PVL Lab Guides

VirtualDynamicsSoft EduVirtualLabs PVL Lab Guides Javier Montenegro Joo 1

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PVL Lab Guides

Javier Montenegro Joo

Physics Virtual Lab, PVL

This document contains some Physics virtual experiments executed with the PVL. The PVL has been successfully used by its creator, Javier Montenegro Joo, to teach physics to Engineering and Physics students at two universities in Lima Peru. Other universities in other countries are also using the PVL. The PVL has also been used

by its creator to train teachers on the use of computers to teach Physics.

The PVL is used as a teaching aid in the classroom to demonstrate topics covered during the exposition of the theory and also as a virtual laboratory of physics in the

physics lab sessions, where every student executes the PVL in a computer.

This document contains the lab guides of some of the experiments executed with the PVL, not all experiments that have eventually been executed with the PVL have been

included. The PVL operates under two different modes, Research and Learning. When

executing the experiments here described, the PVL must be in Research mode.


Advantages of using simulation software

Software allows extreme experimentation, this is, experimenting with parameters not possible with conventional lab equipment. (like different values of g, the acceleration of the gravity).

Trajectories of motion leave tracks so comparison is possible and easy.

Time evolution can be controlled (set time pace, pause, freeze), thus permitting detailed analysis of experiments. This is impossible in real life experiments.

The Doppler Effect module allows the "visualization" of the sound waves as these are emitted by a source in motion. Changing parameters in this module, permits experimenting with different cases of the phenomenon, resulting in a thorough understanding of it, because the sound frequencies in front and behind the moving source can be "seen", something impossible in real life. Simultaneous simulation and visualization allows visualizing the compression and expansion of the sound waves in front and behind the moving source.

Much more experimental data is obtained with software.

Since for a given experiment every student may work with his/her own data, all students in the lab produce similar graphs, but not equal.

The experiment equipment always works as expected, there are no missing pieces, and there are no problems when setting up the equipment.

Neither Physics lab instructors nor students need any kind of special training to operate the software.


Index

Mechanics Experiment PVL Module

1

Analysis of the graphs of Position vs Time in the Uniform Rectilinear Motion.

Mechanics – Analysis of Position vs time graphs

See

2

Analysis of the graphs of Velocity vs Time in the rectilinear accelerated

motion.

Mechanics – Analysis of velocity vs time graphs.

See

3 Curves of Free Fall Mechanics – Free Fall See

4 Relationship between Cannon Elevation Angle and Range

Mechanics - Parabolic motion – Cannon: The Projectile Flight

See

5 Vertical Shot from a Moving Cannon

Mechanics - Parabolic motion – Vertical shot from a moving car

See

6 Cannon bullet hits free-falling object

Mechanics - Parabolic motion – Cannon bullet hits free falling object.

See

7 Cannon elevation angles to hit the same point when shooting up an incline

Mechanics – Parabolic motion – Projectile shot up an incline by a

cannon

See

8 Friction on a tilted plane

Mechanics - Static Friction Coefficient

See

9 Finding the Acceleration of Gravity from the crossing of falling bodies

Mechanics - Falling bodies.- Crossing of parabolic and vertical

falls

See

Waves and Oscillations Experiment PVL Module

1 Behavior of an oscillating spring.

Waves and Oscillations – SHM - Oscillating Spring

See

2 Frequency behavior in the SHM

Waves and Oscillations – SHM – Position, Velocity and Acceleration

See

3 Simple Pendulum executing a SHM and the Acceleration of the Gravity

Waves & Oscillations – SHM –Simple Pendulum executing a SHM

See

4 Dependence of the period on the CM-to-pivot distance for a physical pendulum

Waves and Oscillations - Physical pendulum

See

5 Frequency of an oscillator in a viscous medium

Waves & Oscillations – The Damped Oscillator

See

Electricity & Magnetism Experiment PVL Module

1 Deflection of a Charged Particle by an Electric Field

Electricity & Magnetism – Charged particle deflection by an Electric

Field

See


Experiment

Analysis of the graphs of Position vs Time in the Uniform Rectilinear Motion.

PVL Module.- Mechanics – Analysis of Position vs time graphs

Objective.- Becoming familiar with the graphs of uniform rectilinear motion and learning how to extract useful information from them. Introduction.- Analyzing a graph of position versus time, information about the motion of a body can be obtained. The average velocity may be calculated by means of

t

xx

tt

xx

t

xVV o

o

o

and the displacement of the body is computed with S = <V> t


Research procedure.- 1 Shift to Research mode. 2 Click the "Random XT" button so a XT graph is randomly generated. 3 For each sector of the generated graph compute the Time interval, the

displacement, Motion direction, Average Velocity. For the whole graph compute the Total Displacement.

4 On a piece of paper make a drawing of the randomly generated XT graph. 5 In order to verify the information resulting from the analysis of the graph, shift to

Learning Mode and by displacing the yellow dots on screen, manually reproduce the original graph. As the graph is reproduced, its corresponding data are displayed and the information previously extracted may be verified.

6 This process must be repeated for other two randomly generated XT graphs Results.- Under Research mode the student randomly gets graphs like the one shown above and makes the computations to obtain the data displayed below the graph, which is not visible under Research mode. Back to Index


Experiment Analysis of the graphs of Velocity vs Time

in the rectilinear accelerated motion. PVL Module.- Mechanics – Analysis of Velocity vs Time graphs.

Objective.- To get familiar with the graphs of one-dimensional rectilinear motion under constant acceleration and to learn how to extract information from them. Introduction.- Information about the motion of a mobile may be obtained from a graph of Velocity vs time (V vs t). If the initial Vo and final V velocities are both known, the acceleration may be computed from

t

vv

tt

vv

t

va o

o

o

Where the resulting sign indicates if the motion is accelerated (plus sign) or decelerated (minus sign). The mean velocity is obtained from averaging V and Vo:

2

ovvVV

and hence the displacement S is calculated by means of S = <V> t


Research procedure.- 1 Shift to Research mode. 2 Click the "Random VT" button so a VT graph is randomly generated. 3 For each sector of the generated graph compute the Time interval, Delta

Velocity, Average Velocity, Displacement and Acceleration. Compute the total displacement for the entire graph.

4 On a piece of paper make a drawing of the randomly generated VT graph 5 In order to verify your computations, shift to Learning mode and by displacing the

yellow dots (A, B, C, D, E and F) on screen, manually reproduce the original graph. As the graph is reproduced, its corresponding data are displayed and the computations previously made may be verified.

6 This process may be repeated for two or three randomly generated VT graphs.

Results.- Under Research mode the student randomly gets graphs like the one shown above and makes the computations to obtain the data displayed below the graph, which is not visible under Research mode.

Back to Index


Experiment Curves of Free Fall PVL Module.- Mechanics – Free fall

Objective.- Generate graphs of behavior of position and velocity for the free fall. Introduction.- The equations of position and velocity for a free falling body are respectively

)2()1()21( 2 tgvtgy where g, the Acceleration of the Gravity depends on the place where it is measured; in general the value of is influenced by the presence of lakes, caves, density of ground, etc. In the earth the average g is 9.8 m/s², but in the moon and in the sun its vale is different.

Research procedure.- 1 Prepare a table to be filled in with 15 values of Time, Squared time, Vertical Position Y,

Delta Y, Velocity V and Delta V.

t t² y Delta y v Delta v

1

2

15

2 Use the Free Fall module of the PVL to simulate and generate data for g = 9.8 m/s² during 15 s in steps of 1 s. The module generates data for Time t, Position Y, Delta Y, Velocity V and Delta V.

3 Use equations (1) and (2) to generate data corresponding to other two values of g out of the range of g in the module. Make two new tables.

4 With the collected data make plots of Y vs t, V vs t, Delta Y vs t, Delta V vs t, and Y vs t² . At the end there must be five graphs with three curves each.

5 Get the slopes of the straight lines resulting in the plots of V vs t, these must be equal to the corresponding value of g.

6 Get the slopes of the straight lines resulting in the plots of Y vs t², these must be equal to g/2 in each case.


Example of results.- An investigation with three values of g (g = 3.0, g = 9.8, g =16) was made, the resulting graphs are:

Back to Index


Experiment

Relationship between Cannon Elevation Angle and Range

PVL Module.- Mechanics - Parabolic motion – Cannon: The Projectile Flight

Objective.- Investigate the dependence of the horizontal range of a cannon, on its elevation angle, for both, fixed cannon shooting velocity and fixed acceleration of the gravity.

Introduction.- The range of a cannon depends not only on its shooting velocity, it also depends on its elevation angle, on the acceleration of the gravity and on the air resistance. In this experiment an investigation of the behavior of the cannon range as a function of its elevation angle will be performed, the acceleration of the gravity and the cannon shooting velocity will remain fixed. Research procedure.- 1 Set to Research mode

2 Put the acceleration of the gravity in g = 9.8 m/s². Set also a shooting velocity Vo for the cannon.

3 Set the cannon elevation angle A, at a low value.

4 Shoot the cannon.

5 Once the projectile lands, measure with the mouse the landing distance R. Take note of A and of R.

6 Increment a little the cannon elevation angle A

7 Repeat steps from (4) to (6), some 15 times

8 Repeat the process above for other two values of cannon shooting velocity Vo and acceleration g of the gravity. Cannon elevation angles A, do not need to be the same as those previously used.

9 Make plots of cannon range R versus cannon elevation angle A.

10 At the end a graph with three curves must result, one for each value of g and Vo.


Example of results.- The plots of range R versus elevation angle A, for three different cases are shown below. Each curve displays the associated values of g and of Vo. It can be seen that the maximum range is achieved when A = 45°, no matter neither the shooting velocity Vo, nor the acceleration of gravity g. It can also be seen that the same range is obtained for A° and for 90° – A°.

Back to Index


Experiment

Vertical Shot from a Moving Cannon

PVL Module.- Mechanics - Parabolic motion – Vertical shot from a moving car

Introduction.- When a shot is vertically upwards made, the projectile is provided only with a vertical velocity Voy, but if this shot is made from a horizontally displacing car, then the velocity of the projectile has also a horizontal component Vox, given by the velocity of the car. Under these conditions, once the projectile is fired, it must behave like the projectile shot with velocity Vo by a cannon with a certain elevation angle A respect to the horizontal. The elevation angle of the projectile shot from the moving car, is

oxoy vvA 1tan

its shooting velocity is 22

oyoxo vvv

and the horizontal range of the projectile is

)2

(2

g

vv

g

vvR

oy

ox

oyox

which means that the plotting of R versus 2Voy/g will produce a straight line whose slope will be Vox, the velocity of the car. Objective.- To determine the horizontal velocity Vox of the car, from where the vertical shot is made. Additionally, to investigate the behavior of the horizontal range R and maximum altitude h of the projectile by making plots of R vs Voy, h vs A°, and R vs A° Research procedure.- 1 Set the module in Research mode. Notice that this simulation module reports the

coordinates (x,y) of the mouse, as this displaces over the screen. 2 Choose a value for Vox, the horizontal velocity of the car, and for the Acceleration of the

Gravity g. These will remain fixed during the research. Even though the user sets these parameters, both are supposed to be unknown.


3 Set Voy, the vertical shooting velocity, at its minimum value 4 Right after Vox and Voy are fixed, the values of the shooting velocity Vo and cannon

elevation angle A°, are automatically calculated. Take note of these values. 5 Start the motion of the cart and then fire (make a shot). 6 Once the projectile lands, use the mouse coordinates to find out the horizontal range R

of the projectile and its maximum height h reached during its flight. Fill in the corresponding row in the table:

Voy Vo A° R h 2 Voy / g

1

2

:

15

7 Increment a little the value of Voy 8 Repeat steps from [4] through [7] until no more increments are possible 9 Make a plot of (1) R vs 2Voy/g, this will generate a straight line, whose slope may be

determined with the help of a ruler; this slope must be equal to Vox, velocity of the car. 10 Additionally make plots of (2) R vs Voy (3) h vs A° (4) R vs A° 11 Repeat the experiment for other two values of g.

Example of results.- The experiment was executed with Vox = 50 and g = 9.8

Back to Index


Experiment

Cannon bullet hits free-falling object

PVL Module.- Mechanics - Parabolic motion – Cannon bullet hits free falling object.

Introduction.- In this experiment a cannon shoots a bullet to a ball being held at an elevated position, but the ball starts to fall freely the moment the cannon makes fire. Eventually the bullet hits the ball at a certain position (Xo,Yo) below the initial position of the ball.

It can be seen in the diagram that the time for the bullet to reach the hit point -whose coordinates are (Xo, Yo) - is the same to that needed by the ball to descend to that point.

Consequently, equating the time t from equations

tvx oxo and 2

2

1gtH o it will result that

2

2

1

ox

o

ov

xgH and since AtgxyHH ooo

then the coordinates (Xo, Yo) of the hit point are related by means of

2

2

2 ox

ooo

v

xgAtgxy

which happens to be the equation for the flight of a projectile fired by a cannon, when the projectile is at the point of (Xo, Yo)

Objective.- To investigate the dependence of the hit point height Yo on the shooting velocity Vo of the bullet.

Research procedure.- 1 Shift to Research mode. 2 Fix an elevation angle A for the cannon and fix also the horizontal position Xo of the

object that begins to fall freely the moment the cannon fires. Set a value for the acceleration of the gravity.

3 Fix the shooting velocity Vo of the cannon, making sure that this velocity is enough for the bullet to meet the free-falling object.

4 Fire the cannon.


5 Once the bullet hits the free-falling object, place the mouse on the hit point and read its coordinates. Take note of the shooting velocity Vo and the vertical position Yo of the hit point.

6 Increment a little the shooting velocity Vo

7 Repeat steps from 4 to 6, some 12 times

8 Repeat the whole process for other two values of the acceleration of the gravity. At the end there must be three sets of Vo and corresponding Yo

9 Plot Yo versus Vo, for the three sets on a single graph. Ultimately there must be a graph with three curves, one for each value of g.

Example of results.-

Another experiment: Find the maximum value of g, beyond which there is no bullet - ball meeting. After fixing A, Vo and Xo, vary g in small steps, from small to large values, until it becomes impossible that the bullet hits the ball.

Notice in the resulting graph that the three lines cross by pairs, this suggests that for every g there are two values of Xo, producing the bullet – ball meeting in the same Yo.

Back to Index


Experiment

Cannon elevation angles to hit the same point when shooting up an incline.

PVL Module: Mechanics – Parabolic motion – Projectile shot up an incline by a cannon.

Introduction.- It is known that for a cannon on a horizontal surface and whose shooting velocity is Vo, the same hitting point R on the horizontal is obtained, when the cannon elevation angles are complementary, this is Ao and 90 – Ao To this property we may refer as the law of complementary angles for the parabolic flight of a projectile shot by a cannon; this may be Mathematically expressed as

)1(90 12

ooo AA

Objective.- To experimentally detect the cannon elevation angles that allow hitting the same point when shooting up an incline. Research procedure.- 1 Set an elevation angle B for the incline, another elevation angle A1 for the cannon, and a

shooting velocity for the cannon.

2 Fire the cannon and take note of the point P where the bullet impacts on the incline.

3 Change only the cannon elevation angle A, fire the cannon, and see if the projectile hits the same point P. Repeat this process until finding a cannon elevation angle A2 that allows making impact on the same point P, as with A1.

4 Repeat steps 2 and 3 with different cannon elevation angles, incline elevation angles and also different shooting velocities.

5 Once you are sure you have found a relationship between the elevation angles of cannon and incline (angles A and B), make a statement like the one in equation (1)

6 Repeat steps from (1) through (5) for other two values of the Acceleration of Gravity.

7 Solve theoretically the same problem and compare your experimental results with the theoretical ones.

Results.-


Experimentally it is found that the cannon elevation angles A1 and A2 to hit the same point on the incline, whose elevation angle is B, are related by means of A2 = 90 – A1 + B This relation is valid for any value of the Acceleration of Gravity and it is effective provided that A1 and A2 are greater than B, the incline elevation angle. Back to Index


Experiment Friction on a tilted plane

PVL Module: Mechanics - Static Friction Coefficient

Introduction.- A way of finding the coefficient of static friction Us between two materials is to place one on the horizontal surface of the other, and slowly tilt upwards the surface and see at what inclination angle A, the block resting on the surface begins to slide downwards.

Objective.- To experimentally study the relationship between the inclination-angle of the tilting surface and the coefficient Us of static friction. To determine the value acquired by the normal force, at the moment the sliding begins. To find the influence of the weight in the value of the static friction coefficient Us. Introduction.- The figure shows the deduction of a relation between the static friction coefficient Us and the critical angle A of surface inclination at which the block begins to slide downwards

Research procedure.- 1 Set to Research mode.

2 Set any weight for the block that will slide over the tilted surface.


3 Set a small value for the coefficient of static friction, Us. This is tantamount to select the material of the block to be placed on the tilting surface.

4 Gradually tilt the surface and see at what angle (A) the block begins to slide down. As the surface tilts, the experimental inclination angle (A) is shown.

5 Once the block has slid over the tilted surface, take note of the values of Us and the corresponding tilt angle A.

6 Using the experimental value of A, calculate the value of the normal force N, at the moment the sliding begins

7 Enter the values of Us, A and N in a table:

Experimental data.-

Weight =

1 2 3 4 … … … … 15

Us

A°

N

Increase a little the value of the coefficient of static friction, Us.

8 Repeat some 15 times, steps from (4) through (7).

9 Repeat the whole process for any other two weights.

10 Make plottings of Us vs A, N vs Us, N vs A. At the end three graphs must result, with three curves each, one curve for each used weight.

Example of results.- The three following graphs show the results of the investigation:


Back to Index


Experiment Finding the Acceleration of Gravity from the crossing of falling bodies

PVL Module.- Mechanics - Falling bodies.- Crossing of parabolic and vertical falls

Introduction.- From the free fall equation y = (1/2) g t² it can be seen that the plotting of y versus t² is a straight line with slope g/2. Objective.- To determine the Acceleration of Gravity g, from the position of the crossing point and elapsed time for two projectiles, one under a free fall and another falling in a parabolic route. Research procedure.- 1 Shift to Research mode. 2 Randomly fix a value for the Acceleration of Gravity g. The value of g however,

is supposed to be unknown along the research, because finding g is precisely the objective of this investigation.

3 Randomly set the position “d” of the block supporting the ball that will fall freely. Positions close to the table allow for more data collection, hence these must be preferred. Take note of the supporting-block position.

4 Set the horizontal shooting velocity Vox at its minimum. 5 Make a shot. Take note of the shooting velocity Vox 6 Carefully wait for the vertically falling ball to meet the ball falling along a

parabola. Click "Pause" or "Abort" when both balls meet. Take note of the


meeting point vertical position and of the elapsed time. Acceleration of the Gravity g =

1 2 3 4 … … … … 15

Vox

y

t

t²

7 Increment the shooting velocity by 5 units 8 Repeat some 15 times, steps (5), (6) and (7). 9 Make a plot of y versus t (a parabola must appear), and a plot of y versus t², (a

straight line must show up). 10 Get the slope of the straight line in the plot of y vs t², its value must be g/2. 11 Repeat the whole process for other two values of g. Make the plotting on the

same graphs, so that comparison of curves of the same kind is possible. Example of results.- With data generated in the simulations the two following graphs were obtained:

Back to Index


Experiment

Behavior of an oscillating spring.

PVL Module.- Waves and Oscillations – SHM - Oscillating Spring

Introduction.- The oscillations of a spring can be approximated by the equation of a Simple

Harmonic Motion, )( tSinAA o where the angular frequency is given by

m

k and then )

1(2

mk

m

k

and therefore the plotting of mvs12 is a straight line whose slope is k, the value

of the elastic constant of the spring. Objective.- Analyze the dependence of the period T of oscillation on the mass m.

Verify that the plotting of mvs12 is a straight line whose slope gives the value of

the elastic constant k of the spring.

Research procedure.- 1 Shift to Research mode 2 Choose a spring to work with; this is, set some value for the elastic

constant k. Also set to 10 the number n of periods. These will remain unaltered throughout all the investigation.

3 Set the value of the mass to some value from 1 to 5 units, take note of it. 4 Oscillate the spring 5 Take note of the oscillation total (elapsed) time.


6 Calculate the oscillation period T, dividing the elapsed time by the number n of periods.

7 Compute the frequency f = 1/T and the angular frequency w = 2 Pi f = 2 Pi / T

8 Calculate w² and 1/m and take note of these values. 9 Increase the mass in 4 or 5 units and repeat some 15 times steps from

(4) through (9). For a given value of k, generated data my be stored in a table like

m t T = t / n f = 1 / T f 2 2 1 / m

1

2

:

15

10 Make a plot of Period T vs Mass m. 11 Make a plot of w² versus 1/m, this graph must be a straight line. 12 Compute the slope of the resulting straight line.

This slope must be the value fixed in (2) for the elastic constant k. 13 Repeat steps from (2) to (12) for other two values of k, using the same

values for the mass. Make all plottings on the same graph, at the end there must be two graphs, one of T

vs m and another of mvs12 , with three curves each.

Example of results.- The two graphs below display the results obtained following the instructions in the Research Procedure:

Back to Index


Experiment

Frequency behavior in the SHM

PVL Module.- Waves and Oscillations – SHM – Position, Velocity and Acceleration

In this module the Simple Harmonic Motion is visualized by means of a freely oscillating spring. The oscillator is not subject to the acceleration of the gravity. Three plots are generated as the spring oscillates: position vs time, velocity vs time and acceleration vs time. The number of periods completed during oscillation can be easily visualized and counted on the x vs t plotting.

This means that the period T of the oscillation may be calculated by dividing the total elapsed time, by the number of oscillated periods, and after that, the oscillation frequency may be computed by means of f =1/T Introduction.- The frequency f of an oscillator describing a simple harmonic motion is given by

m

k

Tf

2

1

2

1

Where T is the period of oscillation, w is the angular frequency, k is the elastic constant and m is the mass. It can be seen that the frequency increases with k and decreases with m. Objective.- Generate graphs of behavior that permit to study the dependence of the SHM, on its mass and also on its elastic constant. Research procedure.- 1 Shift to Research mode. Fix some value for the elastic constant k of the oscillator

and set also a small value for its mass m. Take note of k. 2 Start the oscillations and pay attention to the number of completed periods on

the x vs t plotting. 3 Click “Abort” when the spring has completed an integer number of periods. The

larger the number of periods, the more accurate the results. 4 Carefully count the number n of completed periods on the x vs t plotting.


5 Read the elapsed time and calculate the oscillation period T, by dividing the elapsed time by n.

6 Calculate the oscillation frequency by means of f = 1/ T 7 Take note of the mass m and the corresponding T and f. 8 Increment a little the value of m. 9 Repeat some 15 times, steps from [2] through [8] 10 With the collected values of m and f make a plot of frequency versus mass. 11 Repeat this process for other two values of the elastic constant k.

Make the plotting on the same graph, so that, at the end a single graph with three curves must result. Having two or more curves of the same kind on a unique graph, allows comparison.

Following an analogous procedure, fix the oscillator mass m and varying k, find the corresponding frequency f for each k. Make a plot of frequency versus elastic constant (f vs k). Example of results.-

Back to Index


Experiment

Simple Pendulum executing a SHM and the Acceleration of the Gravity

PVL module.- Waves and Oscillations Oscillations of a simple pendulum executing a SHM

Introduction.- Since the angular frequency W of the simple pendulum executing a Simple Harmonic Motion (SHM), is given by

LgLg 1/ 2

then the plotting of Lversus 12 will be a straight line whose slope is g. In this

way the value of the Acceleration of the Gravity g, may be determined from a set of experiments with different pendulum lengths L, by measuring their oscillation periods T and computing their angular frequencies with

Tf 22

In this experiment the oscillations of the simple pendulum are at the most 14°, which means that the pendulum executes indeed a SHM, oscillation amplitudes beyond 14° are not SHM.

Objective.-

Verify that the plotting of Lversus 12 is a straight line whose slope is the

Acceleration of the Gravity g

Research procedure.- Prepare a 5-column table, to be filled in with the following experimental data: Length L, 1/L, Period T, W, W².


1 Set to Research mode. 2 Choose a value for g, the Acceleration of Gravity. This will remain

constant along the research, and it is supposed to be unknown, finding g is the aim of this experiment.

3 Set a small value for the length L of the pendulum rope. 4 Start the oscillations of the pendulum.

As the pendulum oscillates the Amplitude vs Time plotting is shown on screen. Click "Abort" in order to stop the oscillations when an integer and not so small number of periods has been oscillated.

5 When the oscillation finishes, take note of the total time. 6 On the plotting of Amplitude vs Time shown on screen, count the number

n of oscillated periods. 7 Find the oscillation period T, by dividing the total time by n. 8 Increment the length L of the pendulum by a short amount. 9 Repeat some 15 times, steps from (4) through (8). 10 Make a plot of Lversus 12 , this must be a straight line. Obtain the

slope of the resulting line which must be equal to the value of g previously fixed in (2).

11 In order to verify if this method for finding g is valid for other cases, repeat the whole process for two other values of g. The lengths L of the pendulum do not need to be equal to those used previously.

Example of results.- The plotting below is the result of executing the Research Procedure for three different values of g, as expected the slopes of the lines on the graph are close to the predefined values of g:

Back to Index


Experiment

Dependence of the period on the CM-to-pivot distance for a physical pendulum

PVL module.- Waves and Oscillations - Physical pendulum

Introduction.- Though the period of oscillation of a physical pendulum describing small oscillations (at most 14°), is given by

)1(2mgb

IT

some insight may be obtained if this equation is used to investigate oscillations of the physical pendulum of any amplitude.

Objective.- To investigate the dependence of the period T on the CM-to-pivot distance b, and verify that the period of oscillation of the physical pendulum is independent of its mass. Research procedure.- 1 Shift to Research mode. 2 Choose a value for the mass of the pendulum, and another for g, the

Acceleration of the Gravity. Though the amplitudes of oscillation have no influence in this investigation, fix the amplitudes in at least 60°, because in this way it is easier to count the oscillated periods depicted on screen. These three parameters will remain constant throughout the research.

3 Fix the value of b, the distance from the CM to the pivot, at its minimum 4 Oscillate the pendulum. The oscillation amplitudes vs time (A vs t) will be

depicted as a sinusoidal on screen, on real time. The time is also reported as it evolves.

5 Pause or abort the oscillations when an integer number of periods have been oscillated. Including fractions of a period introduces errors. More accurate results are obtained when the number of periods is rather high, so avoid small number of periods.

6 On the A vs t plotting count the number n of oscillated periods. 7 Dividing the Total time by n, get the period T of oscillation 8 Tabulate the experimental values of b and T. 9 Increment in 0.05 the value of b, and repeat steps from (4) to (9) until no more

values of b are possible.


10 Repeat steps from (1) to (9) for other two pendulums with different masses. 11 With the collected data make plots of T versus b.

At the end there must be a graph with three curves, a curve for each used mass. Example of results.- The fig. shows the behavior of the period of oscillation T versus the distance b from the CM of the pendulum to its pivot. The plotting makes evident the fact that the period of oscillation is independent of the mass of the pendulum.

Back to Index


Experiment

Frequency of an oscillator in a viscous medium

PVL module.- - Waves and Oscillations -The Damped Oscillator

Introduction.- The oscillations of a damped oscillator are mathematically represented by the second order differential equation

02

2

x

m

k

dt

dx

m

b

dt

xd

where x is the amplitude of the oscillation, m the mass of the oscillator, k its elastic constant, and b is the viscosity of the medium where the oscillator is immersed.

When there is no damping the (free) oscillator oscillates with a frequency mko ,

and when it is immersed in a medium that imposes a damping mbD 2 , its

oscillation frequency becomes 22 Do .

The solution of the differential equation above is

)()( 2 tSineAtx mtb

and it represents oscillations that decrease as time passes by. It can be seen that the oscillations x(t) depend on both, the viscosity b and the mass m of the oscillator. Objective.- To study the dependence of the oscillation frequency of an oscillator on the viscosity of the medium and, on the mass of the oscillator. Research procedure.- Studying the frequency of oscillation as a function of the viscosity.- 1 Shift to Research mode 2 Set some value for the elastic constant k of the oscillator and also for its mass. 3 Choose a low value of the viscosity b.


4 Start the oscillations. 5 When the oscillations stop, the experimental frequency w of the oscillations is

reported. Take note of b and w. 6 Increment a little the value of b. 7 Repeat some 12 times steps from 4 through 6. 8 Make a plot of w vs b. Studying the frequency of oscillation as a function of the oscillator mass.- 1 Shift to Research mode 2 Set some value for the oscillator elastic constant k, and for the viscosity b. 3 Choose a low value for the mass m of the oscillator. 4 Start the oscillations. 5 When the oscillations stop, the experimental frequency w of the oscillations is

reported. Take note of m and w. 6 Increment the value of the oscillator mass m. 7 Repeat some 12 times steps from 4 through 6. 8 Make a plot of w vs m. Example of results.-

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Experiment

Deflection of a Charged Particle by an Electric Field

PVL module.- - Electricity & Magnetism - Charged Particle Deflection

by an Electric Field Introduction In this experiment, a particle of mass m and charge Q is shot

with velocity ov into a

transversal uniform electric field E, produced by two parallel plates with opposite charge. Due to the interaction of the electric field with the charge of the particle, this is deviated from its original direction and it finally hits onto a screen placed a distance D away. The deviation h of the charged particle, from its original direction, is given by

2

ovm

DLEQh

This means that plotting 2

ovELDvsh will generate a straight line whose slope

is mQ .

The vertical E field between the plates exerts a force QEF on the charged

particle, so that this experiments a vertical acceleration EmQay )/( , and this

acceleration is associated to a vertical velocity tav yy .

Traveling with velocity ov the time to traverse the parallel plates is oL vLt / , hence

the vertical velocity of the particle when leaving the E-field region, is oy vmLEQv

thus the velocity of the charged particle when leaving the E-field region, will be

oooyx vvmLEQvvvv 2222 )(


Objective.- To study the deflection of charged particles by an electric field and to verify that the

plotting of 2

ovELDvsh is indeed a straight line of slope Q/m

Research procedure.-

1 Shift to Research mode. 2 Fix values for the Electric field intensity E, for the distance D to the screen, for

the mass m and charge Q of the particle. The mass m and the charge Q of the particle are supposed to be unknown; experimentally finding the Q/m ratio is the aim of this experiment.

3 Set the shooting velocity Vo of the particle at a small value.

4 Shoot the particle, it will traverse the electric field between the plates, it will be deviated and will finally hit the screen at a height h. Take note of Vo and of h.

5 Increment a little the shooting velocity Vo.

6 Repeat some 15 times, steps (4) and (5)

7 For each recorded value of Vo, compute ELD/Vo²

8 Make a plotting of h vs ELD/Vo²

9 The plotting of h vs ELD/Vo² must generate a straight line, whose slope should be close to the previously defined Q/m ratio

10 This procedure must be repeated for other two sets of values for E, D, m, and Q. The three plottings have to be made on the same graph, in this way at the end there ought to be a graph with three lines, one for each Q/m ratio.

11 Analyze the dependence of the deviation h on the initial velocity Vo, making plottings of h vs Vo, for the three studied cases.

12 Verify that the final velocity of the charged particle is greater than the initial one, making plottings of V vs Vo, for the three studied cases.

Example of results.- Following the instructions above, the three following cases were studied: (1) E = 1, m = 1, Q = 1, D = 400, Q/m = 1 (2) E = 4, m = 7, Q = 3, D = 600, Q/m = 0.428 (3) E = 5, m = 3, Q = 10, D = 500, Q/m = 3.333 The resulting graphs are shown below:


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