PHYS111Q : Labs
Subsections


3 Newton's Second Law

Introduction

Figure 2: 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 system.
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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 $M$, is subject to two horizontal forces, the friction $\vec{f}$ between the block and track and the tension $\vec{T}$ 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

$\displaystyle a^\mathrm{up}$ $\textstyle =$ $\displaystyle \frac{T^\mathrm{up} + f}{M}$ (4)
$\displaystyle a^\mathrm{down}$ $\textstyle =$ $\displaystyle \frac{T^\mathrm{down} - f}{M}$ (5)

The hanging mass is subject to two vertical forces, the tension $\vec{T}$ in the string and its weight $m\vec{g}$. The Newton's second law equations for upward and downward motion of the hanging mass are
$\displaystyle a^\mathrm{up}$ $\textstyle =$ $\displaystyle \frac{mg - T^\mathrm{up}}{m}$ (6)
$\displaystyle a^\mathrm{down}$ $\textstyle =$ $\displaystyle \frac{mg - T^\mathrm{down}}{m}$ (7)

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 $g$ from your measurements.

Experiments

  1. 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.

  2. Run Logger Pro and use File -> Open ... to open the file Probes & Sensors -> Motion Detector -> Motion Detector.

  3. 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.

  4. 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.

  5. Connect the ends of the string to the paper clip and the cart, and run the string over the pulley.

  6. Hang a 10 g mass from the paper clip.

  7. 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.

  8. Add 10 g to the paper clip, and repeat the process. Make a total of 8 measurements covering a hanging mass range of $m
\approx 10-80$ g in 10 g increments. Remember to include the mass of the paper clip in $m$ when it is present. For masses 50 g and above, replace the paper clip with the 50 g mass hanger.

Analysis

  1. Use theCurve Fit ... button \scalebox{0.7}{\includegraphics{LoggerPro3-curvefit.eps}} to find the accelerations $a^\mathrm{up}$ and $a^\mathrm{down}$ of the cart with each hanging mass.

  2. Answer questions 1 and 2 under ``Questions'' below. It's fine to work as a group on this.

  3. Place your $a^\mathrm{up}$, $a^\mathrm{down}$, $m$, and $M$ measurements in adjacent columns in a spreadsheet. Each row should correspond to one value of $m$.

  4. Use the equations you found in step 2 to calculate $g$ and $f$ for each $m$ value.

  5. Do your $g$ and $f$ values show a dependence on $m$? That is, do you see trends or do they vary randomly about the average? Answer this question qualitatively by using your spreadsheet to graph $g$ vs. $m$ and $f$ vs. $m$. Then, come up with a quantitative way of answering the question. If you get stuck, ask for help.

  6. Use the spreadsheet to calculate a ``best value'' of $g$ and the corresponding uncertainty.

Questions

Go over your answers to these questions with your instructor/TA before you leave lab.

  1. Starting with Eq.'s 4-7, find an expression for $g$ that depends only on $M$, $m$, $a^\mathrm{up}$, and $a^\mathrm{down}$. (In the process of combining equations, you should be able to eliminate $T^\mathrm{up}$ and $T^\mathrm{down}$, and $f$.) If you're not sure how to approach this, ask for help. Show your work.

  2. Using a similar approach, find an expression for $f$ that depends only on $M$, $m$, $a^\mathrm{up}$, and $a^\mathrm{down}$. Show your work.

  3. Is your best value for the acceleration due to gravity consistent with the accepted value? Explain.

  4. Does the frictional force depend on the size of the hanging mass $m$? Give quantitative reasoning for your answer.


Copyright © 2003-2010, Lewis A. Riley Updated Fri Aug 27 11:05:11 2010

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