Objectives

• Measurements in reading scales
• Considerations of experimental uncertainties
• Analysis of vector addition by components (using trig) and by graphical construction

Introduction

Vectors (and vector spaces) is a mathematical concept that has certain properties (primarily related to the two operation of scalar multiplication and vector addition). In Physics we find physical quantities that can be described in terms of magnitude and direction can be represented with a vector.

When a system of forces, all of which pass through the same point, acts on a body, they may be replaced by a single force call a resultant. When the body is in equilibrium the resultant force is zero. Force having both magnitude and direction can be represented by a vector.

This lab uses a force table to investigate vector addition. A number of forces of various magnitudes and directions are applied through strings to a central point. The individual strings indicate the force vector direction while force scales measure magnitude. Analysis of recorded vector directions and magnitudes will look at how closely the resultant force vector is zero (as expected from equilibrium).

References open in a separate window

Prelab

(Pre-Labs must be submitted to the Physics lab on separate paper (not in your notebook)
as you come into the lab. See Lab Points)

*** Steps of this prelab are intended to lead you through understanding vectors ***
       (check your textbook for help in understanding vectors)

1. Begin by preparing this unique data set for yourself of vector magnitude and direction.

• Write down your student number
• Write down the last three non-zero digits of your student number as a value for "C".
  (for example 203304007 would yield 347)
• Calculate the vector magnitude components as:  M_x = C,  M_y = C/2
• Calculate the vector position components as:  P_x = -C/3,  P_y = -C/4
• Unit for C is Newton

2. Draw the vector on graph paper with an appropriate scale including the origin (pick some up from the Physics lab - check the counter just outside the lab door or as a last resort you may remove a page from the back of your notebook.)

• Axis
  Start by outlining an xy-axis to the left and bottom of the graph paper. The axis labels are "x-position" and "y-position".
• Scaling
  Determine and add a scale for plotting your vector on the graph with the origin included as part of the graph. Build the scale around your vector. Appropriate scaling will use intervals that are simple (such as in 1's, 2's, 4's or 5's), covering most of the page and showing the vector as large as possible. Note, scaling intervals in multiples of 3 is a poor increment to use.
• Vector
  Draw the vector on your graph. The vector is drawn as an arrow starting at location (P_x , P_y) with a length and direction as indicated by the magnitude (M_x , M_y). Note the magnitude components are not the end location of the arrow head but rather need to be added to the vector position position to locate the arrow head..
• Title
  Add a title to your graph. All graphs have titles. Use a descriptive title (avoiding using "versus" in the title).

3. Calculate the magnitude of the vector from the vector components of M_x and M_y (using Pythagoras). Indicate the resultant value on your vector graph. Note, magnitude has units. 

4. Calculate the angular direction of the resultant vector (using trig). Note, angle by convention is usually determined from the positive x direction. Indicate the angle you determined on your vector graph.

Procedure

Each student will gather and analyze their own unique data. After one student has recorded their data, move the magnetic mounts to a new position for the second student's measurements.

For each measurement with the force scales, show a sketch in your notes of the partial force scale and indicate position on the scale read for the measurement. Use the scale labeled "N" (Newton). Of note, the force scale is a difficult scale to read because of the increments used

1. Prepare the setup as shown in the figure using three scales attached by strings to a common center. Avoid a symmetrical arrangement (prepare with clearly different angles between scales). A lamp shining down is used to help record the string locations from their shadows.

Three vectors are represented by the three strings pulling from a common center point. Vector direction given by the string direction from the center point. Vector magnitude is given by the spring scale reading in Newtons.

2. Accurately sketch on graph paper the location of the strings and record in your notebook a sketch of what each scale shows. The sketch will be used in the analysis to measure direction of the force vectors.

3.  Consider the accuracy of your measurements. Do this by clearly recording in your notes each aspect of an uncertainty and the amount you estimate for the uncertainty with any further notes related to this estimate.

• What would you estimate as an the uncertainty (quote as ± ) in reading the scale? How accurate do you estimate the scales to be?
• How accurate are your angular measurements? Consider the uncertainty in reading the protractor scale and and the accuracy of locating the strings.

Analysis

Component Method

1. Label on your graph paper each sketched string as vector A, B, C with their measured magnitudes.

2. Determine the xy-vector components of each string.
Do this by first measuring, with a protractor (and recording in your notes), the angle each string vector makes to the horizontal. Then calculate the x and y component magnitudes using trig (with the angle and the measured force associated with that string). Finally add the sign of positive or negative direction.

3. Determine the xy components of the resultant vector from adding the A, B, C vectors.
Resultant x-component is the sum of the individual x-components determined in step 2. Similarly for the resultant y-component. Pay attention to the sign of the individual components.

4. Determine the magnitude and direction of the resultant vector. State the resultant vector magnitude in units of newtons.

5. Consider the uncertainties for each step of the process. Record in your notes accuracy issues with each step and an estimate of how large an uncertainty is associated with it.

Graphical Method

1. Sketch the direction of each string onto separate clear acetates.
Do this by aligning the acetate over your graph paper with the data sketch and draw a line indicating the direction of a string. Use a separate acetate for each of the three strings. Clearly label the acetates relating them to the relevant string and include the force magnitude of the string.

2. Draw a scaled vector on each acetate using 1cm=1N
Using an overhead felt pen, trace the direction of your string line for the length required. Be as accurate as possible in drawing the vector length and direction. Indicate direction with an arrow head.

3. Overlay each acetate, combining vectors head to toe
Tape the series of acetates to a graph page (into your notebook) with careful attention to position and orientation (vector direction must not change).

4. Measure the magnitude and direction of the resultant vector using a ruler and protractor. State the resultant vector magnitude in units of newtons.

5. Consider the uncertainties for each step of the process. Record in your notes accuracy issues with each step and an estimate of how large an uncertainty is associated with it.

Lab Report

(Lab Reports must be submitted to the Physics lab at the end of your lab period.  The lab report may be submitted on a separate piece of paper or as a page in your notebook.  See Lab Point)
(Lab Reports must be submitted to the Physics lab at the end of your lab period.  The lab report may be submitted on a separate piece of paper or as a page in your notebook.  See Lab Point)(Lab Reports must be submitted to the Physics lab at the end of your lab period.  The lab report may be submitted on a separate piece of paper or as a page in your notebook.  See Lab Point)

This lab report investigates the accuracy of measurements with the force scales. Two force scales pull in opposing directions. The magnitude of force read on the scales is expected to be the same.

Prepare an experiment as shown in the figure with two scales of the same type to report on the accuracy of these scales. Test the setup in several different ways for independently reading the two force scales with the intent of comparing the readings. Take notes well in preparation of writing a procedure to your work. Be honest with the tests, perhaps even having your partner reading the second scale.

Run half a dozen trials of different spring extensions recording the scale readings.

For the analysis, calculate the difference between the two scale readings for each trial. Then determine an average difference in the readings (this being a systematic error). Then determine the maximum variation in the differences and quote it as an uncertainty (this being a random error).

Write a concise conclusion as to the accuracy of the scales considering random and systematic errors.