3 Newton's Second Law
Free body diagrams of the cart / hanging mass
system. Two diagrams are needed, because the direction of the
frictional force on the cart depends on the direction of motion of the
You will investigate Newton's Second Law applied to a system moving in
one dimension subject to multiple forces -- a cart moving along a
horizontal track connected
by a string over a pulley to a hanging mass. Free body diagrams of
the system are shown in Fig. 2. The string connecting
the cart and hanging mass remains under tension and does not stretch
or compress significantly. It follows that, even though the cart moves
horizontally and the hanging mass moves vertically, the cart and
hanging mass have velocities and accelerations of the same
magnitude. We can therefore treat this as a one dimensional system.
When you make measurements, your motion detector will be to the left
in Fig. 2. Hence, we will work in a coordinate system in
which the positive direction corresponds to motion of the cart toward
the pulley and downward motion of the hanging mass.
The cart, with mass , is subject to two horizontal forces, the
friction between the block and track and the tension
in the string. Assuming that the magnitude of the frictional
force on the cart is constant, Newton's second law describing the cart
in each case takes the form
The hanging mass is subject to two vertical forces, the tension
in the string and its weight . The Newton's second
law equations for upward and downward motion of the hanging mass are
You will measure the acceleration of the system as it moves in both
directions with several different hanging masses. You will then use
Newton's second law to determine both the magnitude of the frictional
force and the acceleration due to gravity from your measurements.
- Place the motion detector on the track near the end opposite the
pulley, and aim it at the cart. Important note: Do not place
the motion detector too close to the magnetic bumper. The magnets
can cause the motion detector to malfunction.
- Run Logger Pro and use File -> Open ... to
open the file Probes & Sensors -> Motion Detector -> Motion
- The analysis of your measurements will depend on the assumption
that the track is level (horizontal). Ideally, the acceleration of
your cart rolling on the track without the hanging mass and
without friction is zero. With friction, the cart should
have a small, acceleration, always opposing the velocity, and the
magnitude of this acceleration should be the same rolling
in either direction. Use the motion detector and Logger Pro
to determine whether or not your track is level, and use the
adjustable feet to remove any tilt you detect.
- Measure the mass of the cart and the paper clip and determine
the corresponding uncertainties. The paper clip has a small enough
mass that it is difficult to measure precisely. Instead of measuring
the mass of a single clip, measure the mass of a group of them (at
least 10), and divide by the number of clips. The uncertainties are
important. If you aren't sure how to determine them, ask for help.
- Connect the ends of the string to the paper clip and the cart,
and run the string over the pulley.
- Hang a 10 g mass from the paper clip.
- Start collecting data with Logger Pro, and launch the
cart away from the pulley, keeping your hand in place so that you
can catch it before it crashes into the end of the track when it
returns. Repeat the process until you are able to capture a
high-quality measurement in which the cart travels a reasonably long
distance (at least 50 cm) before turning around.
- Add 10 g to the paper clip, and repeat the process. Make a
total of 8 measurements covering a hanging mass range of
g in 10 g increments. Remember to include the mass of the
paper clip in when it is present. For masses 50 g and above,
replace the paper clip with the 50 g mass hanger.
Use theCurve Fit ... button
to find the accelerations and
cart with each hanging mass.
questions 1 and 2 under
``Questions'' below. It's fine to work as a group on this.
- Place your ,
, , and
measurements in adjacent columns in a spreadsheet. Each row should
correspond to one value of .
- Use the equations you found in step 2 to
calculate and for each value.
- Do your and values show a dependence on
? That is, do you see trends or do they vary randomly about the
average? Answer this question qualitatively by using your
spreadsheet to graph vs. and vs. . Then, come up with
a quantitative way of answering the question. If you get
stuck, ask for help.
- Use the spreadsheet to calculate a ``best value'' of and the
Go over your answers to these questions with your instructor/TA
before you leave lab.
- Starting with
Eq.'s 4-7, find an expression for
that depends only on , , , and
. (In the process of combining equations, you
should be able to eliminate and
and .) If you're not sure how to approach this, ask for
help. Show your work.
- Using a similar approach, find an expression for that depends
only on , , , and
. Show your
- Is your best value for the acceleration due to gravity
consistent with the accepted value? Explain.
- Does the frictional force depend on the size of
the hanging mass ? Give quantitative reasoning for your answer.
Copyright © 2003-2010, Lewis A. Riley
Updated Fri Aug 27 11:05:11 2010
This work is licensed under a Creative Commons License.