• Experiment 12. The net force on a friction car released on a ramp is proportional to the 'effective' slope of the ramp, for small slopes.

• Experiment 13. The force of air resistance on a falling coffee filter is proportional to the square of the velocity of the filter (?).

• Experiment 14. When an object is immersed in water the water exerts a buoyant force equal to the weight of the water displaced.

You have been supplied with a mass set and a crude but effective and precise balance. With this balance you can with reasonable accuracy determine the mass of an object by placing it on one side of the balance and adding masses from the mass set to achieve a balance. The balance is very sensitive to small changes in mass, capable of detecting changes on the order of .01 grams.

The mass set consists of masses of .1 gram, .2 gram, .3 grams, .5 gram, 1 gram, 2 grams, 3 grams, 5 grams, 10 grams, 20 grams, 30 grams, 50 grams and 100 grams. Using various combinations of these masses you can obtain any mass from .1 gram to over 200 grams, in increments of .1 gram. Each mass is accurate to within +-.5%.

• Note that 1 gram is, within very narrow limits of precision, the mass of a 1 cm cube of liquid water at 20 degrees Celsius.

Place the balance beam on the stand, balanced on the edge of the dulled blade, with the damping cylinder partially immersed in water.

• Be sure that the beam is resting as precisely as possible at the vertex of the angle formed at the notch near its center, as demonstrated on the video clip.
• Place a cup of water beneath the damping cylinder, with the water depth adjusted so that the cylinder will be about halfway immersed when the beam is horizontal.
• As shown on the video clip, adjust the position of the washer on the beam until the damping cylinder is about halfway immersed in the water and the beam is horizontal. Mark the position of the washer or accurately measure it from the nearest end of the beam so that in case it moves, it will be easy to reposition. Use a small amount of thin tape to help hold the washer in its place.
• Place a ruler or other measuring device (the pendulum stand might be convenient for this purpose) at the position of the damping cylinder, and note the position of the top of the beam relative to this measuring device.

Add enough water to the cup to increase the water level by approximately 1 cm.

• What happens to the position of the beam?
• Can you explain why the position of the beam changes, why it changes in the direction it does (i.e., up or down), and why the change is equal to, greater than or less than the change in the water level (as the case may be)?

Now we will observe what happens to the position of the beam when various masses are added to the balance at the position of the damping cylinder, and also when the same masses are added at a position opposite to the position of the damping cylinder.

• Place the .1 gram mass on the beam, at the position of the damping cylinder. By how much does the reading on the measuring device change? By how much would you expect the reading to change for a .2 gram mass?
• Record the mass added and the position of the beam, as determined by the measuring device.
• Record also the position of the beam in the absence of the added weight.

• Remove the .1 gram mass, note the reading on the measuring device, and add the .2 gram mass.
• How much displacement corresponds to the addition of the .2 gram mass?
• Record the mass added and the position of the beam, as determined by the measuring device.

• Repeat twice more, first with the .3 gram and then with the .5 gram mass.
• Now, place the .1, .2, .3 and .5 grams masses, in turn, on the opposite side of the balance at the same distance from the balancing point as the damping cylinder. For each mass, record the reading on the measuring device.

The position of the beam in the absence of any added mass will be called the equilibrium position of the system. We will construct a graph to determine the displacement of the beam as a function of the mass added at the position of the cylinder.

• You have recorded the position of the beam at equilibrium and with the addition of 4 masses at the position of the damping cylinder and 4 masses at the position opposite that of the damping cylinder. Place these data in a table.
• For each line of your table, calculate the displacement of the beam from its equilibrium position. The displacement will be either positive or negative, depending on whether the reading on the measuring device is greater or less than the reading at equilibrium.
• Sketch a graph of y = beam displacement from equilibrium vs. x = added mass.
• If the graph appears to be linear, sketch the straight line corresponding to the behavior of the system. If the graph does not appear linear, sketch a smooth curve to fit the data as closely as possible without attempting to go through any point except the origin (i.e., don't go out of your way to actually 'hit' any point; come as close as possible on the average to the points you have graphed without making nonsensical wobbles to accomodate experimental errors which have nothing to do with the behavior of the actual system).
• Using your graph, estimate the beam displacement corresponding to the addition of .15 gram, .25 gram, .4 gram and .7 gram at the position of the cylinder, and opposite to the position of the cylinder.
• Place these masses on the beam at the appropriate points and record beam positions. Compare with your predictions.

Now you will use the balance to accurately determine the masses of various objects.

Hang the 'balance pans' at opposite ends of the beam, at equal distances from the balancing point.

• To check the symmetry of your pan placements, take two of the largest washers supplied with the lab kit and place one at the center of each pan. See if the equilibrium position of the system changes, and if so how much. Then exchange the washers between the pans and again note any change in the equilibrium position of the system. How do your results determine whether the pans were positioned symmetrically about the balancing point?
• If necessary, adjust the positions of the pans until you can verify symmetry by the preceding procedure.
• With the pans unloaded, record the position of the beam with respect to the measuring device. This will be the equilibrium position for the balance pans.

Measure the masses of various objects.

• Place a pencil or pen in one pan, and in the other place a combination of masses from the mass set to bring the system back as near as possible to its equilibrium position. Add all masses in such a way that the balance pans continue to hang symmetrically about a vertical line.
• Note the total of the masses added from the mass set. If the balance pans have been placed at equal distances from the equilibrium position, and if they have been kept symmetric about a vertical line, then the total of these masses will be equal to the mass of the pencil or pen.
• Repeat this procedure to measure as accurately as possible the mass of each washer supplied with the lab kit (if there are more than 3 of a given type of washer, randomly select 3 washers of that type). Record these masses for future reference.
• Determine the mass of your friction car, and record this mass for future reference.
• Determine the masses of four randomly selected paper clips and record these masses for future reference.

Answer the following questions:

• Why is it essential that the balancing pans be placed at equal distances from the balancing point?
• How much would the result of weighing a 40 gram object be affected if one pan was positioned 29 cm from the balancing point, and the other 28 cm from this point?
• If when determining the mass of a small washer, the beam position could be brought to within .1 cm of its original position by a balancing mass of 1.21 grams, then what might be the mass of the washer?
• Why does the beam not balance at its equilibrium position when unequal masses are added at the two ends?
• Why does the beam not balance at its equilibrium position when equal masses are added at unequal distances from the balancing point?

We first obtain the data necessary to determine the acceleration of the system vs. the number of suspended washers.

• We begin with approximately 20 small washers in the friction car. One by one we remove washers from the car and suspend them over the pulley, continuing until the car begins to accelerate as a result of the gravitational force on the suspended washers.
• We will use the pendulum as our timer, synchronizeing the motion of the pendulum with the motion of the car.
• Determine the time required for the system to accelerate from rest through a known distance under the influence of this gravitational force. Record the pendulum length, the number of quarter-cycles, the distance traveled by the car and the number of suspended washers.
• Add two washers and repeat, allowing the car to accelerate from rest through the same distance.
• Continue adding two washers at a time and repeating the process.

We determine the acceleration of the system for each number of suspended washers.

• From the pendulum length and number of quarter-cycles, determine the time necessary to accelerate from rest.
• From the time to accelerate from rest and the distance, determine the average velocity, final velocity and acceleration of the system for each number of suspended washers.

Next, plot the acceleration of the system vs. the number of suspended washers and interpret the results.

• Plot an accurate graph of acceleration vs. number of suspended washers and determine how well a straight line can be made to fit the data.
• Explain why the x intercept of this graph can be interpreted as the number of washers necessary to overcome the frictional resistance of the system.
• Create another table for the acceleration of the system vs. the number of suspended washers, but this time subtract the number of washers that corresponds to the x intercept of the graph from the number of suspended washers. Call the resulting quantity the net number of accelerating washers.
• Plot acceleration vs. the net number of accelerating washers.
• Assuming that gravity exerts the same force on each suspended washer, how well do your results confirm Newton's Second Law, which states that the acceleration of a system of constant mass is directly proportional to the net force on the system?
• Describe another experiment that would be necessary to completely validate Newton's Second Law.
• The frictional characteristics of the pulley and that of the wheels of the friction car are approximately, but not exactly, the same. Assuming that the ideal graph, which results when the frictional characteristics of the pulley and the wheels of the car are the same, is perfectly linear, then how might the differing frictional characteristics affect the linearity of the graph as more and more washers are suspended over the pulley (which results in a greater influence by the frictional characteristics of the pulley and a lesser influence by those characteristics of the car).

Use your results to predict the acceleration of the entire system when the net force is equal to that of the gravitational force on the mass of the system:

• Using the shelf-standard balance and the weight set, determine the mass of a single washer and of the friction car (or, if these quantities have already been determine, simply list the results).
• How many washers would be required to equal the mass of the entire system? According to your graph of acceleration vs. net number of accelerating washers, what acceleration would be expected if the system was subjected to the force of gravity on this number of washers?
• If the system is simply dropped and allowed to fall freely to the floor, it will accelerate in response to the force exerted by gravity on its mass. What will this acceleration be?

Repeat this experiment using weights suspended over a pulley (an Atwood Machine):

• Suspend 30 paper clips and two large washers over a pulley, with 15 clips and a single washer on each side, as indicated on the video clip.
• One at a time, transfer paper clips from one side to the other, until the system begins accelerating when released.
• Measure the distance the system falls from rest and the time required.  Spin the pulley and repeat twice more, for a total of three measurements with the pulley in a different initial position for each.
• Transfer another paper clip and repeat.
• Continue until the time of fall is less than 1.5 seconds.
• Calculate accelerations and graph acceleration vs. force.  Force is measured in units of paper clip weights; each transfer of a paper clip results in an increase in net force of 2 paper clip weights (the weight a paper clip is added to one side and removed from the other, resulting in a force difference equal to two paper clip weights).
• The x intercept of your graph corresponds to frictional force.  For each data point, subtract frictional force from total force to get net force and plot acceleration vs. net force.
• Determine from your graph whether, within experimental uncertainties, acceleration is proportional to net force (i.e., if the graph of acceleration vs. net force is a straight line through the origin).
• Determine the mass of a paper clip and of a large washer, using the shelf standard balance.
• Use your graph to show that the acceleration of the system is in the same proportion to the acceleration of gravity as the difference of the masses on the two sides to the total mass of the system, when the difference of the masses is corrected for friction.

• Create a table for position measurement vs. suspended mass, and add a column for the displacement from equilibrium.
• Construct an accurate graph of y = pendulum displacement from equilibrium vs. x = suspended mass. Fit a straight line to this graph.
• Your graph should, within experimental uncertainties, give you a straight line through the origin. Use the slope of this graph to estimate the displacement from equilibrium corresponding to a suspended mass of 150 grams, and to a suspended mass of 200 grams. From these estimates, predict the position of the pendulum due to each suspended mass.
• Suspend masses of 150 grams and 200 grams from the pulley and see if your predictions are correct.

Now check out the proportionalities involving forces and distances.

• The force exerted on the pendulum bob by the tension in the thread is in fact somewhat less than the force exerted on the suspended mass by gravity. This is due to the friction associated with the pulley. This friction reduces the force by approximately 4%, so that the actual force exerted on the pendulum will be approximately .96 of the force exerted by gravity on the suspended weight. Thus the effective suspended masses can be regarded as .96 of the actual suspended masses.
• Make a table of pendulum displacement from equilibrium vs. effective suspended masses.
• Based on this table, create another table of [ pendulum displacement / length of string to measurement position ] vs. [ effective suspended mass / mass of pendululm bob ].

• If we let x = pendulum displacement at measurement position, L = length of pendulum string to measurement position, m = effective suspended mass and M = mass of pendulum bob, then this last table shows (x / L) vs. (m / M).
• Does your table, within experimental uncertainty, confirm the hypothesis that pendulum displacement as a proportion of pendulum string length (i.e., x/L) is identical to the ratio of masses (m / M)?

• Construct a graph of effective suspended mass m vs. pendulum displacement x. Fit a straight line to your graph and determine its slope `dm / `dx.
• If it is in fact true that x / L = m / M, then x / m = L / M so that the slope `dx / `dm must be equal to the unchanging ratio L / M.
• How well do your results confirm the hypothesis that x / L = m / M?

Use the proportionality x / L = m / M to predict various displacement and suspended masses.

• Use the proportionality to predict the displacement x corresponding to a suspended mass of 70 grams. Recall that m stands for the effective suspended mass, which for this pulley with its 4% coefficient of friction is .96 of the actual suspended mass.
• Hang a 70 gram mass from the hook and see how well the observed displacement corresponds with your prediction.
• Use the proportionality to predict the effective mass m that will lead to a displacement of 2.3 cm from the equilibrium position.
• Hang the appropriate mass (i.e., one whose effective mass is equal to that you predicted) from the hook and see how well the observed displacement corresponds to the desired 2.3 cm displacement.

Justify the following statements: 

• Since the weight of an object is equal to the force exerted on it by gravity, which is proportional to mass, then if F is the force of gravity on the effective mass and W the weight of the pendulum bob, m / M = F / W and therefore x / L = F / W.  That is, the displacement is in the same proportion to the length of the pendulum as the applied force to the weight of the pendulum.
• If a force F is applied to the pendulum in the horizontal direction, then the displacement of the pendulum from equilibrium will be F / W times the length of the pendulum.
• The length and displacements of the pendulum as measured in this experiment were not the actual length and actual displacements of the pendulum.  The actual length should have been measured to the center of the sphere, and the displacements should have been measured at this position.  However, the ratios x / L obtained in the experiment are identical to the x / L ratios that would have been obtained had these quentities been measured at the center of the sphere.

Analysis of Errors

• Estimate the errors in your effective masses and in your measurements of pendulum position, and represent your errors on the appropriate graph.
• Discuss the implications of your errors.

Note that the results obtained here are only valid as long as the displacement of the pendulum from equilibrium is small compared to its length.

• If displacement is less that .1 of the length, then predictions of forces or displacements are accurate to within +-.5%, or +-.005 of the actual quantities.

Click here for Physics 121 Short Version

We can measure forces using a rubber band and a scale by which to measure the stretch of the rubber band. The force vs. stretch curve is not linear for a rubber band. For a spring which is light compared to its strength, a force vs. stretch curve would be very nearly linear; however we choose here to use the rubber band in part because it is readily available and cheap, and in part for the very reason that it is nonlinear, and is nonlinear to a degree that forces us to account for it.

We begin by using the rubber band to displace a simple pendulum of known mass and length through a variety of displacements.

• Using the largest pendulum bob, construct a pendulum whose length is at least 1 meter, and set up a measuring device by which to measure the displacement of the pendulum. Record the length of the pendulum to the point of measurement, and ulcer record the equilibrium position of the pendulum.
• Take two pieces of thin string, each about 50 cm long, and tie a hook to each end of each piece. You can use small paper clips to make the hooks.
• You have two kinds of rubber bands. Hook one of the thinner rubber bands and the two pieces of string together, with the rubber band between the strings.
• Attach one end of the string to the tab on the surface of the pendulum bob, using the hook at that end of the string.

• When you have finished, construct a table of rubber band length vs. pendulum position. Add a column for rubber band stretch and one for force.
• In the column for rubber band stretch, fill in the magnitude of the stretch of the rubber band in excess of its length when it begins to stretch.
• In the column for force, determine the magnitude of the force corresponding to the appropriate pendulum displacement. (The weight of the pendulum is equal to the force exerted on it by gravity, which is W = m g; this weight will be in Newtons when m is in kg and g in m/s^2.  The pendulum mass in grams should be marked on the pendulum. If it is not you may assume a mass of 1100 grams.).

We sketch a graph of force vs. rubber band stretch.

• Construct an accurate graph of force, in Newtons, vs. rubber band stretch, in meters.
• Submit a description of this graph, and speculate on the reasons for its shape.
• Keep your graph for future reference.

Repeat the experiment for two rubber bands hooked 'in parallel' (i.e., both stretched between the same two hooks).  Repeat all steps except the parts using the parallel and series combinations.

• Based on your observations and your graphs, how does the force exerted by two identical rubber bands 'in parallel' compare to that exerted by one?
• Why do you think it should be so?

Repeat the experiment for two rubber bands hooked 'in series' (i.e., one attached to the end of the other, like a chain; you might make a second hook for this purpose).  Repeat all steps except the parts using the parallel and series combinations.

• Based on your observations and your graphs, how does the force exerted by to identical rubber bands 'in series' compare to that exerted by a single rubber band?
• Why do you think it should be so?

We proceed to calibrate the single rubber band.

• Using the program CALIBRT (download from the homepage if necessary; run by clicking on the name of the program in Explorer), enter your data and store your calibration.

• The program CALIBRT is designed to calibrate any measuring device.  We use it here for rubber bands.

• The program will ask you for the zero-point reading.  This is the reading just before the rubber band began to stretch, when the force (the 'quantity your are measuring') is essentially zero.
• The zero-point reading may vary the next time you insert a rubber band into the apparatus; the program that later converts your data according to the calibrations will compensate for this variation by asking you for your zero-point reading everytime you use it.

• You then enter the remaining points, according to instructions.
• When you name your calibration file, you should write down the name of the file and the type of rubber band you calibrated.

We calibrate another rubber band by hanging weights.

• As described on the video clip, suspend the thin rubber band next to a vertical scale (e.g., the pendulum stand).
• Hang masses of 5, 25, 50, 75, 100, 125, 150, 175 and 200 grams from the rubber band (but do not let the stretch exceed 1/3 the length of the rubber band) and record the positions of the end of the rubber band.
• Make a table of force in Newtons vs. stretch in meters.
• Construct a graph of force vs. stretch.
• Compare the shape of this graph with that obtained for the thickest rubber band.
• Run the CALIBRT program and save your calibration of this rubber band.

Using the same procedure, calibrate this rubber band.  Be sure to mark the rubber bands for future use. 

The CALIBRT program also reads your previos calibration file and converts new data to the appropriate units. 

• Calibrt creates a table under the filename you give it, which you can open and cut-and-paste into your document.
• Calibrt also creates a comma-delimited text file under the same filename but with an .XCL extension, of your data which can be opened in a spreadsheet.
• Calibrt creates one more file, under the filename with an MTH extension, which can be imported into DERIVE.
• All these files will be in your C:\CALIB724 directory.

Answer the following questions:

• According to the CALIBRT program, what force would be associated with each rubber band, given the following displacements:  1.2 cm, 1.9 cm, 2.6 cm, 3.3 cm, 4.8 cm, 6.6 cm, 8.2 cm and 10.3 cm?
• How well do the values given by the CALIBRT program agree with those you would estimate from your graph?

Analyze the errors in this experiment.

• First measure the maximum unstretched length of the rubber band--the length when the 'slack' is gone but no significant tension yet exists.
• Using a thin rubber band suspended at top by a hook made from a paper clip, hang weights of 10, 25, 50, 75, ... grams from another hook at the bottom of the rubber band. For each mass accurately measure the length of the rubber band.
• For each length determine the stretch of the rubber band -- i.e., its length in excess of its maximum unstretched length.
• Sketch a graph of suspended mass or weight vs. rubber band stretch.
• Fit a smooth curve to your graph (a sample graph is shown below).

In this experiment we suspend a weight from two mutually perpendicular strings, neither in the vertical or the horizontal direction. We measure the tension forces in the strings by means of force balances with calibrated rubber bands, and infer the components of the weight in a coordinate system defined by the strings.

First set up the experiment and obtain data.

• First determine the size of the rubber bands to be used. They must be strong enough to support a 200-gram mass, while still stretching enough to give useful stretch measurements, but not stretching by more than 1/3 their original length.  If necessary you can use 'double' rubber bands (i.e., two rubber bands in parallel), being sure to take account of the doubling in your calculations of force.

• Suspend a known mass of approximately 200 grams from two force balances, as shown in the video clip.
• Adjust the directions of the strings until one makes an angle of approximately 10 degrees with horizontal (corresponding to a slope of around .15 with horizontal) and the other is perpendicular to it. It is essential that the two directions be as close to perpendicular as possible.
• Note the lengths of the rubber bands.
• Using a sheet of paper, make the markings you will use to determine the actual angles of the forces exerted (see the video clip). On the paper record the lengths of the rubber bands and the known mass.
• Repeat this process for angles near 20 degrees (slope approx. .3), 30 degrees (slope approx. .6) and 40 degrees (slope approx. .85).

Determine the actual tensions, and the angles of the tensions with horizontal.

• Use your graph of mass or weight vs. rubber band stretch to determine the forces corresponding to the rubber band stretches.
• Use tan^-1 (`dy / `dx) to find the direction of each string, as in the video clip.

For each trial, determine the angles and magnitudes of the weight components, and show that the vector sum of the weight components is equal in magnitude and direction to the weight of the suspended mass.

• The weight components are equal and opposite to the two forces of tension.
• From the angles and magnitudes of the weight components, determine the horizontal and vertical components of each.
• Determine the sum of the horizontal and vertical components of the weight components.
• Show that, within the limits of experimental error, the weight components are indeed equivalent to the weight vector.

Answer the following questions:

• How should this experiment and demonstration convince you that the weight of the suspended object can be regarded as the resultant of two components in two mutually perpendicular directions?
• How do the relative magnitudes of the two components change as the angle of the component nearest horizontal changes from 10 degrees to 40 degrees?  What would happen if the angle continued to 50, 60, 70 and 80 degrees?
• What would be the tension forces if one tension was vertical and the other horizontal?

Analyze the errors inherent in this experiment.

When an object is suspended from two calibrated rubber bands, with the tops of the rubber bands attached to two pencils protruding over the edge of a tabletop, the magnitudes of the supporting forces are inferred from the calibration graph(s), and the directions of the supporting forces from the length of the rubber bands and the separation of the pencils. From the magnitudes and angles of the two forces we determine the horizontal and vertical components of these forces and compare the sum of the vertical components with the weight or mass of the suspended object; we compare also the magnitudes of the horizontal components of the forces.

We first suspend an appropriate mass vertically from two identical rubber-band supports.

• Place a single thin pencil so that its end protrudes over the edge of a level tabletop. Place a weight (a book would work well) on the other and of each pencil to keep it in place.
• We will refer to a pair of rubber bands connected in series by a paper clip in the middle as a 'rubber-band support'.

• Construct two rubber-band supports, using either the thicker or the thinner rubber bands.

• Use paper-clip hooks to hang the two rubber-band supports from the end of the pencil.
• From the other ends of the two supports use paper-clip hooks to suspend a weight which will stretch the rubber bands by approximately 10% of their original length.

Obtain data for support length vs. pencil separation

The first figure below shows the configuration of the system with lengths L of the rubber-band supports and the separation S of the pencil ends.

• With pencil separation 0 (i.e., both weights hanging from a single pencil end), measure the lengths L of the two rubber-band supports, from the pencil to where the hook at the other end supports the hanging weight.  Ideally these lengths will be the same or very nearly the same (this should be the case unless the rubber bands are not identical, which could be the case if they have been over-stretched).
• Move the pencils to separation S = 10 cm and measure the new lengths L of the rubber-band supports. Record this information, including the separation of the pencils.
• Repeat for pencil separation of 20, 30, 40, ... cm, until one or more of the rubber-band supports has exceeded its unstretched length by 30%.

Determine tension forces and angles for the rubber-band supports

The second figure above depicts the tension forces which support the hanging mass or weight.

• For each length of each support, use your calibration graph (similar to that shown below) to determine the corresponding tension in the rubber band.

• Tension is the force tending to pull the rubber band back to its original length.
• If your calibration graph is in grams of supported mass vs. stretch, you may treat grams as force units as long as you understand that a gram is really a unit of mass, not force, and what we choose to call a gram unit of force really refers to the force exerted by gravity on a 1-gram mass.

• Hand-sketch a reasonably accurate picture of the system for each separation.

• You need not actually measure distances to scale, but you should make a reasonable attempt to estimate relative distances so that the angles in your sketches are reasonably close to the actual angles of the system.

• For each sketch construct a right triangle for each rubber-band support, with the support (including the paper clips at the ends) forming the hypotenuse and a vertical leg extending from the hanging mass upward.

• The horizontal leg will have a length equal to half the separation of the two pencils. Determine the length of the vertical legs using the Pythagorean Theorem.

• Determine the angle made by each support with horizontal by finding tan^-1 ( y / x ) , where y is the length of the vertical leg and x the length of the horizontal leg.

Sketch and Analyze Force Vectors

• Sketch the force vectors corresonding to the tensions and find the x and y components of the forces.
• For each pencil separation, sketch the force vector corresponding to the right-hand rubber-band support. The length of the vector should represent the amount of force, and the angle should be the angle found from the preceding analysis.
• Label each vector with its angle and the magnitude of the force.
• Determine the horizontal and vertical components of each force vector, using your knowledge that Fx = F cos(`theta) and Fy = F sin(`theta) (see figure above).

• For each separation, add up the y components of the two supporting forces, and also add up the x components of these forces.

• Note that the x components are in different directions, so one will be negative while the other is positive, while the y components are both in the same direction.

Answer questions:

• What should the x components add up to, and why?
• What should to y components at up to, and why?

The 'effective' slope of a ramp is the slope in excess of the (small) slope necessary to overcome friction -- the slope at which the friction car moves down the ramp at a constant velocity.  

By observing the displacement of a pendulum by the net force accelerating the friction car down a ramp, we conclude that for small slopes the net force exerted on a car coasting down a ramp is proportional to the 'effective' slope of the ramp.

We begin by determining the slope at which the car moves down the ramp at constant velocity.

• By changing the slope and vibrating the ramp slightly, as demonstrated on the video clip, determine the heights of the ramps at two ends for the slope at which the car tends to move down the ramp at constant velocity.
• Calculate the 'constant-velocity' slope.

We next determine the forces required to prevent the car from accelerating down the ramp for various slopes in excess of the constant-velocity slope.

• Using the light pendulum bob, set up a pendulum and a means of measuring the displacement of the pendulum.
• Using the procedures developed in the previous force-on-a-pendulum experiment, obtain the data that will permit you to determine the force required to prevent the car from accelerating down the incline for various slopes.

• Organize your data into a table.

We convert our data to force vs. slope information.

• From your data table, obtain the force in Newtons and the slope corresponding to each trial.
• For each slope, subtract the constant-velocity slope to obtain the effective slope.
• Present your force vs. effective slope data in a table.

We now plot force vs. effective slope and interpret the results.

• Construct an accurate graph of force in Newtons vs. effective slope.
• Sketch the straight line that best fits the data.
• Determine whether this straight line indicates a force vs. effective slope proportionality.

• Recall that a proportionality is indicated when, within the limits of experimental error, the straight line passes through the origin and the data points.

Compare the slope of the graph with the weight of the car, and draw conclusions about accelerating forces and accelerations.

• From the mass of the car determine its weight. Is the weight, within experimental error, equal to the slope of the graph?
• Can we as a result conclude that, for small slopes at least, the force required to restrain the car from accelerating down the incline is equal to the weight of the car multiplied by the effective slope of the incline?
• Can we conclude that for small slopes, the net force accelerating a car coasting down that slope is equal to the weight of the car multiplied by the effective slope of the incline?
• How can we therefore conclude that the acceleration of a friction car down a small slope is equal to the acceleration of gravity multiplied by the effective slope of the incline?

We begin by dropping a single filter and doubled filter, as shown on the videotape.

• Two identical filters will be used. In one filter we will place a single crumbled-up filter, effectively creating a 'doubled' filter identical to the other except with twice the mass.
• The single filter and the doubled filter should present the same area to air resistance as they fall. To check for possible differences, the crumpled filter will be switched from one of the original filters to the other.
• We drop the doubled filter from a fixed height of at least 1 meter, and preferably higher. We adjust the height from which the single filter is dropped until, when the two filters are dropped simultaneously, they strike the floor simultaneously.
• Having found the desired height, switch the crumpled filter and repeat the experiment, using the same heights. If the filters still strike the floor simultaneously, then we conclude that the difference in terminal velocities is due only to the difference in the mass and not to the surface area exposed to air resistance.
• Since the filters take equal times to reach the floor, the ratio of their average velocities will be the same as the ratio of the heights from which they are dropped.

We next drop the single filter and a tripled filter, then the single filter and a quadrupled filter.

• The procedure is identical to the one used for the doubled filter. For the tripled filter we add two crumpled filters to a single filter.
• Again we will find the heights from which the single filter reaches the floor in the same time is the tripled filter. And again we will switch the crumpled filters and repeat the drop from the same heights to validate our results.
• Again the ratio of the velocities of the single and tripled filters will be the ratio of the distances fallen.
• We repeat for a quadrupled filter, this time adding three crumpled filters to a single filter.

We make a table and graph for force vs. terminal velocity.

• Let w0 stand for the weight of the single filter. Then the weights of the doubled, tripled and quadrupled filters will be 2 w0, 3 w0 and 4 w0.
• Let v0 stand for the average velocity of the single filter and express the velocities of the doubled, tripled and quadrupled filters as multiples of v0. For example, if the doubled filter is found to travel at 1.6 times the average velocity of the single filter, then its velocity will be expressed as 1.6 v0.
• Make a table for weight vs. average velocity. Then sketch a graph of the table.
• Explain the following: Since at terminal velocity the air resistance on the filter is equal to its weight, and since we assume that terminal velocity is obtained very quickly so that the average velocity of a falling filter is very close to its terminal velocity, our table of weight vs. average velocity will constitute a table of air resistance vs. velocity.

Attempt to linearize the data using a power-function transformation of the velocity.

• Attempt to linearize the table by either squaring (taking the 2d power) or taking the square root (taking the .5 power) of the average velocity, whichever results in the more nearly linear graph.
• Test to see whether a nearby power (e.g., the 1.8 or the 2.2 power if the 2d power more nearly linearized the data; the .4 or .6 power if the .5 power did a better job) does a better job of linearizing the data.

Determine the power-function relationship between terminal velocity and force.

• If we let F stand for force and vTerm for terminal velocity, then if p is the power of the transformation of vTerm required to linearize the data, it follows that F is proportional to vTerm ^ (1/p).
• What power of vTerm do you conclude is proportional to the force F of air resistance?

Answer the following:

• Why do we expect that the results will be more accurate when the filters are dropped from greater heights?
• How could you design and experiment to observe the effect of the non-negligible time required for a single filter to reach terminal velocity on the average velocity at different heights?

Set up the shelf-standard balance in a horizontal configuration with water reaching a marked point on the damping cylinder.

• On the damping cylinder place two marks, one about 1.5 cm below a point halfway between the bottom in the top of the cylinder, and another 3 cm above the first.
• Place a cup beneath the balance so that it is possible to fill the cup and cover either mark.
• By adjusting the water depth and the position of the washer on the balance, put the system in a configuration where the beam of the balance is horizontal and water reaches the lower of the two marks.

Add weights and add water to achieve a horizontal configuration with water reaching a higher marked point on the damping cylinder.

• Now using your mass set, add weights to the beam at the position of the damping cylinder and add water to the cup until the beam is again horizontal and the water level reaches the second mark on the cylinder.
• Note the total mass added, and from this total mass determine the weight that was added at the position of the cylinder.

Determine the weight of the displaced water and compare with the weight added at the position of the cylinder.

• The damping cylinder has a diameter of .25 inch. From this information determine the additional volume of the water displaced when the additional 3 cm of the cylinder was immersed.
• At 1 gram / cm^3, what mass of water was displaced by the additional 3 cm of the cylinder? What is the weight of the displaced water?
• How does the weight of the displaced water compare with the weight of the masses that were added at the position of the cylinder?

Answer the following questions:

• What evidence does this experiment give us that when an additional length of the cylinder is immersed, there is an upward buoyant force exerted on it?
• To what extent do your results support the hypothesis that, when the additional 3 cm of the cylinder were immersed, an upward buoyant force was exerted on the cylinder which was equal to the weight of the additional water displaced?
• Why was it important to have the beam horizontal for both of the measurements made in this experiment?

We now determine the weight of a bolt and its apparent weight when immersed in water, from which we infer the buoyant force on the bolt and the density of the bolt.

• Carefully unscrew one of the bolts embedded in the blue Styrofoam disk in your kit. Tie a thread around the bolt and suspend it from one end of the beam on your shelf-standard balance. By adding masses at an equal distance on the other side of the balance, determine the mass of the bolt and infer its weight.
• Position a cup of water so that the bolt is immersed in the water. Find the mass necessary to balance the bolt when it is subject to the buoyant force of the water, and infer its apparent weight in water.
• From the actual weight and the apparent weight of the bolt, determine the buoyant force on the bolt.
• This buoyant force is equal to the weight of the water displaced. What is the ratio between the weight of the bolt and the weight of the water it displaces?
• What therefore is the mass density of the bolt, in grams / cm^3?

Repeat this experiment with various household objects, determining the mass density of each.

• In your own words, describe how this procedure works to give us the mass density of an object.

Devise an experiment of your own, using the same apparatus with procedures similar to those used above, to show that when an object floats, the buoyant force on the object is equal to the weight of the object.