Physics Report #1

7/31/2019 Physics Report #1

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Lab #1

Measurements

Date of Investigation : 17 JULY 2012

Date of Submission : 23 JULY 2012

Report by : 1. Calvin Ng Kah Joon

2. Samuel Wong Chang Bing

Physics Report


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For an example, an acceleration of 22ms2

, 45 from the X-axes in the first quadrant will

register an arrow connecting a point 22 units away from the origin with vertical and horizontalcomponents of 2 units each as illustrated inFigure 1.3.

A vector diagram, as shown in Figure 1.4, can be used to determine the resultant of two vectors or

in this case break up a vector into its x andy components for analysis and easy manipulation.

Apart from drawing vector diagrams, arithmetic methods together with trigonometric treatments

and triangle rules can be used to work with vectors.

In the above example, if the resultant acceleration of the two vectors x and y, each of magnitude 2

which acts perpendicularly against one another is desired, the Pythagorean Theorem can be applied

to find the resultant acceleration. The Pythagorean Theorem states:

In any right triangle, the area of the square whose side is the hypotenuse is equal to the

sum of the areas of the squares whose sides are the two legs(the two sides that meet at a

right angle).[2]

or, it may be expressed in the Pythagorean Equation:

a2

+b2

=c2

where c represents the length of the hypotenuse while a and b represents the that of the two legs.[3]

On the other hand, if the magnitude of the acceleration and its constituent direction are given, thevertical and horizontal components can be found using the trigonometric equations:

y=asin (x)x=acos(x)

Reference:1. http://en.wikipedia.org/wiki/Derived_unit 2.http://en.wikipedia.org/wiki/Pythagorean_theorem 3. Judith D. Sally, Paul Sally (2007)."Chapter 3: Pythagorean triples".Roots to research: a vertical development of mathematical

problems. American Mathematical Society Bookstore. p.63.ISBN0-8218-4403-2.

Figure 1. 3 : Acceleration on XY plane

2

2

(2,2)

a

y

xFigure 1.4: Vector Diagram

y = 2 units

x = 2 units

22unitsa =

45
http://en.wikipedia.org/wiki/Derived_unithttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://books.google.com/books?id=nHxBw-WlECUC&pg=PA63http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0-8218-4403-2http://en.wikipedia.org/wiki/Pythagorean_theoremhttp://books.google.com/books?id=nHxBw-WlECUC&pg=PA63http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0-8218-4403-2http://en.wikipedia.org/wiki/Derived_unit


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Hypothesis:

The greater the degree of the angle of inclination, the greater the acceleration of the golf ball along

the plane, in they direction, and in thex direction.

Diagram:

Materials:

The materials used were a ring stand, 2 C-clamps, a golf ball, a retort stand with clamp, a plywood

board, a wood block, a meter rule, and a stopwatch.

Method:

An inclined plane was set up as shown in the diagram, ensuring that the ramp was secure and will

not fall or slide away. Alternatively, for gentler slope, the board was rested on the wooden block.

The dimensions of the ramp, length, height and base length were measured. The golf ball was rolled

down the ramp, and the time taken to reach the bottom of the ramp was determined using a

stopwatch. The experiment was repeated using various angles 10, 20, 30, 40, 50, and 60. Each

experiment with varying angles was repeated three times and the average time taken was calculated.

The acceleration of the golf ball down the ramp, and in the horizontal and in the vertical direction

were determined. The data was recorded in appropriate data tables. Graphs of acceleration against

angle of inclination were plotted and analysed.

Diagram 1: Arrangement of apparatus

Golf ball

C-clamps

Woodenblock

Woodenplane


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Graph 1.1: acceleration along the inclined plane against the angle of inclination

0 10 20 30 40 50 60 70 80 90

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

Graph of acceleration along the plane vs

( )

accelerationalongtheplane(m/s)


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Graph 1.2: acceleration along the horizontal direction against the angle of inclination

0 10 20 30 40 50 60 70 80 90

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

Graph of horizontal acceleration vs

( )

horizontalacc

eleration(m/s)


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Analysis:

From Table 1, it is clear that the smallest angle of inclination results in the lowest acceleration down

the plane. In contrast, the greatest angle, 60, results in the highest magnitude of acceleration in the

direction down the plane. Hence, we can infer that the acceleration down the plane is dependant on

the angle of inclination.

Also, the gradient ofGraph 1.1 decreases marginally to the right of the graph. This subtle change in

trend of the graph should be caused by the increasing angle of inclination .

Graph 1.2 is a peculiar case, for a clear decrease in magnitude of acceleration is seen towards the

end of the graph. The first thought of the careless observer will surely be 'bad data' and the urge to

discard this corrupted inconsistency will be great. But that was not the case. By comparing the data

with the expected theoretical value, we notice that it follows the trend of the theoretical value. It

appears that the behaviour of this graph is also dependant on . Well, no surprises there.

Moving to Graph 1.3, it is observed that the graph of the vertical acceleration against angle of

inclination has a sigmoid shaped curve. In the first half of the graph, there is a gradual increase inthe rate at which acceleration becomes greater. However, after approximately 45, the slope of the

graph starts to decrease.

Careful observation of the data in Table 1 reveals an abashment which no one would want to see in

the collected data. Further investigation is required to identify the nature of this error, be it a

systematic, or is it just another random happening. Either way, the occurrence of the absurd value of

10.37 ms-2 for an acceleration due to gravity on a planet which gravitational acceleration is no more

than 9.81ms-2 must be addressed.

In order to make further inference and deduction, a comparison between the actual theoretical

values and the data yielded from the experiment is needed. What follow in the next page are graphs

of theoretical equations and with the associated experimental values plotted together for easy

comparison.


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The experimental data appears to be

positively biased. All the data obtain

from the lab seems to be greater than the

theoretical value except for the first

point. Also, the absolute differencebetween experimental data and

theoretical value increases to the right.

Similar to the graph above, the

experimental data follows the general

trend of the theoretical graph although

there is a biased deviation in value. What

caused this apparent systematic error?

That is the question.

The experimental data obeys the general

trend of sigmoid shaped graph of the

theoretical equation. However, there isan increasing difference in absolute value

between the two plots. It seems

reasonable to infer that this upward trend

is caused or partially caused by the

increasing angle of inclination.

*Note that in the above theoretical graphs, frictional force is not taken into account.

Graph 2.3: horizontal acceleration against

Graph 2.1: acceleration down the plane against

Graph 2.2: vertical acceleration against


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Discussion:

Before anything can be said about the experiment, it is necessary to understand the kinematics and

mechanics involved. Due to the gravitational pull of the planet, our object is set to accelerate

downwards perpendicular to the surface of the Earth. This acceleration is independent of the mass

and its magnitude is approximately 9.8 ms -1. The following diagrams(which aim to break down this

vector into its components which are more useful for our cause) illustrate the mechanics involved:

Diagram 2.1 illustrates the weight of the

golf ball acting downwards in the vertical

direction. However, due to the inclined

plane, it is clearer to break up this force

into two distinct components which are

perpendicular to the plane and parallel to

the plane.

After the weight is broken down into its

two components, we can remove it from

the free body diagram, as shown in

Diagram 2.2. The perpendicular

component of the weight results in a

normal reaction between the golf ball and

the impenetrable inclined plane. A normal

force which is equal in magnitude but in a

different direction is formed. These two

forces cancels out each other, as

illustrated inDiagram 2.3.

In Diagram 2.3, all that is left is one of

the components of the weight of the ballwhich is parallel to the plane. This

component is the sole force responsible

for the acceleration of the ball. As

illustrated, it is further broken up to its x

and y components to determine the

vertical and horizontal acceleration.

Diagram 2.2

mg sin()

mg cos()

FN

Diagram 2.3

mg sin()

x

y

Diagram 2.1

mg

mg sin()

mg cos()


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With this, we arrived at a simple model to build our discussion on. The acceleration down the plane

is governed completely by the equation:

a =P gsin ()and from here, we get the equations for thex andy components, which are:

a =X gsin ()cos()and,

a =Y gsin ()sin ()

These equations for acceleration were used in generating the theoretical graphs inAnalysis in which

the theoretical values and experimental data were compared.

The values of the accelerations calculated in Table 1 were computed using a derived form of the

basic kinematics equations which is:

a=2s

t2

The experimental data was quite consistent with the theoretical values with a minor deviation that

appears to be positively biased.

The systematic error shown must had occur during the taking of measurements and may be due to

the method in which the time taken by the golf ball to travel down the inclined plane was recorded.

During the experiment, the first person releases the golf ball on the count of three while the partner

starts the stopwatch upon hearing the first person saying three. This method was agreed upon at

the start of the experiment because that arrangement seems more convenient then. However, this

might be the cause of the systematic error which appeared in the data. The proposed explanation is

that the person with the stopwatch did not start the stop watch at the same instant the first person let

go of the golf ball. This delay is due to the human reaction time. Therefore, the recorded time will

be shorter than the actual value thus yielding a higher acceleration. This condition worsens at higher

angles of inclination because while the human reaction time remains relatively constant, the actual

time taken becomes shorter, the percentage deviation increases.

Conclusion:

The acceleration of the golf ball increases as the angle of inclination of the wooden plane increases.

The vertical acceleration also increases when the angle of inclination increases. However, the

horizontal acceleration of the golf ball increases until about 45 and then decreases.

The hypothesis is accepted.

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