Physics 105 Laboratory Manual

Department of Physics

Physics 105 Laboratory Manual

Prepared to the Department of Physics by:

Hamed S.Hamadnah, M. Sc.

Anas Ababneh, Ph. D.

1


A

Graph Plotting

Name: Lab. Partners Section: Group No. Date: Instructor:

Grade 2

EXPERIMENT: A- Graph Plotting Purpose To practice data plotting and data analysis. Frequently, a graph is the clearest way to represent the relationship between the quantities of interest. A graph indicates a relation between two quantities, x and y, when other variables or parameters have fixed values.

Components of a graph You need to consider the following components to make your graph accurate and readable. ✓ Title: A title should be given to the whole graph. The title of the graph is written at the top of the graph. The title should be clear, descriptive, and self-explanatory. ✓ Axes: Take the independent variable as the abscissa (x-axis) and the dependent variable as the ordinate (y-axis) (unless you are instructed otherwise) ✓ Labels: Label each axis to identify the variable being plotted and the units being used. Mark prominent divisions on each axis with appropriate numbers. ✓ Scale: Choose a convenient scale for each axis so that the plotted points will occupy a substantial part of the graph paper, do not choose a scale which is difficult to plot and read. Graphs should usually be at least half a page in size. ✓ Points: Identify plotted points with appropriate symbols, such as crosses or empty circles. ✓ Fitting: If the experimental data is theoretically a straight line, draw the best straight line through the points with a straight edge. There are mathematical techniques for determining which straight line best fits the data, but for the purposes of this lab it will be sufficient if you simply make a rough estimate visually. The line should be drawn with about as many points above it as below it, and with the `aboves' and `belows' distributed at random along the line. (For example, not all points should be above the line at one end and below at the other end).

Straight line Analysis Often there will be a theory concerning the relationship of the two plotted variables. A linear relationship can be demonstrated if the data points fall along a single straight line.

The general form of a straight line equation is Where Yo is the y-intercept. THE SLOPE

.

In plotting linear relationships, you frequently will be asked to find the slope of the line you have fitted to the data (this slope usually has a physical meaning). The slope is given by

To calculate an accurate slope, • Read a point on the line near each end of the line. • Do not use data points unless they are on the fitted line. That is, choose widely separated points only on the line for the calculation of a slope. INTERCEPTS The y-intercept of a line is that point on the abscissa, or x-axis, at which a line crosses that axis. The y-intercept is defined in the same manner

3

Simple exercise to practice data plotting and data analysis The following data was collected on the position of a car x as a function of time t , as detected by a police radar detector:: 0.00 0.67 t (min)

1.21

1.82

2.41

3.01

3.61

4.21

2.48

6.51

7.62

8.94

11.98

12.73

13.16

X (km) 1. 2. 3. 4. 5. 6.

3.40

Plot a graph of x versus t. Draw the best straight line fit of the data Write the equation of the best line that describe the relation between x & t What is expected value of x if t=10 min? The physical significance of the slope and intercept? Describe the motion?

4

Figure (A-1): Exercise of graph plotting

5

Figure (A-2): Exercise of graph plotting

6


1

Experimental Errors and Uncertainty


Grade 7

1

Experimental Errors and Uncertainty

All measurements contain error. An experiment is truly incomplete without an evaluation of the amount of error in the results. You will learn to use some common tools for analyzing experimental uncertainties. The methods you will learn here are sufficient for many applications.

Types and Sources of Experimental Errors Experimental errors arise for many sources and can be grouped in three categories: 1. personal: • Associated with personal bias or carelessness in reading an instrument, in recording observations, or in mathematical calculations. 2. systematic: • Associated with particular measurement techniques. Improper calibration of measuring instrument • Is the ―same error each time. This means that the error can be corrected if the experimenter is clever enough to discover the error. 3. random error: • Unknown and unpredictable variations • Unbiased estimates of measurement readings • Is a ―different error each time. This means that the error cannot be corrected by the experimenter after the data has been collected.

Significant Digits It is important that measurements, results from calculations, etc. be expressed with the appropriate amount of digits. A real-life example concerns the average weight of an adult. If someone expressed this value as 75.234353556 kg., that would not make much sense. If it were expressed as 75 kg. Similarly, when you present your results for a given experiment, the figures should contain no more digits than necessary. 1- Exact factors have no error (e.g., e, π ) 2- All measured numbers have some error or uncertainty , this error must be calculated or estimated and recorded with every final expression in a laboratory report , the degree of error depends on the quality and fineness of the scale of the measuring device 3- Use all of the significant figures on a measuring device. For example, if a measuring device is accurate to 3 significant digits, use all of the digits in your answer. If the measured value is 2.30 kg, then the zero is a significant digit and so should be recorded. 4. keep only a reasonable number of significant digits - e.g., 136.467 + 12.3 = 148.8 units - e.g., 2.3456 ± 0.4345634523 units → 2.3 ± 0.4 units

NOTE: hand-held calculators give answers that generally have a false amount of precision. 5. Round these values correctly; again as a rule, the final answer should have no more significant digits than the data from which it was derived.

8

READING ERROR (DIRECT MEASUREMENTS): Scientists make a lot of measurements. They measure the masses, lengths, times, speeds, temperatures, volumes, etc .Use half of the smallest division (e.g. in case of a meter stick with millimeter divisions, the reading error is 0.5 mm). When they report a number as a measurement the number of digits and the number of decimal places tell you how exact the measurement is. For example: 121 is less exact than 121.0, the difference between these two numbers is that a more precise tool was used to measure the 121.0. The total number of digits and the number of decimal points tell you how precise a tool was used to make the measurement.

Reporting measurements: There are 3 parts to a measurement: The measurement (x), the uncertainty ( Δx) and the unit. The result of the measurement of a quantity x should be presented as (unit) No measurement should be written without all three parts. The last digit in your measurement should be an estimate Example: 5.2 ± 0.5 cm. Which means you are reasonably sure the actual length is somewhere between 4.7 and 5.7 PROPOGATION OF ERROR (INDIRECT MEASUREMENTS) In the majority of experiments the quantity of interest is not measured directly, but must be calculated from other quantities. Such measurements are called INDIRECT. For example, the volume of a sphere calculated from the measurement of its radius, or the density of the sphere’s material calculated from the ratio of the mass and volume of the sphere. As seen above, the quantities measured directly are not exact but have errors associated with them. In order to calculate the uncertainty of an indirect measured quantity we need first to calculate the uncertainties of the directly measured quantities involved, and then apply the appropriate formula for propagation of errors. These formulas depend on how the indirect measurement quantity depends on the direct measured quantities. The three rules you need for this lab are: Rule 1: If two or more direct measured quantities are being added or subtracted to yield an indirect measured quantity y: If or then where , , etc. are the uncertainties of the direct measured quantities the uncertainty of the indirect measured quantity y. Rule 2: If two direct measured quantities are being multiplied or divided: If

or

then

is called relative error Rule 3: Multiplying or dividing by a pure (whole) number: We multiply or divide the uncertainty by that number. If

or

then

Where c is constant

9

,

,

etc and

is

How to Use a Caliper? Although the vernier caliper can be used to measure a variety of different dimensions, the way in which it is read is always the same. The reading on a vernier caliper is a combination of the value shown on the main scale and the value indicated on the sliding vernier scale. Figure (1-1 ) caliper Step 1 - Look at main scale When taking a measurement, you should first read the value on the main scale. On a metric vernier caliper, this will be given in millimetres (mm).The smallest value that can read from the main scale is 1mm (indicated by a single increment).The value on the main scale is the number immediately to the left of the 0 marker of the vernier scale. In the example in figure (1-2) its 13 mm

Figure (1-2 ) The reading on a vernier caliper

Step 2 - Look at vernier scale Next, look at the vernier scale. The vernier scale of an metric vernier caliper has a measuring range of 1 mm. In the example we will be looking at, the vernier scale is graduated in 50 increments. Each increment represents 0.02mm. In the example in figure (1-2) its 0.42 mm However, some vernier scales are graduated in 20 increments, with each one representing 0.05mm. some vernier scales are graduated in 10 increments, with each one representing 0.1mm. Step 3 - Add both values together When reading the vernier scale, identify the increment that lines up most accurately with an increment on the main scale. This value will make up the second part of your measurement. To get your total reading, add both the value from the main scale and the value from the vernier scale together. In the example in figure (1-2) its 13 mm+ 0.42 mm the reading is 13.42 mm.

10

How to Use a micrometer? Micrometer Screw Gauge is used to measure smaller length or distance such as the diameter of a sphere more accurately A micrometer screw gauge has two scales: the main scale and the vernier scale

Figure (1-3 ) The reading on a micrometer • •

The main scale of a micrometer is calibrated in mm. The calibrations of the main scale of micrometer screw gauge vary depending on the range of measurement that the micrometer screw gauges are meant to measure. The vernier scale has 50 equal divisions. Each division is obtained by dividing 5 by 10. The vernier scale moves a distance of 0.5mm along the main scale when it makes 1 revolution by turning round once. One division on the vernier scale equal to 0.5/50 which equal to 0.01mm on the main scale. The accuracy of a micrometer screw gauge is 0.01mm or 0.001cm. This means that the smallest distance that a micrometer screw gauge can measure is 0.01mm or 0.001cm. (

see figure (1-3))

11

procedure 1- Paper A4 1. Take a paper of certain size, say A4. Measure the dimensions of the paper (length and width) using a typical ruler. 2. Make sure to record the reading error of the ruler which would be the same for the length and width. 3. Calculate the circumference of the paper , Calculate the error in C( Δc) . , Calculate the error in A( ΔA) .

4. Calculate the area of the paper:

Tabulate your data in table (1-1)

Length,

ΔL

∆W

Width,

L (cm)

Circumference,

Area,

C (cm)

A (cm2)

W(cm)

Table (1-1) Data for paper A4 2- Density of sphere 1. Take a sphere of certain volume. Measure the diameter of the sphere (d) using a micrometer. Make sure to record the reading error of the micrometer(Δd). 2. Measure the mass of the sphere (m), and record (Δm) 3. Calculate the volume of the sphere Calculate the error in v( ΔV) . 4. Calculate the density of the sphere ρ= m/V Calculate the error in ρ ( Δρ) .

The uncertainty in the density is

Repeat calculation of density using caliber to measure Diameter of the sphere Tabulate your data in table (1-2)

Trial

Mass (M) g

d (cm)

Volume (V) Density ( cm3 g/cm3

using micrometer Using Caliber

Table (1-2) Data for sphere

12

) g/cm3


2

VECTORS USING THE FORCE TABLE


Grade

13

2

VECTORS USING THE FORCE TABLE

PURPOSE In this experiment you will investigate the general properties of vectors noting their resolution into components and their additive properties. The type of vector which will be used to illustrate these properties will be the vector representing forces The force table is used to experimentally determine the force which balances two other forces. This result is checked by adding the two forces by using their components and by graphically adding the forces. APPARATUS Force table; Three clamp-mounted pulleys Ring with three strings attached ;three hangers; Set of masses; metric ruler; protractor. The force table is an apparatus that allows the experimental determination of the resultant of force vectors. The rim of the circular table is calibrated in degrees. Forces are applied to a ring around a metal peg (pin) at the center of the table by means of strings. The strings extend over pulleys clamped to the table and are attached to hangers Figure (2-1): The Force Table The direction of the forces may be adjusted by moving the position of the pulleys. The magnitude of the forces is adjusted by adding or removing masses to the hangers. The forces used on the force table are actually weights. THEORY A scalar is a physical quantity that possesses magnitude only. Examples of scalar quantities are mass, time, and temperature. A vector is a quantity that possesses both magnitude and direction; examples of vector quantities are velocity, acceleration and force. A vector can be represented by an arrow pointing in the direction of the vector; the length of the line should be proportional to the magnitude of the vector. The RESULTANT of two or more vectors is a single vector which produces the same effect. For example, if two or more forces act at a certain point, their resultant is that force which, if applied at that point, has the same effect as the two separate forces acting together. The EQUILIBRANT is defined as the force equal and opposite to the resultant. 14

Vectors can be added either graphically or analytically. In this experiment, three methods for vector addition will be used: the experimental method (using the force table), the graphical method (head to tail method) and by the component method. 1- Experimental Method

Two forces are applied on the force table by hanging masses over pulleys positioned at certain angles. Then the angle and mass hung over a third pulley are adjusted until it balances the other two forces. This third force is called the equilibrant ( ) since it is the force which establishes equilibrium. The resultant ( ) is the addition of the two forces and is equal in magnitude to the equilibrant and in the opposite direction (see Figure 2). So the equilibrant is the negative of the resultant:

Figure (2-2). The Equilibrant Balances the Resultant 2- Component (or Analytical) Method Two forces are added together by adding the x- and y-components of the forces. First: the two forces are broken into their x- and y-components using trigonometry , where Fx is the x-component of vector and is the unit vector in the x-direction. Fy is the y-component of vector

and is the unit vector in the y-direction To determine the sum of

and

, the

components are added to get the components of the resultant : and To complete the analysis, the resultant force must be in the form of a magnitude and a direction (angle):

3- Graphical Method (Tail to the head) Two forces are added together by drawing them to scale using a ruler and protractor. The second force ( ) is drawn with its tail to the head of the first force ( is drawn from the tail of

). The resultant ( ) to the head of

.

See Figure (2-4). Then the magnitude of the resultant can be measured directly from the diagram and converted to the proper force using the chosen scale. The angle can also be measured using the protractor.

Figure (2-4): Adding Vectors Head to Tail

15

PROCEDURE PART ONE: COMPONENTS OF A FORCE. 1. Arrange identical masses of 200 g at positions of 40° and 220°. These masses should balance. 2. Now, remove the mass at 40°and by trial and error find the amounts of mass that can be hung at 0° and 90° to balance the mass at 220°. These are the x- and ycomponents of the force of 200 g at 40o. Figure (2-3): Resolving a vector into its components Record these in Table-1. 3. Calculate the x- and y-components of the vector with a magnitude of 200 g and direction of 40° as shown in Figure2-3 and record them in Table-1. PART 2. ADDITION OF TWO VECTORS. Experimental Method To find the resultant of two forces: a 2 N force at 20o and a 3 N force at 50o. 1. Place one pulley at 20o and the other at 50o. 2. Place a mass of 200g (including hanger) On string over the pulley at 20o , and Place the mass of 300g (including hanger) On string over the pulley at 50o 3. With a third pulley and masses, balance the forces exerted by the two forces. The forces are balanced when the ring is centered around the central metal peg (pin) NOTE: To balance the forces, move the third pulley around to find the direction of the balancing force, and then add masses to the third hanger until the ring is centered. 4. The balancing force obtained in (step 3) is the equilibrant force. The resultant has the same magnitude as the equilibrant. To determine the direction of the resultant, subtract 180o from the direction of the equilibrant. Record your results in Table-2 under Experimental Method. Component (or Analytical) Method Express each given vector in component notation, add them up and calculate the resultant vector. From your result, express the resultant vector in magnitude and direction. Record your results in (Table-2) under Component Method Graphical Method (tail-to-head) On a separate piece of paper, construct a tail-to-head diagram of the vectors of Force F1 and Force F2. Use a metric rule and protractor to measure the magnitude and direction of the resultant. Record the results in Table

16

Part 1. COMPONENTS OF A FORCE VECTOR . Table-1: Summary of the Data in Part-1, the components of the force vector F=0.2N @ 40o

x-component (at 0o)

y-component (at 90o)

M=………..

M=………..

Fx=

Fy =

Fx=

Fy=

Experimental

Computational % error

PART 2. ADDITION OF TWO VECTORS . Experimental Method Table-2: Summary of the Results to part-2, Addition of two vectors Graphical Experimental Method Method Forces

( ) =-( )

Sources of Errors:

17

Error %

Show your calculations in this page.

Please clean up your worktable for the next class. 18


(F1=

N,

θ1= )

(F2=

19

N,

θ2=

)


(F1=

N,

θ1= )

(F2=

20

N,

θ2=

)


(F1=

N,

θ1= )

(F2=

21

N,

θ2=

)


(F1=

N,

θ1= )

(F2=

22

N,

θ2=

)


3

One Dimension Motion (constant acceleration)


Grade 23

3

One Dimension Motion (constant acceleration)

Purpose The purpose of this experiment is: 1. Studying the motion of body moves with constant acceleration in one dimension. 2. Determination of the acceleration of a body moves in straight line sliding down an inclined plane.

Apparatus G1 and G2 are the photo gates that are used to measure t X is the distance between the two gates The Expected value of acceleration is

a = g sin (θ) The ExperimentaL value of acceleration is

Figure ( 3-1) The apparatus for this experiment

Theory A cart of mass MO on an incline will roll down the incline as it is pulled by gravity. The force of gravity (MO g) is straight down. The component of that is parallel to the inclined surface is

MO g sin Using Newton's second law

1 So

2 24

The one-dimensional equation of motion of constant acceleration is

3 Where

x0 : the initial position v0 :the initial velocity x:distance travelled by the cart t : the time that is needed to move the distance x a : the acceleration To determine the acceleration experimentally, you will release the cart from rest and measure the time (t) for it to travel a certain distance (x), If x0 and v0 are zero then

4 The acceleration a can be calculated using the relation

5 Velocity after time t also can be calculated using the relation

6 The average velocity can be calculated using the relation

7 Figure (3-2) shows graphs of one dimension motion (constant acceleration)

Figure (3-2) graphs of one dimension motion (constant acceleration)

25

Procedure: 1. Set up the experiment as shown in figure 1 let the angle of inclined plane (θ= 15 o) 2. Choose a distance between the two gates (G1 & G 2) x=10cm. 3. Turn the pump on 4. Let the cart move with initial velocity (v0 =0), and record the time that the cart needed to move the distance x 5. Find the acceleration of the system using Eq. (5) 6. Repeat. steps 2 - 5 five times using different values of X 7. find v and vaverage 8. Tabulate your results in table (3-1)

Data Analysis Use results in table 1 to 1. Plot a graph of x versus t2. 2. Draw the best straight line fit of the data 3. Write the equation of the best line that describe the relation between x & t2 4. Determine the acceleration of the cart from the slope. 5. Determine the expected value of acceleration of the cart from equation 2.

Note: The relation between x & t2 is

So If you draw the best line fit, The slope will equal 7. Compare your result with the accepted value of a by calculating the percent error (%error) 8. What is expected value of t if x=70 cm? 9. What is the Expected value of the slope if θ =30o ? 10. Plot a graph of x versus t 11. Plot a graph of v versus t 12. Plot a graph of a versus t 13. . Compare your graphs with figure (3-2) 14. What is expected value of a if θ=60o?

26

M0" cart mass" =…………(kg) , X (m)

θ=

t (s)

Average t(s)

t2 (s2)

a(m/s2)

(m/s)

average

(m/s)

Table (3- 1): Data Table for One Dimension Motion experiment.

Slope = …………… Using slope

a =……………

Expected value of

a =……………………….

Error % =………………………………. The equation of the best line that describe the relation between x & t2

If x=70 cm then the expected value of t is …………… The expected value of a if θ=60o

What happened to the slope if M is doubled?

27

Figure (3-1):The best line that describe the relation between x & t2

28

Figure (3-2): a plot of X vs t

29

Figure (3-3): a plot of v vs t

30

Figure (3-4): a plot of a vs t

31

Figure (3-5):

32


4

Motion in two dimensions (Projectile Motion)


Grade

33

4

Motion in two dimensions (Projectile Motion)

Purpose The purpose of this experiment is to: 1. To predict the landing point when the projectile is fired from a height y at a zero angle 2. Determine an experimental value of the acceleration due to gravity g and comparing with the theoretical prediction.

Equipment: Small sphere, meter stick, photo gate tools, timer, plastic ruler, tape, small weight and carbon paper

Apparatus The apparatus for this experiment is shown in Figure (4-1)

Figure (4-1) The apparatus for this experiment Where: G1 and G2 are the photo gates that are used to measure t D is the distance between the two gates The magnitude of velocity of the projectile can be measured using The projectile is fired with ϴ = 0 Y is height where projectile is fired X is the horizontal distance

Theory Projectile motion is an example of motion with constant acceleration when air resistance is ignored. An object becomes a projectile at the very instant it is released (fired, kicked) and is influenced only by gravity. The x- and y-components of a projectile’s motion are independent, connected only by time of flight, t. Consider an object is launched at an angle θ= 0 , with a velocity = vo ( vox=vo, voy=0) from a height = y and hit the floor at a distance X . you can easily show that

34

Eq. (1) Procedure 1. Let a ball start roling From a height h Use a smooth groove that the ball can roll down. Find a starting point on the launcher that gives the ball a reasonable velocity. 2. Place 2 photogates G1 & G2 "D "cm apart on the path of the ball. This is the distance "D" in the diagram above. Record this distance. 3. Carefully measure the vertical distance "Y". Record this distance. 4. Find the approximate position where the projectile hits the floor. Tape a piece of scrap paper at this location, and put a piece of carbon paper face down over it to mark the landing spot of the projectile. 5. Launch the projectile several times.

a. For each launch, measure the time it takes the ball to roll the measured horizontal distance on the table" D", and record the rolling times (t) in the data table(4-1). Pay attention to units! By measuring

t

&

D

you can calculate the initial velocity of

projectile b. You should get a group of reasonably-close-together spots on the "target" paper. If the spots are wildly far apart, you need to adjust your launcher or launching technique to get more consistency. 6. Locate the point on the floor directly below the edge of the table top where the ball leaves the table. Measure the distance from this point to the center (average) of your landing positions. This is the range of the projectile ("x" in the diagram above). 7. Change your launcher so that your projectile is launched at a different speed and repeat.

Data Analysis Use results in table 1 to 1. Plot a graph of vo versus x. 2. Draw the best straight line fit of the data 3. Write the equation of the best line that describe the relation between vo &

x

4. Determine the acceleration due to gravity from the slope.

Note: the relation that connect vo & x is

If you draw the best line fit , The slope will equal 5. Compare your result with the accepted value of 9.8 m/s2 by calculating the percent error (%error) 6. What is expected value of vo if x=60 cm ? 7. What is the Expected value of the slope if y =40 cm ?

35

D=…….. (m) , y=………(m) t(s)

t(s)

θ=zero o

X (m)

(m/s)

Table (4-1): Data Table for Projectile Motion experiment

Slope = ……… g=…………… Error % = ………… The equation of the best line that describe the relation between

If X=60 cm the expected value of

o is ……………

What happened to the slope if M is doubled? Explain?

36

o & X

Prove the equation

Please clean up your worktable for the next class. 37

Figure (4-1): The best line that describe the relation between

38

o

& X


39

o

& X


5

Newton's Second Law


Grade 40

5

Newton's Second Law

Purpose 1- Confirm the validity of Newton’s 2nd Law by analyzing the motion of two objects (glider and hanging mass) on a horizontal air-track 2- To study the relation between the acceleration of a system and the total mass of the system when the force applied on the system is constant. 3- To study the relation between the acceleration of a system and the force applied on the system when the total mass of the system is constant. 4. Determine an experimental value of the acceleration due to gravity g, and compare the result with the accepted value g=9.8 m/s2

Apparatus The apparatus for this experiment is shown in Figure 1

Figure ( 5-1) Setup of Newton's Second Law experiment F is the magnitude of the force acting on the system G1 and G2 are the photo gates that are used to measure t X is the distance between the two gates The Expected value of acceleration is

Theory The notion of a system, that is a group of objects that are treated as one, is critical in physics. In situations where objects move together, it can be easier to study them as one system rather than as separate objects. In this lab, we will be examining the motion of two objects connected by a 41

string: a glider on a nearly frictionless air track and a mass hanging vertically from the other end of the string. Figure (5-1) In this experiment you will use Newton’s second law to understand the motion of this system. In particular, we are going to determine the acceleration of the system and use that information to determine the (unknown) mass of the glider, and determine the acceleration due to gravity Figure (5 -1) shows a system consists of three masses ( M" Hanging mass" and Mo" cart mass) and m is the mass added on the cart. We can easily proof that the acceleration of the system is given by

1 Where F=Mg, so if M and Mo remain constant and change m by adding masses on the cart. The force acting on the system F will remain constant but the acceleration will be changed.

Procedure Part 0ne: The force applied on the system is constant. 1. Set up the experiment as shown in figure 1 with M= 50 g, and the distance between the two gates (G1 & G 2) x=50cm 2. Turn the pump on "why?" 3. Let the cart move with initial velocity (v0 =0), and record the time that the cart needed to move the distance x 4. Find the acceleration of the system using relation

5. Add additional loads on the cart (m= 50 g) , let the cart move with initial velocity (v0 =0), and record the time that the cart needed to move the distance x, find the acceleration . 6. Repeat the procedure for m = 100g, 150g and 200 g. 7. Tabulate your results in table (6-1)

Data Analysis Use results in table (5-1) to 1. Plot a graph of m versus 1/a. 2. Draw the best straight line fit of the data 3. Write the equation of the best line that describe the relation between m & 1/a 4. Determine the acceleration due to gravity from the slope. 5. Determine the mass of the cart (M0) from the intercept.

Note: the relation between m & 1/a is

If you draw the best line fit , The slope equal (M+MO)

, and the y-intercept equal -

6. Compare your result with the accepted value of 9.8 m/s2 by calculating the percent error (%error) 7. What is expected value of a if m=700 g ? 8. What is the Expected value of the slope and intercept if MO =300 g and M= 90g?

42

Part two: The total mass of the system is constant 1. Set up the experiment as shown in figure 5- 1 with M= 10 g and m= 50 g. let the distance between the two gates ( G1 & G 2) x=50cm, calculate the value of F, F=Mg. 2. Turn the pump on " why?". 3. Let the cart move with initial velocity (v0 =0), and record the time that the cart needed to move the distance x 4. Find the acceleration of the system using relation

5. Transfer loads equal (10 g) from the mass on the cart to the hanging mass so that M and F increased but the total mass of the system remains constant.(M= 20 g, calculate F) 6. let the cart move with initial velocity (v0 =0), and record the time that the cart needed to move the distance x, find the acceleration of the system . 6. Repeat the procedure in step 5 and step 6 until M reaches 60 g. 7. Tabulate your results in table (5-2)

Data Analysis Use results in table (6-1) to 1. Plot a graph of F versus a. 2. Draw the best straight line fit of the data 3. Write the equation of the best line that describe the relation between F & a 4. Determine the acceleration due to gravity from the slope. 5. Determine the mass of the cart ( M0) from the slope.

Note: the relation that connect F & a is If you draw the best line fit, The slope equal 6. Compare your result with the accepted value of Mo by calculating the percent error (%error) 7. What is expected value of F if a=300 cm/s2 ?

43

Part 0ne: The force applied on the system is constant. M=………(kg) , F=………(N), M0(cart mass) =…………(kg), m (Kg)

t (s)

Average t(s)

a (m/s2)

x=…………(m) 1/a (s2/m)

0

Table (5-1): Data Table for Newton's Second Law experiment

Slope = ……………..

intercept=……………………

g=……………………

Mo=………………………

Error % = …………………………………………… The equation of the best line that describe the relation between m & 1/a

If m=700 gm the expected value of a is …………… What happened to the slope and intercept if both M0 and M are doubled? Explain?

44

Figure (5-1): The best line that describe the relation between m & 1/a

45

Part two: The total mass of the system is constant M=………(kg) , F=………(N), M0(cart mass) =…………(kg), M (g)

F(N)

t (s)

x=…………(m)

Average t(s)

10 20 30 40 50

Table (5-1): Data Table for Newton's Second Law experiment

Slope = …………….. Mo=……………………… Error % = …………………………………………… The equation of the best line that describe the relation between F & a

If a=300 cm/s2 the expected value of F is …………… What happened to the slope and intercept if both M0? Explain?

46

a (m/s2)

Figure (5-2): The best line that describe the relation between F & a

47


6

THE COEFFICIENT OF FRICTION


Grade 48

6

THE COEFFICIENT OF FRICTION

Purpose The goal of this experiment is to: 1. Measure the coefficient of static friction for contacting surfaces by measuring the angle of response for a block on an inclined plane. 2. To determine if, within experimental error, the coefficient of friction is independent of the mass of the block. 3. To measure the coefficient of kinetic friction for contacting surfaces by the constant speed method.

Apparatus Adjustable inclined plane apparatus, Wood block, Plastic block. .

Figure-1: Apparatus for experiment

Theory Friction can be defined as the resistance to motion between contacting surfaces. When one surface slides (or attempts to slide) over another surface, there is generally some friction between them, which produces a force that opposes the sliding motion. There are two kinds of friction, in terms of their effects: static friction, which keeps two surfaces “stuck together” (stationary with respect to each other), and kinetic friction, which opposes an ongoing sliding motion. When you want to push a heavy object, static friction is the force that you must overcome in order to get it moving. The magnitude of the static frictional force, satisfies Where

is the coefficient of static friction, a dimensionless constant that depends on the object

and the surface it is laying upon. N is the normal force between the contacting surfaces. 49

From this equation it is clear that the maximum force of static friction, , that can be exerted on an object by a surface is Once the applied force exceeds this threshold the object will begin to move and a kinetic friction force, exists. In general and so the object accelerates once it starts to move. The magnitude of the kinetic Figure(7-1): Plot of frictional force versus frictional force is given as follows applied force Where is the coefficient of kinetic friction and is approximately constant. Figure 1 is a plot of frictional force versus applied force Block on an Inclined Plane

• Static friction A common example of a static friction force is that of a stationary mass on an incline. Figure (7-2) depicts the free-body diagram of this case. A block is initially at rest on top of an adjustable inclined plane apparatus. The plane is initially horizontal; a hinge allows its angle to the horizontal to be adjusted between 0o(horizontal) and about 70o.

Figure (7-2): Free-body diagram of a mass on an incline The plane is slowly raised. At first, static friction keeps the block stuck to the incline. As the angle increases, the component of the gravitational force downward along the incline increases, and the upward static friction increases in step, so that the net force on the block remains zero. But the static friction can increase only to a certain size; when the angle gets big enough, the downward force exceeds the maximum static friction, and the block breaks loose and starts sliding downward. We call the angle at which this happens the angle of repose, The angle of repose is defined as the angle at which an object just starts to slide down an inclined plane.

50

What is the coefficient of static friction

between the block and the plane?

Figure (7-2) shows a picture of the situation and a free-body diagram (

).

The x-axis is parallel to the plane and pointing upward along it; the y-axis is perpendicular to the plane and pointing upward. Since the block does not move in the y-direction, the net force in that direction must be zero. Hence When θ= θr , the block begins to slide.

This is the point at which the downward force in the x-direction is equal to the maximum force of static friction. Hence Notice that θr depends only on the constant

.

In particular, notice that the weight of the block w does not appear in the equation. This suggests that the angle of repose of the block will not depend on its weight. It is this that you will be testing in this lab.

• Kinetic Friction It can also be shown that when an object slides down an incline at constant velocity, the coefficient of kinetic friction is given by: Where θk is the angle at which the block moves with constant velocity.

Procedure Part 1. Static friction 1. With the adjustable section horizontal, place the block near its upper end (farthest from the hinge). 2. Slowly and carefully raise the adjustable section, until the block just begins to slide. Read and record the angle of elevation of the adjustable section . Note: Please note "slowly and carefully" in the preceding paragraph. Remember that there are both static and kinetic friction, with . If the adjustable section is at an angle θ such that , then the block will not start sliding if it is at rest; but a slight nudge or vibration will start it moving, and it will then continue to slide. You need to avoid giving it that slight nudge, and this requires a certain amount of care. If you think that you have accidentally started the block moving at θ , then you should discard the measurement and take a new one.

51

3. Repeat the same procedure three times, recording the different measurements of

. Record

your results in Table (7-1). For consistent results, use the same face of the block each time, and start the block from the same place on the board each time. 4. Find the average of the trails and use it to find . 5. Add 500 g to block and repeat the previous procedure to find

.

6. Record your results in Table (7-1). 7. Repeat the previous procedure for another block (you can call it Block #2) and Record your results in Table (7-1).

Part 2. Kinetic friction 1. Slowly and carefully raise the adjustable section until the block slides at constant speed. Note the block will not move by itself but you need to tab it each time you raise the angle.

Note: With our apparatus, it is not possible to measure a constant speed, but it is possible to obtain a close approximation of constant speed. 2. Record the obtained angle

in Table-(7-2).

3. Repeat the same procedure three times, recording the different measurements of For consistent results, use the same face of the block each time, and start the block from the same place on the board each time. 4. Find the average of the trails and use it to find µk. 5. Record your results in Table (7-2). 6. Add 500 g to block and repeat the previous procedure to find µ

K

. Record your results in

Table(7-2). 7. Repeat the previous procedure for another block (you can call it Block #2) and Record your results in Table(7-2). Note: You should have to tap the block each time you increase the angle in order to get it start moving.

52

1. Coefficient of Static Friction

µs

Table (6-1): Data for computing the coefficient of static friction for Block #1 & Block #2.

M (kg)

(degree)

average

(N)

block #1 without weight block with+500gm

#1


#2

CONCLUSION:

2. Coefficient of Kinetic Friction

µk

Table (6-2): Data for computing the coefficient of Kinetic friction for Block #1 & Block #2.

M (kg)

(degree)


#1


#2

CONCLUSION:

53

average

(N)


7

Conservation of Energy


Grade 54

7

Conservation of Energy

PART ONE: (ROLLING OF A SLOID SPHERE) Purpose In this experiment we will attempt to 1- Confirm the validity of conservation of energy 2- Determine an experimental value of the acceleration due to gravity g and comparing the measured value the theoretical prediction. 3- Find the ratio between the translational and rotational Kinetic energy of rolling solid sphere.

Equipment: Small sphere, meter stick, photo gate tools, timer, string, plastic ruler, smooth groove


Figure (7-1) The apparatus for this experiment G1 and G2 are the photo gates that are used to measure t D is the distance between the two gates

Theory Figure (7-1) shows a sphere of mass M and Radius R starts rolling from height H with initial velocity = 0 . At this height the solid sphere will have potential energy U given by the equation

1 As the sphere goes down both linear and angular velocity increase because potential energy is transferred to Kinetic energy, When the sphere reaches the horizontal reference (H=0) all potential energy is transferred to translational and rotational Kinetic energy (K=KT+KR) The law of the Conservation of Energy state that 55

2

3 4 Where KTran: is linear Kinetic energy Krota: is the rotational Kinetic energy M: is the mass of the solid sphere I: is the moment of inertia of the sphere about an axis passes through center of mass : is the velocity of the sphere.

ω: is the angular velocity of the sphere Procedure 1. Set up the experiment as shown in figure (8-1) 2. Measure M the mass of the sphere, R the radius of the sphere, and the distance D between the two gates ( G1 & G 2) , and record them in table(8- 1). 3. Calculate the moment of inertia of the sphere about an axis passes through center of mass using the relation

4. Let the sphere start rolling with initial velocity (v0 =0) from height H=5 cm

5. calculate the potential energy of the sphere using the relation

6. record the time" t " that the sphere needed to move the distance D. 7. Find the linear velocity and angular velocity of the sphere using the relations

8. Calculate the linear Kinetic energy and the rotational Kinetic energy using the relations 56

7. Calculate the ratio between linear Kinetic energy and the rotational Kinetic energy.

8. Repeat the procedure for H = 10cm, 15 cm , 20 cm and , 25 cm . 9. Tabulate your results in table (8-1)

Data Analysis Use results in table 1 to 1. Plot a graph of

versus H.

2. Draw the best straight line fit of the data 3. Write the equation of the best line that describe the relation between

& H


Note: the relation that connect V2 & H is

If you draw the best line fit, The slope equal 5. Compare your result with the accepted value of 9.8 m/s2 by calculating the percent error (%error) 6. What is expected value of V if H=50 cm ? 7. What is the Expected value of the slope if M =300 g and r= 3 cm?

57

PART TWO : A MOVING CART Purpose 1- Confirm the validity of conservation of energy 2- Determine an experimental value of the acceleration due to gravity g and comparing the measured value the theoretical prediction.


Figure (7-2) The apparatus for this experiment

Theory Figure (7-2) shows a hanging mass of mass M at height H above the ground connected to a cart with mass Mo and Length L with a string , both M and Mo are at rest. With initial velocity = 0. At this height the hanging mass will have potential energy U given by the equation As the hanging mass goes down linear velocity increases because potential energy is transferred to Kinetic energy. When the hanging mass reaches the ground all potential energy is transferred to Kinetic energy The law of the Conservation of Energy state that

Procedure 1. Set up the experiment as shown in figure (7-2) with measure the values of and record them in table(7- 2).

Mo,M and L

3. Let the hanging mass start moving with initial velocity (v0 =0) from height

H=10 cm ,

calculate the potential energy of the hanging mass using the relation 58

4.. record the time" t " that the cart needed to move the distance D when the hanging mass hits the ground. 5.. Find the velocity of the cart and the velocity of hanging mass when the hanging mass hits the ground. using the relation 6. Calculate the Kinetic energy of the hanging mass and the cart and find the summation. 7. Repeat the procedure for H = 20cm, 30 cm , 40 cm and , 50 cm .

9. Tabulate your results in table (7-2)

Data Analysis Use results in table 1 to 1. Plot a graph of V2 versus H. 2. Draw the best straight line fit of the data 3. Write the equation of the best line that describe the relation between

& H


Note: the relation that connect

& H is

If you draw the best line fit, The slope equal 5. Compare your result with the accepted value of 9.8 m/s2 by calculating the percent error (%error) 6. What is expected value of V if H=50 cm ? 7. What is the Expected value of the slope and intercept if MO =300 g and M= 90g?

59

Part one : Rolling Sphere M=…………(kg) R=………….(m) , I=………………(kg.m2) H (m)

U

t (s)

ω

(s) (m/s)

(Joule)

2

(m/s)2

Krota

KTrans

KTotal

Ratio

(Joule)

(Joule)

(Joule)

KT/KR

Table (8-1): Data Table for Conservation of Energy experiment

Slope = …………….. g=…………………… Error % = ……………………………………………..

The equation of the best line that describe the relation between

If H=50 cm the expected value of

2

& H

is ……………

What happened to the slope if the radius of the sphere is doubled? …………………………………………………………………………………………… …………………………

60


61

& H

Part Two :

M=…………(kg) R=………….(m) , I=………………(kg.m2) H (m)

U

t (s)

(s)

(Joule)

KM (m/s)

(Joule)

(m/s)2

KMo

KTotal

(Joule)

(Joule)

Table (8-1): Data Table for Conservation of Energy experiment

Slope = …………….. g=…………………… Error % = ……………………………………………..

The equation of the best line that describe the relation between

If H=50 cm the expected value of

&H

is ……………

What happened to the slope if the radius of the sphere is doubled? …………………………………………………………………………………………… …………………………

62


63

& H


8

SIMPLE PENDULUM


Grade

64

8

SIMPLE PENDULUM

Purpose The purpose of this experiment is to use the apparatus for a simple pendulum to: 1. Investigate the dependence of the period T of a pendulum on the length L and the mass M of the bob. 2. Determine an experimental value of the acceleration due to gravity g.

Apparatus The apparatus for this experiment consists of a support stand with a string clamp, a small spherical ball with a string, a meter stick, and a timer. The apparatus is shown in Figure (8-1). The pendulum that we will use in this experiment consists of a small object, called a bob, attached to a string, which in turn is attached to a fixed point called the pivot point.

Figure(8-1): The apparatus of the Simple Pendulum experiment

Theory A simple pendulum consists of a bob suspended from a string whose weight is insignificant compared to the bob. When initially displaced, it will swing in a plane back and forth under the influence of gravity over its central (lowest) point. The motion of the pendulum is nearly periodic. Several of the bob’s parameters can be varied: the mass of the bob, length of the string, and amplitude of the motion

Figure(8-2): Schematic diagram of the Simple Pendulum and its periodic motion

65

Oscillations of the pendulum: One oscillation is the motion taken for the bob to go from position B (initial position) to A (mean position) to C (the other extreme position) and back to B (see Figure-2). Period of the pendulum: The time taken to complete one oscillation by the pendulum. This is denoted by T. The length of a pendulum: is the distance from the pivot point to the center of mass of the bob. The force that causes the pendulum to oscillate is the gravitational force (Figure-3). The period of the simple pendulum is given by Figure(8-3): Force components acting on the mass bob of a simple pendulum. Where

the length of the pendulum and

g is is the acceleration of gravity

Procedure Part one: 1. Set up the pendulum as in the Figure (8-1), and adjust the length L to about 20 cm. The length of the simple pendulum is the distance from the pivot point to the center of the ball. 2. Displace the bob from its equilibrium position by a small angle (