Download LAB 3: FORCE AND ACCELERATION Study of Newton`s Second

Brown University
Physics Department
Physics 0030
Lab 3
LAB 3: FORCE AND ACCELERATION
Study of Newton’s Second Law of Motion
An object which is free to move horizontally without friction is subjected to a series of
known forces and its acceleration is measured for each force. The measured accelerations
are to be plotted graphically as a function of the applied force.
REFERENCES:
Physics 0030 Guide to Laboratory Measurements; A study of Free Fall Velocities;
Kestin and Tauck, University Physics Vol. 1, Chapter 4 (section 4-6).
BASIS OF THE EXPERIMENT
A linear air track provides a nearly frictionless path for a glider resting on it (Fig. 1). The
glider is pulled by a string which passes over a pulley at the end of the track, to a hanger
to which known weights can be attached. These weights with the hanger provide the
variable applied force.
Glider
Pulley
Track
Air from
Blower
Weights
FIGURE 1
Suppose we let:
mh = mass hanging on the string (including weight hanger)
mt = mass on track (glider plus any weights resting on it)
M = m h  mt
F = total applied force = mh g
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Brown University
Physics Department
Physics 0030
Lab 3
T = tension in string (acting horizontally toward pulley, on mt , and acting vertically,
upward, on mh )
a = acceleration of mt (horizontal, toward pulley), and also
a = acceleration of mh (vertical, downward)
Note that mt and mh must move together – that is with the same velocity and
acceleration – as long as the string connecting them remains the same length throughout
their motion.
Fig. 2 shows the forces acting on mt and mh . The equations governing the motion are:
(1)
mt a  T
(2)
mh a  F  T
Pulley
mt
T
a
Adding these and substituting
T
mh
(mt  mh )a  F
gives
Ma  F
which can be written
F=mhg
(3)
a  F /M
FIG. 2
Equation (3) is what we wish to investigate. It expresses Newton's Second Law of
Motion, as applied to the composite system, consisting of glider + weights + hanger +
connecting string, while Equations (1) and (2) are expressions of Newton's Second Law
applied to mt and mh separately.
It is T which actually pulls the glider, and to find T from F we would need to make use of
the Second Law as expressed in Eq. (2). However, without using the second law we
know that M  mh  mt is the mass of the composite system and F  mh g is the only
force applied to it; T is an internal force within the two-body system, not a force applied
to it. For the two-body system we know the left- and right-hand sides of Eq. (3)
independently, once we measure the acceleration a and the masses, and so can test the
equation.
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Brown University
Physics Department
Physics 0030
Lab 3
PLAN OF THE EXPERIMENT
In the experiment, M remains constant while movable pieces of mass (loose weights) are
transferred from glider to hanging support. Each such transfer decreases mt and adds to
mh (and therefore increases F without changing M). For each value of F, a is constant
during the motion.
Since M remains constant, Eq. (3) becomes the expression of a linear relationship
between F and a in this experiment. It predicts that if each acceleration a is plotted as a
function of the corresponding force F  mh g , the observed values will lie on a straight
line passing through the origin, with a slope equal to 1/M.
The principle of the acceleration measurement is identical to that used in the free fall
experiment (Reference 2). Here it is the glider's acceleration that is measured, so to base
it on photobridge measurements, the bridges are mounted on a rod suspended
horizontally over, and parallel to, the track (Fig. 3). The acceleration is constant (one of
our basic assumptions) for a given value of masses because it derives from the
gravitational force acting on those masses; it is not of course, equal to the constant
gravitational acceleration of free fall at any time.
Backstop
Glider
L
STOP
STOP
C
START
U
TUC
TUL
S
dUL
FIG. 3
The glider rest position ( x  0, t  0, v  0) is at Z. To make the correspondence with
Reference 2 clearer, call the photobridge nearest to Z the U (for upstream) photobridge.
Call the middle photobridge the C photobridge, (it was the M photobridge in Experiment
1) and the L photobridge is the last photobridge, nearest the pulley.
If the glider leaves a fixed point on the track, moving to the right, it will interrupt each of
the beams in order.
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Brown University
Physics Department
Physics 0030
Lab 3
The upstream (U) interruption starts the two timers.
The interruptions at C and L turn off the UC and UL timers respectively.
The C photobridge position is adjusted until the UC time reading is one-half the UL time
reading.
The rest of the acceleration measurement involves measuring distances. These
measurements are made readily using the metric scale that is permanently mounted on the
track. Initial position and photobridge positions require passing a perpendicular line in
space down to the track scale.
The position of the C photobeam (at the time midpoint between U and L) relative to the
leading edge at Z of the resting glider is the distance S in this experiment, at which we
find the instantaneous velocity. The value of that instantatenous velocity v(tc), at the time
midpoint between U and L, is equal to the average velocity of the glider in moving from
the U to the L photobeam; it is therefore simply the distance UL between these two beams
divided by the time registered on the UL timer. With these two numbers, S and v(t c ) , we
can then deduce the acceleration of the glider from
v(t c )  v (t u , t L ) 
DISTANCE UL
, and
TUL
a  [v 2 (t c )]2 / 2S ,
[Alternatively, we might have left the C photobeam at an arbitrary position between U
and L and determined the acceleration as in Exp 2.]
PROCEDURE
Several such acceleration measurements will be made, for different masses mh but
constant total mass M.
Three small weights are provided, which can be distributed between glider and hanging
support in four different ways (number of weights on hanger support = 0, 1, 2, 3). For
each of the four distributions of weights you should measure the system's acceleration
(that is, the glider's acceleration), at least twice.
First determine the total mass of the system by weighing glider, weight-hanger and the
three free weights, all together. This total mass (M) remains constant. You will also
need to measure the hanger weight and the individual weights to determine mh for
various arrangements.
To Guard Against Some Potential Problems in Setting up Your System:
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Brown University
Physics Department
Physics 0030
Lab 3
Practice starting and stopping the glider to get a feel for the speeds involved and the
techniques required. Practice releasing the glider from rest (starting at the backstop
position and letting go without imparting any initial velocity). Pick a U photocell
position near to the rest position, and an L position near the pulley, note the measured
transit time from U to L and repeat several times. If the transit times are not essentially
the same, you are giving the glider some initial velocity in the process of releasing it.
With all three movable masses loaded on the hanging support (maximum applied force),
determine how much room you need to stop the glider near the pulley end, without
allowing it to bump into the stop at the end of the track, and without jamming the glider
down onto the track. Place the L photobridge to give yourself this room. Record the U
and L positions, and leave them there throughout the experiment.
With all three movable masses loaded on the hanger, determine the half-time point C of
the glider passing through the photobridge array. Do this at least twice before changing
the weights, and calculate the corresponding values of acceleration; if there is
inconsistency make further adjustments and measurements until the inconsistency is
eliminated and you have at least 2 runs in reasonable agreement. (Note the potential
problems described in the preceding two paragraphs.)
Repeat the last procedure, including at least two runs, with one of the masses transferred
to the glider. Repeat again for two masses transferred, then for all three. Remember, you
need to specify mh for each mass arrangement.
NOTE: It is best for one to start with four loose weights on the hanger initially. This
ensures that on the fourth and final trial there will be enough weight on the hanger to
allow the glider to accelerate down the track at a sufficient rate. Leaving only the weight
of the hanger itself causes the tension in the string to be too slack, and thus will be
susceptible to vibration from the air blowing out of the air track, causing a non-uniform
acceleration. [Do not attempt to take data with no weights on the hanger.]
ANALYSIS OF DATA
For each mass combination calculate the applied force, F  mh g , and for each run,
calculate the value of the measured acceleration. Plot the measured acceleration as a
function of the applied force ( F  mh g ). You will have at least two measurements of the
acceleration for each of the four values of F. You can find the uncertainty for acceleration
by propagating the reading errors in the measurements, not by calculating the standard
deviation – the standard deviation is not applicable since there are only 2 or 3 trials for
each setup. Draw the best-fitting straight line through these points and calculate its slope.
Estimate the uncertainty in the slope. (See Fig. 4). You may watch the TA video Exp 3
Uncertainty Video in lab wiki for reference.
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Brown University
Physics Department
Physics 0030
Lab 3
9
(x2, y2)
8
7
y2-y1
y
6
5
(x1, y1)
4
x2-x1
3
0
1
2
x
3
4
5
You can approximate the uncertainty in the slope form the maximum
and minimum slopes in the range as δm = ( mmax - mmin )/2.
Fig. 4
DISCUSSION
Equation (3) predicts that your graph is expected to be a straight line, with slope
(a / F )  1 / M passing through the origin. Within the experimental errors: Is your graph
linear? Does it pass through the origin? Calculate M from the slope, estimate the
uncertainty in this value, and compare with the value of M determined by weighing. Are
the two consistent?
What are the possible systematic effects which might have influenced your results?
Some possible effects which we have ignored are:
Remaining friction between track and glider
Friction in the pulley
Track not quite level
Mass of string, which we have ignored
String pulling glider possibly not exactly parallel to track.
How would these different effects act to systematically change your result? How might
you test to see if they are important?
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