Introduction

This lab seeks to confirm the conservation of linear momentum. We will give a steel ball an initial velocity by rolling it down a ramp. It will then collide with and become embedded in a wooden block. The momentum of the system before and after the collision will be compared.

Measure the ball's velocity, $v_b$

Place the block on the platform, where it will be safely out of the way. Remove the slider guide and slider from the box and clamp a strip of waxed paper to the floor of the box. Allow the steel sphere to roll down the track from its highest point. It will fall into the box and leave an imprint. Since the very end of the track is horizontal, we can use basic kinematics to determine the velocity of the ball as it leaves the ramp.

Determine the height $b$, through which the sphere falls; be aware that the track is a channel, and the lowest point of the sphere is below the upper edges of the channel.

What is your uncertainty in this measurement?

Make ten or more trials, and find the average value of the distance $d$, the distance the ball travels before hitting the ground.

What is the uncertainty in a single measurement of $d$?

Standard Deviation, $\sigma$

Another tool for reporting the error of a measurment is to use the standard deviation. This is a better tool than simply reporting the uncertainty of a measurement using the methods we have so far. The mathematical definition of the standard deviation of the individual measurements: $x_1, x_2, x_3, \ldots$, is $$\sigma_x = \sqrt{\frac{1}{N}\sum\left(x_i - \bar{x}\right)^2}$$ $x_i$ is the value for a particular measurement, and $\bar{x}$ would be the average of all $N$ measurements, i.e. the mean. Essentially, it compares each measurement to the average of the measurements. A small $\sigma_x$ implies that the measurement is closer to the mean value.

An example experiment where we can use this might be finding little $g$, as you did a few weeks ago. Let's say you did 5 experiments that measured the acceleration due to gravity. Your results are:

You could then calculate the mean ($\bar{g} =9.7$) and then do the summation shown in the equation above. That gets tedious, so we generally use a calculator function to do this. For example, go to WolframAlpha and enter the following code:

   StandardDeviation[{9.9, 9.6, 9.5, 9.7, 9.8}]

Or, another way would be enter these values into an excel spreadsheet then use the STDEVA() function to find the standard deviation of the cells where your data is. Here is an example file. Both methods should give you a standard deviation of $\sigma_{g} = 0.1581$.

The final step is to then find the standard deviation of the mean $\sigma_{\bar{x}}$, or the standard error. This is defined as: $$\sigma_{\bar{x}} = \frac{\sigma_x}{\sqrt{N}}$$ where $N$ is the number of measurements. This is the value you will report when sharing the results of the experiment. For example, with the 5 measurements taken above for $g$, we will have a $\sigma_{\bar{g}} = \frac{0.1581}{\sqrt{5}} = 0.071$ You can then use this value to report the results of your experiment: $$g = 9.7 \pm 0.071 \; \textrm{m/s}^2$$ 9.7 m/s² is the mean value, and 0.071 is the standard error. A useful feature of the standard deviation method is that as you make more measurements, you can decrease the standard error, (provided there are no systematic errors, such as a mis-calibrated scale).

Now, redo your uncertainty in the measurement of $d$ using the standard error of the mean method. You'll probably want to set up an excel file to do this. Enter your results the box below:

From these data, calculate the time of flight, and the velocity of the center of the sphere as it leaves the track.

Collide 'em

Determine the mass of the steel sphere. Also record the mass of the wooden block, which is inscribed on the block. Mount the slider guide and slider in the box. Suspend the block at the appropriate level such that the faces of the block are parallel to the corresponding faces of the box. The block should hang freely, with a gap of about 1/8 inch between it and the track. The block must be perfectly still while awaiting the arrival of the sphere.

Measure the distance $h$.

What is your uncertainty in the measurement of $h$?

Notice that when the block swings, the suspension prevents rotation. The horizontal distance $x$, through which the block swings after impact must be measured (see below), and the vertical distance $y$ is calculated from: $$y = h - \sqrt{h^2-x^2} $$

Using the conservation of energy, we can use this change in height to figure out the velocity of the block/mass system, $v_{b+m}$, immediately after the collision.$$v_{b+m} = \sqrt{2gy} \tag{1}$$

To determine $x$, first position the slider so that it is barely touching the stationary block, and record this distance setting as read from the slider guide. Next, perform a few trial runs until you succeed in positioning the slider such that for ten successive impacts the block at the end of its swing sometimes just barely flicks the slider, and at other times just barely fails to reach it. Note the position of the slider and calculate $x$.

Calculate the linear momentum of the system before impact ($mv_b$) and after impact ($[m + M] v_{b+m}$ ), and compute the relative difference (in %).