Chapter 6, Problem 76PQ

Chris, a recent physics major, wanted to design and carry out an experiment to show that an object’s mass determines its inertia. He used an ultrasound device to measure acceleration of a low-friction cart attached to a hanging block to provide the same force on the cart during each run (Fig. P6.76A). Chris varied the mass of the cart by varying the number of lead rods placed in it.

Chris used Newton’s second law ${\displaystyle \sum F_x=F_T=M{a}_x}$ to predict his results. He reasoned that because ${\stackrel{\to }{F}}_T$ is the same for each run, the cart’s acceleration should be inversely proportional to its mass:

$a_x=\frac{F_T}{M}=\frac{\text{constant}}{M}$    (1)

Chris’s goal was to show that his data fit Equation (1). He decided to analyze his results by plotting a_x as a function of 1/M; Equation (1) predicted that he should get a straight line, passing through the origin with a slope equal to the tension (red line in Fig. P6.76B):

Chris ran several trials for each run, averaged his results and estimated the error. He then plotted his data (green line in Fig. P6.76B). Chris was excited to see that he correctly predicted that the data fell along a straight line:

$a_x=\left(0.27\text{N}\right)\frac{1}{M}-\left(0.048\text{m}/{\text{s}}^2\right)$

According to the straight-line fit to the data, the slope of the line is 0.27 N, which was close to the weight of the hanging mass and therefore close to the tension in the string. Chris, though, was disappointed to see that the line had a negative intercept. Mathematically, as $M\to \infty$, $\frac{1}{M}\to 0$. Chris was confused because he believed that as the mass increased, the cart’s acceleration should approach zero. He was quite sure that he did not discover some new property of inertia or mass. After convincing himself that he was not being careless in the laboratory and that his data were correct, he started to search for an explanation for the discrepancy between his prediction and his data. Help Chris find an explanation.

FIGURE P6.76 A. Chris’s experimental apparatus.

B. Chris’s prediction (red line) and experimental results (green line).