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VELOCITY AND ACCELERATION






1. PURPOSE

The quantitative study of motion is a key element of physics. The simplest motion to describe is the motion of an object traveling at a constant speed in a straight line. In this lab you will furnish your own constant speeds and utilize the Smart Pulley for acquiring and processing data.



3. THEORY

The concept of speed and know that the speed of an object is measured in units such as miles per hour, kilometers per hour, or meters per second. The speed is the ratio of the distance traveled to the time required for the travel. We define the average speed as the total distance, x traveled during a particular time divided by that time interval t;

Average speed = total distance traveled / time interval for interval =


The definitation deals only with the motion itself, in the same, other definitions in kinematics are restricted to properties of the motion only. If the average speed is the same for all of a trip, then the speed is constant.

In reality, motion is usually not restricted to one dimension, and we must take account of the direction as wheel as the speed of an object’s motion. The name for the quantity that describes both the direction and the speed of motion is velocity. Even though we are considering only one-dimensional motion, we must still take account of direction) for example, positive versus negative, or east versus west), so we will use the term velocity. Suppose a car is located at point x1 at a time t1, and at another point x2 at a later time t2. Then the car’s average velocity v over the time interval is


v = (final position – initial position)/(final time – initial time) =
The average velocity is the displacement divided by the time elapsed during that displacement. In general, a bar over a symbol (as in v) indicates the average value of that quantity, in this case the average velocity. Note that the average velocity can be either positive or negative. The difference between speed and velocity is more than just an algebraic sign; it involves the difference between the total distance traveled (for speed) and the net change in position (for velocity).

If the velocity of a moving body does not change with respect to time, the body’s motion is called “ uniform “. The instantaneous velocity of a moving particle at a particular time t is given by



(4.1)

where x is the displacement vector.

We defined the average velocity of an object as its change in position divided by the time elapsed, v=Δx/Δt. This tells s how the objects position changes with time. It is reasonable to define a quantity that indicates how the object’s velocity changes with time. We define the average acceleration, a, as the change in velocity divided by the time required for the change. The average acceleration can be written as
(4.2)
According to Newton’s first law, an object set in motion on a perfectly smooth, level, frictionless surface continues to move in a straight line with constant velocity. If the velocity of a moving object changes in time either in magnitude or direction, the object is said to be in accelerated motion. The instantaneous acceleration at time t is given by

(4.3)

According to Newton’s second law, when a force is applied to an object it experiences an acceleration which is proportional in magnitude to the applied force, in the direction of the force. This relation is expressed as



(4.4)

where m is the mass of the object.



3. EXPERIMENTAL PROCEDURE


Part A: Constant Velocity

  1. Set up the apparatus as show in Fig. 4.1, in an area where you can pull the thread 50-100 cm in a straight line. Connect the Smart Pulley to your Apple II.

  2. Insert The Smart Pulley software disk drive and start up the computer.

  3. The computer will ask you to specify how the Smart Pulley is connected. Ask your instructor for the correct response, select it, then press RETURN.

  4. When you have gotten to the main menu, select option M, the motion timer. In this mode, the computer will measure and record up to 200 time intervals as your pulley spins.

Hint: To avoid getting extraneous times in your data when using option M, make sure your set up is ready to go and the red LED on the Smart Pulley is off before you press RETURN.

Pull


Universal clamp

Eraser


Universal clamp

Figure 4.1: Equipment Setup


  1. Now press RETURN. Let one person pull the thread at a constant speed; another should press on the eraser to establish enough tension to turn the pulley. As the thread runs out, press RETURN to halt the timing process. RETURN.

  2. When the computer finishes its calculations, it will display the measured times. Press the space bar on the keyboard to scroll through the data. When you reach the bottom of the table, press RETURN to move to the next menu.

  3. At the next menu, choose option G to enter the grapping mode, then chooseA to tell the computer you are using the Smart Pulley to monitor a linear motion. When you get to the grapping menu, choose D to select a distance-time graph.

  4. In the next menu, choose G, the press the space bar so your graph will have a grid. Also press P followed by SPACE BAR so your graph will not have point protectors. Pressing RETURN starts the actual graphing routine.

  5. Examine the graph, then press RETURN. You will be shown a new menu. If your graph shows reasonably constant speed, press T to see the data.

  6. Now choose option A from the same menu, so you can alter the style of the graph. Choose a velocity-time graph by pushing V to display the velocity and time information. Record the first 25 velocities in your data table.


Part B: Acceleration

  1. Set up the apparatus as shown in Fig. 4.2, in an area where the cart move 1-2 meters in a straight line. Connect the Smart Pulley to your Apple II.

Universal clamp

Eraser


Move motion

Figure 4.2: Equipment Setup


  1. When you have gotten to the main menu, select option M, the motion timer. In this mode, the computer will measure and record up to 200 time intervals as your pulley spins.

Hint: To avoid getting extraneous times in your data when using option M, make sure your set up is ready to go and the red LED on the Smart Pulley is off before you press RETURN.

  1. Now press RETURN on the computer. Release the mass hanger which fall downward, pulling to cart across the table. Stop the timing just before the mass hanger reaches the floor by pressing RETURN.

  2. When the computer finishes its calculations, it will display the measured times. Press the space bar on the keyboard to scroll through the data. When you reach the bottom of the table, press RETURN to move to the next menu.

  3. At the next menu, choose option G to enter the grapping mode, then choose A to tell the computer you are using the Smart Pulley to monitor a linear motion. When you get to the grapping menu, choose D to select a distance-time graph.

  4. In the next menu, choose G, then press the space bar so your graph will have a grid. Also press P followed by SPACE BAR so your graph will not have point protectors. Pressing RETURN starts the actual graphing routine.

  5. Examine the graph, then press RETURN. You will be shown a new menu. If your graph shows reasonably constant speed, press T to see the data.

  6. Now choose option A from the same menu, so you can alter the style of the graph. Choose a velocity-time graph by pushing V to display the velocity and time information. Record the first 25 velocities in your data table.


4. DISCUSSIONS AND CONCLUSIONS

    1. Construct a graph showing, Distance (vertical axis), Time (horizontal axis). Construct a second graph showing Velocity (vertical axis) versus Time. Be prepared to discuss the two graphs.

    2. In your write-up, include a description of the motion, a description of the graphs that you obtained, and try to generalize on what the different shapes of graphs mean of the motion they describe.

    3. Sketch a curve of velocity versus time for the displacement-time curve. Sketch the acceleration-time curve also.

    4. Sketch graphs to represent the following assumptions: (a) A car driven fro 1 hour at a constant speed of 37 km/h, (b) A person runs as fast as possible to the corner mailbox and immediately runs back as fast as possible.

    5. Determine the average velocity and the average acceleration using the graphs.


5. QUESTIONS

  1. What are significant sources of error in this experiment?

  2. Theoretically what should be the shape of the graph of part A? Is it so? If not, what factors may have caused this deviation from the expected shape?

  3. Considering the time intervals to be errorless, calculate the percentage error in the velocity measured by you?

  4. If the maximum error in the time intervals is %10, what is the % error in the measured acceleration?

  5. In trying to determine an instantaneous velocity, what factors (timer accuracy, object being timed, type of motion) influence of the measurement? Discuss how each factor influences the result.

  6. Can you think of one or more ways to measure instantaneous velocity, or is an instantaneous velocity always a value that must be inferred from average velocity measurements?

  7. Can you think of physical phenomena involving the earth in which the earth cannot be treated as a particle?

  8. Each second a rabbit moves half the remaining distance from his nose to a head of lettuce. Does he ever get to the lettuce? What is the limiting value of his average velocity? Draw graphs showing his velocity and position as time increases.

  9. Average speed can mean the magnitude of the average velocity vector. Another meaning given to it is that average speed is the total length of path traveled divided by the elapsed time. Are these meanings different? If so, give an example.

  10. When the velocity is constant, does the average velocity over any time interval differ from the instantaneous velocity at any instant

  11. Can an object have an eastward velocity while experiencing a westward acceleration?

  12. Can the direction of the velocity of a body change when its acceleration is constant?

  13. Can a body be increasing in speed as its acceleration decreases? Explain.



GENERAL PHYSICS

PART A: MECHANICS

EXPERIMENT – 5

FREELY FALLING OBJECT





1. PURPOSE

The purpose of this laboratory is to determine the acceleration of gravity by timing the motion of a freely falling object.


2. THEORY

The most common example of motion with (nearly) constant acceleration is that of a body falling toward the earth. In the absence of air resistance we find that all bodies, regardless of their size, weight, or composition, fall with the same acceleration at the same point on the earth’s surface, and if the distance covered is not too great, the acceleration remains constant throughout the fall. This ideal motion, in which air resistance and the small change in acceleration with altitude are neglected, is called “free fall”. The acceleration of a freely falling body is called the acceleration due to gravity and denoted by the symbol . Near the earth’s surface its magnitude is approximately 9.8 m/sec2, which 980 cm/sec2, and it is directed down toward the center of the earth.

Up to now, the relationships between kinematics quantities such as velocity and acceleration were not dependent upon any property of nature, but rather on how they were defined. Here, for the first time, we have introduced a quantity, the acceleration of gravity, which reflects a property of nature. We cannot calculate the acceleration of gravity from just our knowledge of the kinematical relationships but rather it must be measured. The value we measure depends on the coordinate system and, hence, the units of measurement. But the fact that all things fall with the same acceleration (in the absence of air friction) is a consequence of natural law.

The acceleration of gravity near the earth’s surface is slightly different at different location on earth. The acceleration depends on latitude because of the earth’s rotation. It also depends on altitude. But for any given location, the acceleration there is the same for all objects.

The force of gravity at the same rate. Strictly speaking, such experiments must be conducted in a vacuum so that the force of air resistance does not affect the results. For relatively small, smooth bodies of considerable density, however, the error introduced by conducting such experiments in the atmosphere is quite small.

In any motion problem it should be apparent that three variables- distance, rate, and time- are involved. If the motion uniform, or if the concept of average velocity is used, the motion can be described by the simple equation

x=vt (5.1)

where x is distance traveled in time t and v is the average velocity for the time interval t. When motion is non-uniform, that is, where velocity is changing, acceleration is said to take place. If the acceleration is uniform, as from a constant force such as the force of gravity, the acceleration can be defined as the average rate of change of velocity and it is given by the following equation:


(5.2)
where v2-v1 represents the change in velocity which occurs in time t. If a body starts from rest (i.e.,v=0) and is uniformly accelerated by a constant force for a time interval t, the total distance it will travel is given by the equation
(5.3)
For the case of a body falling from a height h under the influence of the acceleration of gravity g, becomes
and v2-v2o =2gh. (5.4)

In this experiment, the “ picket fence” included with the Smart Pulley system has evenly spaced black bars on a piece of clear plastic. When dropped through the photo gate, the bars interrupt the light beam. By measuring the distance between bars, and using the time measurements of the Smart Pulley, the acceleration of the freely picket fence can be calculated.

Note: On using the ”Picket Fence”


  1. When performing free-fall experiments, place a soft pad under the experiment to cushion the fall of the “Picket Fence”, or make sure to catch the bar to keep it from breaking.

  2. For accurate results drop the “Picket Fence” through the Smart Pulley Photo gate vertically as shown in Fig. 5.1.

photogate

Release area

Figure 5.2: Picket Fence


Photogate

Figure 5.1: Picket Fence

To achieve vertical alignment of the “Picket Fence” hold it between your thumb and forefinger, centered at the top of the bar, before releasing (See Fig. 5.2).




3. EXPERIMENTAL PROCEDURE

  1. Set up the apparatus as in Fig. 5.3. Measure d, the distance between the leading edges of adjacent bars on the picket fence, as shown. Record d.

Δd
Figure 5.3: Equipment setup

  1. Connect the Smart Pulley to your computer. Make sure the proper connections have been made before going on. Insert the Smart Pulley software disk into your computer disk drive and start up the computer.

  2. The computer will ask you how the Smart Pulley is connected. Ask your instructor for the correct response, select it, then press RETURN.

  3. From the Main Menu, select option M, but do not press RETURN.

  4. Hold the picket fence in the gap between the arms of the photogate, as shown in Figure 3. Position the picket fence so that the photogate beam passes through a clear area, so the LED on top of the photogate is not lighted.

  5. Now press RETURN. Drop the picket fence, being sure to catch it before it hits the floor. Press RETURN again to halt the timing process of the computer.

  6. When the computer finishes its calculations, it will present you with a menu of data analysis options. Choose G to move to the graphing function, then choose C to tell the computer that you are monitoring the motion of the picket fence. When you get to the graphing menu, choose V which will give you a velocity-time graph.

  7. You will now be asked to specify the style of the graph you want. Select R,G and S. (Remember, you must use the space bar so that ON appears to the left of each selection). The letter S indicates that statistical data will be displayed along with the graph. At the top of the graph you will see three numbers. They are:

M= slope of the graph

B= y-direction

R= correlation coefficient (how close the graph is to a straight line)




  1. If your graph is a good straight line (as theory says it should be), record the slope of the graph, which is the acceleration, in Table 5.2. Its units are meter/sec2.

  2. When finished looking at the graph, press RETURN. You will now be given several choices. If you are pleased with the graph you obtained, you should press T to get a readout of the data from your experiment. Copy the velocities and times into Table 5.1 or follow instructions for printing the data out on a printer.

  3. Repeat the experiment at least 5 times. Select X to return to the Main Menu, then repeat steps 4-8. You need to record velocities and times for only one of your runs, but record the acceleration for each run.


4. DISCUSSIONS AND CONCLUSIONS

  1. Use your data (from one run) to construct a velocity (vertical axis) versus time graph.

  2. Average the acceleration from all of your runs.

  3. Calculate the slope of your velocity-time graph. Analyze how close the several values for the acceleration of gravity were to each other. Analyze how close your average value was to the standard value of 9.80 m/sec2.


5. QUESTIONS

  1. What can be the sources of errors in your results?

  2. Do you think that precise determinations of “” on an area might give some evidences for the underground resources at that location?

  3. Do you expect any dependence in the value of “” on latitude and altitude of the location where the experiment is performed?

Distance per interval= d: ...... (assumed to be 0.050 m)

  1. A ball thrown vertically upward rises to a maximum height and then falls to the ground. What are the ball’s velocity and acceleration at the instant it reaches its maximum height?

  2. A professor drops one lead sinker each second from a very high windows.(a) How far has the first sinker gone when the second one is dropped? (b) Does the distance between the first and second sinker remain constant? Explain your answer.

  3. The equation v2-v2o =2gh was used to calculate the acceleration. Under what conditions is this equation valid? Are those conditions met in this experiment?

  4. Could you use the relationship to determine the force acting between the earth and the moon? Explain.



Table 5.1. Free-Fall Velocities and Times

Interval

Velocity

Total Time

Interval

Velocity

Total Time

1

2

3



4

5

6



7

7

8



9

10








11

12

13



14

15

16



17

18

19



20

21










Table 5.2. Free-Fall Acceleration

Trial

Acceleration

1

2

3



4

5

6



7






GENERAL PHYSICS

PART A: MECHANICS

EXPERIMENT – 6

SPEED OF PROJECTILE






1. PURPOSE

To determine the initial velocity of a projectile directly, using the Smart Pulley Photogate, and also by examining the motion of the projectile.


2. THEORY

Projectile motion adds a new dimensions, literally, to experiments in linear acceleration. Once a projectile is in motion, its acceleration is constant and in one direction only-down. But unless the projectile is fired straight up or down, it will have an initial velocity with a component perpendicular to the direction of acceleration. This component of its velocity, since it is perpendicular to the applied force of gravity, remains uncharged. Projectile motion is therefore a superposition of two relatively simple types of motion: constant acceleration in one direction, and constant velocity in an orthogonal direction.

A projectile is defined as any object in motion through space or through the atmosphere which no longer has a force propelling it. Thrown balls, rifle bullets, abd falling bombs are examples of projectiles. Rockets and guided missiles are not projectiles while the propellant is burning, but become projectiles once the propelling force ceases to exist.

Consider an object, like a golf ball, projected horizontally (Fig.6.1). For the y (vertical) component of motion, the initial velocity is zero and the acceleration is that of gravity, giving


(6.1)
x

y

Figure 6.1: A photograph of a golf ball illuminated that given an initial horizontal velocity.


We use the minus sign because the acceleration of gravity is downward and we have chosen the upward direction to be positive in the figure. The projected objects starts ay y=0 and falls to negative values of y.

The horizontal component of motion has an initial velocity but is not accelerated, so
x = vot (6.1)

Solving this equation for t and inserting the value of t into the equation for y, we get


(6.2)
This equation has the same form as the equation for a parabola. In both cases the factor that is multiplied by x2 on the right-hand side is a constant for a particular problem. Thus, we conclude that projectile motion is parabolic.

Now consider an object which is at the origin of the coordinate system at time t=0, and which has the initial velocity v0 making an angle  with the positive x-axis. There is no acceleration along the x-axis therefore the horizontal component of v0 remains constant in time and is


vx=v0cos (6.3)
Since there is acceleration along the negative y-axis, ay=gsin, the y-component of v0 will change with time and is given by
vy=v0sin- ayt (6.4)
The two coordinates of the object’s position at any time t can be obtained by integrating equations (6.3) and (6.4).
x=(v0cos)t (6.5)

y=(v0sin)t-ayt2 (6.6)


The two equations above are called parametric equations of the path of motion.
3. EXPERIMENTAL PROCEDURE

  1. Set up the apparatus as shown in Fig. 6.2. Attach the photogate to your computer, insert the software disc, and turn on the computer.

Smart Pulley Photogate

Support Rod

Figure 6.2: Equipment Setup


  1. Place a piece of paper on the table, under the photogate. Remove the ramp, and use it to push the ball slowly through the photogate, as shown in Fig. 6.3. Determine the point at which the ball first tiggers the photogate timer-this is the first point at which the LED turns ON- and mark it on the paper. Then determine the point at which the ball last triggers the timer, and mark this point also. Measure the distance between as d. Replace the ramp as in Fig. 6.2.

  2. Choose option G, the gate function, from the Main menu. Now move the ball to a starting point somewhere on the ramp. Mark the starting position with a pencil so you will be able to repeat the run, starting the ball each time from the same point. Hold the ball at this position using a ruler or block of wood. Make sure the timer is not actively timing. Release the ball so that it moves along the ramp and through the photogate. Record the time in Table 6.1.

  3. Repeat the trial at least five times with the same starting point and average the times you measure. Divide your distance d by the average measured time to calculate v0, the velocity with which the sphere leaves the ramp.

  4. Use a plumb bob to determine the point directly below where the ball leaves the edge of the table. The distance from the floor to the top of the table at the point where the ball leaves should be measured and recorded ad dy.

  5. Now release the ball from your original starting spot, and note where it lands on the floor. This can be accurately determined by having the sphere hit a piece of carbon paper lying over a piece of plain paper. The impact will leave a clear mark for measuring purposes. Repeat this at least 5 times.

  6. Measure the average distance from the point directly below the ramp to the landing spot of your ball. Record this distance as dx.


Photogate

Mark with a pencil

LED comes ON




Photogate

Mark with a pencil

LED goes OFF





Figure 6.3: Measuring Δd.
4. DISCUSSIONS AND CONCLUSIONS

The horizontal velocity of the sphere can be determined using the equations for projectile motion and your measured values for dx and dy. Calculate v0 in this manner and compare it to the value you obtained using the photogate timer. Report the two values and the percentage difference.



5. QUESTIONS


  1. What are the possible sources of error in this experiment?

  2. For one of the trajectories, calculate the percentage error in v0, R and tR assume the percentage error in the measurement of time to be 10%.

  3. Is tR=2tH? Why?

  4. Is the displacement of the mass along the x-axis constant for each time interval? Why?

  5. For a given initial velocity, what should be the angle  to make R maximum?

  6. Will a ball dropped from rest reach the ground quicker than one launched from the same height but with an initial horizontal velocity?

  7. In projectile motion when air resistance is negligible, is it ever necessary to consider three-dimensional motion rather than two –dimension.

  8. At what point in its path does a projectile have its minimum speed? Its maximum?

Table 6.1: Measuring time


Trial

Time

Vertical height, dy=……
Average horizontal distance, dx=……
Horizontal velocity, v0=……
Percentage difference=……


1




2




3




4




5




Ave. time




V0 (Average)






GENERAL PHYSICS

PART A: MECHANICS

EXPERIMENT – 7

ACCELERATION OF A LABORATORY CART ( NEWTON’S SECOND LAW)





1. PURPOSE

In this experiment, you will investigate the changes that occur with different masses hanging from the thread and with different masses being moved by the resulting forces.


2. THEORY

The study of the causes of motion is called dynamics. The laws that govern the motion of an object were described by Newton in 1657- known as Newton’s laws. The laws are physical interms of force and mass. Newton’s first law describes what happens when the near force acting on an object is zero. In that case, the object either remains at rest or continues in motion with constant speed in a straight line. If the net force on an object is zero then the objects acceleration is zero. If  =0 then =0. And so the object remains at rest or at constant velocity. We used Newton’s 1st law in static,  =0.

Newton’s second law describes the change of motion that occurs when a nonzero net force acts on the object. The original translation of Newton’s second law was, The alternation of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. Elsewhere in the Principia Newton was clear that by “motion” he meant the product of the velocity and the mass. For the moment, it is sufficient to use Newton’s identification of mass as the “quantity of matter”. Then the second law:

The rate of change of momentum with time is proportional to the net applied force and is in the same direction:


=∑ (7.1)
Where is the net force – that is, the vector sum of all forces acting on a body- and the change in the momentum Δ(m) is in the direction of .

In the majority of real situations, the mass of an object does not change appreciably, so the change in momentum is just the mass times the change in velocity. Then



=m (7.2)

The rate of change in momentum of a body is proportional to the net force on the body. In equation from the 2nd law states.

=m (7.3)

This leads to the definition of force interms of the acceleration of mass.


3. EXPERIMENTAL PROCEDURE

  1. Set up the apparatus as shown in Fig.7.1 connect the Smart Pulley photogate to your computer, and start up the computer.

mw

Universal clamp

Eraser

mc



Figure 7.1: Equipment setup


  1. Place a total of about 200 grams of mass on top of the cart and record the total mass of cart plus added mass as mc in your Table 7.1. Place about 20 grams on the mass hanger. Including the mass of the mass hanger, record the total as mw.

  2. Move the cart backwards until the mass hanger almost touches the pulley. With the mass motionless, select M on the main menu.

  3. Now press RETURN on the computer. Release the mass hanger which will fall downward, pulling the cart across the table. Stop the timing just before the mass hanger reaches the floor by pressing RETURN.

  4. When the computer finishes converting the times, choose G which will move you to the graphing function. When you get to the graphing, select A which will interpret the timing as a linear motion. Choose V which will give you a velocity-time graph.

  5. The next choices give you the style of graph wanted. Choose S to indicate that statistical data will be displayed along with the graph and R to plot the regression or best fit line. To choose these, move the curser to the choices and push the SPACE BAR changing the “OFF” to an “ON” next to the choices. When completed, press RETURN to have the computer plot the graph.

  6. At the top of the graph you should see three numbers. They are:

M--- The slope of the graph

B --- The y-intercept



R --- The correlation coefficient (how close to a straight line it is)

  1. If the value for R is 1.00 or not less than 0.98, the graph is statistically a good straight line. This indicates that the acceleration is constant. Record the slope of the graph, the acceleration. Its units are in meter/sec2. Study the graph as long as you wish, and when finished, press RETURN. Press ESC until you move to the main menu to make another run.

  2. Change the applied force (due to mw) by moving masses from the cart to the hanger. This changes the force without changing the total mass. Record your new values in the data table. Repeat steps 3-8 at least five times using different values for mw.

  3. Now change the total mass, yet keep the net force the same as in one of your first five runs. Add mass to the cart, keeping the hanging mass the same. Record your new mass values, and the accelerations that you obtain. Repeat at least five times.


4. DISCUSSIONS AND CONCLUSIONS

    1. Calculate the net force acting on the cart for each trial that you performed. The net force is the tension in the string (if friction is neglected), which can be calculated as:

Fnet=(mwmc)/(mc+mw)

    1. Also calculate the total mass that was accelerated in each trial: (mc+mw).

    2. Graph the acceleration versus the applied force for cases having the same total mass. Graph the acceleration versus total mass for cases with the same applied force. What relationships exist between the graphed variables?

    3. Calculated the theoretical acceleration using Newton’s 2nd Law: Fnet=ma. Compare the actual acceleration with the theoretical acceleration, determining the percentage difference between the two.

    4. Discuss your results. In this experiment, you measured only the average acceleration of the object between the two photogates. Do you have reason to believe that your results also hold true for the instantaneous acceleration? Explain. What further experiments might help extend your results to include instantaneous acceleration?


5. QUESTIONS

    1. Analyze the sources of error in the performance of the experiment.

    2. If a loaded elevator weighs 3 tons, what force of tension in the hoisting cable (N) will be required it upward at a uniform rate of 6 m/s2?

    3. According to Newton’s laws, an external force is needed to stop a car when brakes are applied. Where is this force and what is its origin?

    4. A person on an upward-moving elevator is throwing darts at a target on the elevator wall. How should she aim the dart if the elevator has (a) constant velocity, (b) constant upward acceleration, (c) constant downward acceleration.

    5. When a moving car is slowed to a stop with its brakes, what is the direction of its acceleration vector? Describe the path of a ball dropped by a passenger during the time the car is showing down.

    6. A horizontal force acts on a mass that is free to move. Can it produce an acceleration if the force is less than the weight of that mass.



Table 7.1: Acceleration of a laboratory cart

Trial #

mc

mw

Experiment

Acceleration



Applied Force

Total mass

Theory Acceleration

%Difference

1

2

3



4

5

6



7

8

9



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GENERAL PHYSICS

PART A: MECHANICS

EXPERIMENT – 8

THE INCLINED PLANE AND SLIDING FRICTION





1. PURPOSE

To study the relationship between force, mass and angle and compare the mathematical solution with data taken directly from a scale model and to investigate some of the properties of sliding friction-the force that resists the sliding motion of two objects when they are already in motion


2. THEORY

Force is the cause of motion. Inertia is that property of mass which resists a change in motion. Newton’s first law of motion states that a body at rest or in motion will continue at rest or in motion at the same speed and in the same direction, unless acted upon by an unbalanced force. When a force acts on a body (Newton’s second law), the change in motion produced (i.e., acceleration) is produced to the force acting and inversely proportional to the mass of the body. This law may be stated as



= (8.1)
this equation may be written as

(8.2)
which is the mathematical statement of Newton’s second law. One important example of Newton’s second law is the expression fro an object’s weight. The weight of an object on earth is the gravitational force exerted on it by the earth. As a result, we know that near the earth’s surface, when we neglect air resistance, the acceleration is the same for all falling bodies. This constant acceleration is known as the acceleration of gravity, g, and has the standard value 9.807 m/s2. When an object is dropped near the earth’s surface, it is accelerated by the gravitational force (equal to its weight) with an acceleration g. Thus, by Newton’s second law, the weight w becomes

W = mg (8.3)


We see in this expression the relation between mass and weight: Weight is a force proportional to the mass of a body and g is the constant of proportionality.

When an object such as brick rests on the ground, the gravitational force continues to act on the brick, even though it is not accelerating. According to Newton’s second law, the net force on the brick at rest must be zero. There must be another force acting on the brick that opposes the gravitational force. This force is provided by the ground (Fig.8.1a). The force provided by the ground is perpendicular to the surface of contact and is known as the normal force. If the brick rests on an inclined surface, the gravitational force mg acting on the brick is still directed downward. The normal force N acts perpendicular to the surface, and since the surface is inclined, the normal force must be inclined (Fig.8.1b). That is, it is the vector sum of mg and N. The brick will then accelerate down the incline at a rate determined by this net force and the brick’s mass.



Normal Force,N

Weight of Brick, W

(a)


N

W

(b)


Figure 8.1: The normal force is always perpendicular to tha contact surface (a) The normal force is equal to the weight of the brick. (b) The normal force is less than the weight of the brick.

The surface of any material, no matter how smooth it may seem to the touch, is actually full of irregularities which oppose the sliding of any other body across it. This force of opposition as one surface slides across another is called friction. Friction is a force which always acts to oppose a change in motion.

Since friction is a reaction force, it follows from Newton’s third law that when there is no force tending to cause a relative motion between two surfaces, there is no force of friction. As shown in Fig.8.2, let A and B or two bodies, and let be a force which tends to cause A slide across B. Let N be the normal force pressing the two bodies together. (In this case N equals the weight of body A.) Note that Ff, the force of friction, acts at the boundary between the two surface and is opposite in direction to F.

If is zero, is zero. As F is increased, increases also, until the condition is reached where motion impends. At this instant, just as the block A would begin to slide, the force of friction reaches its maximum value is said to be the force of limiting friction or the force of static friction .

B

A

N







Figure 8.2: Diagram showing the factors involved in sliding friction
Once the body A begins to move, it will be found that the force of friction diminishes somewhat. This lowered value of frictional force for surfaces where sliding already exists, is called the force of kinetic friction .

For elementary studies of friction in the laboratory, the following statements are nearly, if not quite exactly, true for dry surfaces.



  1. The force of kinetic friction is almost independent of the area of contact, but is directly related to the force pressing the two surfaces together- the normal force.

  2. The force of kinetic friction depends on the nature of the surfaces. This constant of the sliding surfaces is called the coefficient of friction.

  3. The force of kinetic friction is almost independent of the relative velocity of the sliding surface for normal velocity ranges.

Thus, the coefficient of kinetic friction is defined as

k = force of kinetic friction / normal force

or

k = Fk/N (8.4)


3. EXPERIMENTAL PROCEDURE

Part A: Inclined Plane

  1. Weigh the rolling mass on the spring balance and record its mass (M) and weight(W).

  2. Now set up the equipment as shown in Fig. 8.3. Calculate the force that must be exerted by the string and by the ramp, W can be resolved into two components; Wx, a component directed along the surface of the ramp, and Wy, a component perpendicular to the surface of the ramp. The magnitudes of Wx and Wy are easily calculated: Wx=Wsin, and Wy=Wcos. Fx, the force provided by the string, must be equal and opposite to Wx. Fy, the force provided by the ramp, must be equal and opposite to Wy.

Wx

W



Wy



Figure 8.3: Equipment Setup




  1. Adjust the angle of inclination of the ramp to each of the values shown in Table 8.1, below. At each value, record the experimental value of Fx , as read on the Newton scale of the spring balance, in Table 8.1. For accurate results, the string must be parallel to the surface of the inclined plane.

  2. To measure the force of the rolling mass on the inclined plane, set up the equipment as shown in Fig.8.4. Vary the tilt of the inclined plane until the hanging mass and the rolling mass are in equilibrium.

  3. Record the mass (M’) and weight (W’) of the hanging mass, and the angle of inclination () of the inclined plane.

  4. Set up the spring balance and a pulley as shown in Fig.8.5. Adjust the pulley and the spring balance so the string pulls the bracket of the rolling mass at a 900 angle to the surface of the inclined plane. Pull the spring balance up until the force just barely lifts the rolling mass off the inclined plane.

  5. Read the value of Fy on the Newton scale of the spring balance.

M

W

M’





Figure 8.4: Normal Force, Equipment Setup
M

Fx

Fy

Figure 8.5: Measuring the Normal Force

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