
−−−−−−→
ln(ab)=ln a+ln b; ln xn =n ln x
−−−−−−−−−−−−−−−−−−−−→
Let Y=ln y; K=ln k; T=ln t
−−−−−−−−−−−−−−−−−→
ln y = ln (ktn )
ln y = ln k + n ln t
Y = K + nT (a linear equation)
(4)
This gives Y = ln y as a linear function of T = ln t. Thus if we have a set of experimental
data points (ti , yi ), and if our power-law hypothesis is true, then the points (ln ti , ln yi )
should lie on a straight line, with slope n and intercept ln k.
4
Lab procedure
For this experiment, you will use a special ball-drop apparatus, shown in figure 1. The
ball is initially held in a clamp-like mechanism at a fixed height above a drop pad. While
the ball is held in place, an electric current passes through it; when the ball is released,
the circuit is broken and the current stops, which starts a timer. When the ball lands on
the drop pad, it pushes two metal surfaces together, closing a circuit and stopping the
timer. This system is much more precise than manually dropping the ball and timing it
with a stopwatch, since it removes the error due to human reaction time. It of course
introduces its own error, but it is much smaller than using a manual method.
Figure 1: The apparatus.
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Hackernotes: Physics 210 Lab, Wayne
Hacker 2009. All rights reserved.
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You should drop the ball from ten different heights at 10-cm intervals: y1 = 10 cm,
y2 = 20 cm,. . . , y10 = 100 cm. For each height, you should drop the ball ten times and
record the times, so that you have a total of 100 measurements.
Before you start taking measurements, you should calculate the theoretical value of t for
each value of y:
r
2y
t=
(5)
g
You need to do this because the ball occasionally bounces in the mechanism when released, producing incorrect readings as the circuit is broken and then immediately reconnected. These incorrect readings will be very small compared to the theoretical value
of t: on the order of milliseconds, while the theoretical t’s should be on the order of a
tenth of a second. You do not need to record these false readings in your notes, although
if you get any you should mention this fact in your report, along with an explanation of
how they came about and why you discarded them.
You should also make sure that the vertical ruler is straight (it can get twisted); and that
none of the wires are in the path of the ball’s fall. Set the photogate apparatus on the
“gate” setting.
When you position the ball in the release mechanism, use the plastic right triangle to
make sure that the bottom of the ball lines up with the appropriate tick-mark on the
ruler. Don’t forget to estimate the error in your measurement of y, and include this
estimate in your report.
When you release the ball, turn the knob at least 180 degrees. You should try to turn
it quickly; doing so reduces the likelihood that the ball will bounce in the mechanism
and give you a false reading. As you release the ball, make sure that the vertical rod
doesn’t sway. It’s best to use your left hand to steady the apparatus and your right hand
to release the ball (if you’re right-handed). If you forget to stabilize the apparatus, you
should throw out that data point and re-do the drop. You should try to be as consistent
as possible in your technique.
Record the time from the timer (unless you’re discarding that reading), and then don’t
forget to reset the timer to zero.
5
Results and analysis
For each mean drop distance yi,best , calculate the average of the ten measurements of fall
time. Call this mean time ti,best . For brevity, we shall refer to the mean distance and
time as yi and ti . For each yi , calculate the standard deviation of the ten measurements
of fall time. This will give you an estimate of the random error in ti .
Next, for each value of i, calculate ln yi and ln ti . Make a plot of ln yi versus ln ti . If all
has gone well, these points should lie on or very close to a line.
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Hackernotes: Physics 210 Lab, Wayne
Hacker 2009. All rights reserved.
6
Perform a linear regression on the ten points (ln ti , ln yi ). If you’re working in Excel, the
easiest way to do this is by using the slope and intercept functions. These will give
you the parameters n and ln k in equation (4) that most closely fit your ten data points.
According to equation (2), you should get n = 2. What is the fractional error in your
experimental n? Is this comparable to the fractional error in your measurements of yi
and ti ?
Calculate the experimental value of k. According to equation (2), this should have a
value of g/2. Find the actual value of g at or near your lab site—NOAA maintains a
website at which you can enter latitude, longitude, and elevation and get back the local
surface gravity. What is the fractional error in your experimental k? Is this comparable
to the fractional error in your measurements of yi and ti ?
If your points (ln ti , ln yi ) did not fall very close to a straight line, or if your fractional
error in n or k was significantly greater than your fractional error in measuring yi and
ti , try to explain why. Develop hypotheses concerning what might have gone wrong, and
look for patterns in the data that would tend to support or disprove these hypotheses.
Make suggestions as to how the experiment might be modified to test these hypotheses
and to produce better results.