Newton's Second Law

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Newton's Second Law

F = ma (Force = mass x acceleration)

Aim.

To investigate the relationship between force, mass and acceleration,
as detailed in Newton's Second Law.

Hypothesis.

For the experiment method we are doing, I predict that as the mass on
the end of the string increases, increasing the force, the trolley(s)
will accelerate faster.

I also think that the more trolleys there are, the more force will be
required to pull them along at the same speed as previous, and
therefore the acceleration at any given weight, compared to results at
a time when the were less trolleys, will be slower.

This makes sense to me because if the force (weights pulling the
trolley), mass (trolley(s)) and acceleration (of the trolley(s)) are
all proportional, which they must be according to the equation of
Newton's 2nd Law, then if one is increased, at least one or even both
of the others are, or the equation wouldn't work.

Method.

We put a trolley on a desk, attached via a piece of string to a mass
going off the end of the desk, on a pulley. The pulley itself was held
in place by a vice, and was secured very tightly so that it wouldn't
move during the experiment.

We measured a fixed distance, using a ruler, from the end of the
pulley (i.e. as far as the trolley could travel, towards the end of
the desk') to the trolley, and marked it out. The length of this
distance was decided on how long we could make the string, before the
weights would hit the floor as the trolley approached the end of the
table. This gave us the maximum length allowed us by the size of the
desk we were using, which turned out to be a not-very-long 70 cm.

We took note of the weights on the end of the string, held it next to
the top of the pulley, and then released the trolley, which was pulled
along by the weights.

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Using a stopwatch, we timed how long it took for the trolley to travel
the aforementioned distance between its mark and the pulley.

We then repeated the experiment several times, but using either more
weights on the end of the string, or more trolleys, by stacking them
on top of each other, after every set of timings.

Apparatus.

[IMAGE]

Text Box: 1/ Weights on a hook. 2/ Pulley. 3/ Vice 4/ String. 5/ Trolley. 6/ Desk. 7/ 1-metre rule. 8/ Stopwatch.


Text Box:

6

1

7

8

Text Box:


Results.

Weight (N)

No. trolleys / Weights (g)

Distance (m)

Time 1 (secs)

Time 2 (secs)

Time 3 (secs)

Av. time (secs)

Time2

Acceleration (m/s2)

1

1 / 707.5

0.7

1.06

1.06

1.1

1.07

1.15

0.61

2

1 / 707.5

0.7

0.82

0.72

0.78

0.77

0.60

1.17

3

1 / 707.5

0.7

0.57

0.59

0.65

0.60

0.36

1.92

4

1 / 707.5

0.7

0.59

0.53

0.54

0.55

0.31

2.29

5

1 / 707.5

0.7

0.5

0.44

0.5

0.48

0.23

3.04

6

1 / 707.5

0.7

0.47

0.41

0.44

0.44

0.19

3.62

7

1 / 707.5

0.7

0.41

0.41

0.43

0.42

0.17

4.03

8

1 / 707.5

0.7

0.4

0.34

0.38

0.37

0.14

5.02

9

1 / 707.5

0.7

0.37

0.31

0.34

0.34

0.12

6.06

10

1 / 707.5

0.7

0.44

0.4

0.41

0.42

0.17

4.03

1

2 / 707.5 + 688.1

0.7

1.44

1.38

1.47

1.43

2.04

0.34

2

2 / 707.5 + 688.1

0.7

1.16

0.85

0.88

0.96

0.93

0.75

3

2 / 707.5 + 688.1

0.7

0.87

0.63

0.69

0.73

0.53

1.31

4

2 / 707.5 + 688.1

0.7

0.84

0.62

0.62

0.69

0.48

1.46

6

2 / 707.5 + 688.1

0.7

0.60

0.68

0.53

0.60

0.36

1.92

8

2 / 707.5 + 688.1

0.7

0.47

0.47

0.52

0.49

0.24

2.96

10

2 / 707.5 + 688.1

0.7

0.44

0.44

0.43

0.44

0.19

3.67

1

3 / 707.5 + 688.1 + 742.2

0.7

1.75

1.72

1.91

1.79

3.22

0.22

2

3 / 707.5 + 688.1 + 742.2

0.7

1.35

1.31

1.22

1.29

1.67

0.42

3

3 / 707.5 + 688.1 + 742.2

0.7

1.13

0.97

0.87

0.99

0.98

0.71

4

3 / 707.5 + 688.1 + 742.2

0.7

1.00

0.88

0.97

0.95

0.90

0.78

6

3 / 707.5 + 688.1 + 742.2

0.7

0.62

0.60

0.78

0.67

0.44

1.58

8

3 / 707.5 + 688.1 + 742.2

0.7

0.63

0.56

0.50

0.56

0.32

2.21


Analysis.

From simply looking at the results, I can see that the general trend
for each weight of trolleys, seems to be that the acceleration
gradually goes up, as the weight increases. This is a good start to
proceedings, but to analyse the data and see the real patterns
present, it is best to produce some scattergraphs. I will use this
form of data visual presentation because it makes it easy for me to
see correlations between results, and I can then draw useful
conclusions from this.

Text Box:

Correlation Test Result = 0.925604265

This strong positive correlation shows me that the force and
acceleration of the one trolley are closely related. The greater the
force, the faster the acceleration. This, so far, proves Newton's
Second Law, because when one factor on one side of the equation rises,
so must the other, so the whole equation stays in proportion.

Text Box: Indeed, they would be almost perfectly related if it weren't
for the fact that I seem to have an anomly when the weight gets to
10N, but more of that and other problems that occurred in this usually
unreliable experiment, in the conclusion.

Correlation Test Result = 0.992939517

This time, the correlation is very cose to perfect, again implying
that Newton's Second Law is correct, and that my hypothesis, that the
acceleration will increase as does the force, remains intact.

Although I haven't yet plotted a graph with more than one line so I
can compare the three trolleys, it seems to me that the trolleys'
acceleration is declining. The first trolley was going at an
acceleration of around 5 ½ m/s2 when the weight was at 10N, whereas
with the two trolleys here, they are only going at just over 3 m/s2.
Could this be proof that my second hypothesis is correct…?

Text Box:

Correlation Test Result = 0.989742708

Again, the acceleration has increased as the force pulling the
trolleys along has increased. The fact that I have had a very strong
positive correlation for each of the three trolleys proves Newton's
Second Law and justifies my first hypothesis. As well as this, I have
noticed that, at 10N, the three trolleys are only accelerating at just
over 2 m/s2. This proves my second hypothesis as well, because the
acceleration has gradually been slowing as the mass of the trolleys
has been increasing. This provides further proof for the correctness
of Newton's Second Law, because the Force has been the same over the
three phases of the experiment (with different trolley weights), but
the acceleration and the mass can't both go up at the same rate, at
the same time, without the force having to change. So, one of them
(the mass) increases,and the other one (acceleration) has to decrease.

This is highlighted in the following scatter graph, on which all three
lines are plotted:

Text Box:

These are all three scatter graphs represented on one set of axis (I
have removed the anomalous 10N value from tolley 1's data), and they
confirm my second hypothesis, that the acceleration will be slower, at
a given force, for the bigger the number of troleys.

Take an example. According to the trendlines:

When 1 trolley is being pulled along by 6N, it is accelerating at 3.6
m/s2

When 2 trolleys are being pulled along by 6N, they are accelerating at
2.2 m/s2

When 3 trolleys are being pulled along by 6N, they are accelerating at
1.6 m/s2

To conclude my analysis, both of my hypotheses have ben proved with
the help of scatter graphs and results that produced very strong
correlations, and Newton's Second Law (F=ma) has been proven to work.

Conclusion.

This experiment was never meant to work as well as it did for us.
However, even with these good results, they could have been improved
in many ways:

The string we were using was average, household string, and wasn't
quite suited to holding its length when on the receiving end of a 10N
weight. When we did the experiment for trolley 1, using that
particular weight, the string stretched at least 5 cm, meaning it hit
the ground before the trolley finished its journey, and lost a lot of
the acceleration it might have otherwise had. This accounted for the
anomaly which I had to exclude.

We could have used a stronger, man-made fibre string.

A severe restriction to the experiment was the height of the desk, and
therefore the length we could make our trolleys' journeys before the
weights hit the floor. Not only did this low height mean we couldn't
see if any patterns would come up after the trolley had been
travelling for quite a few seconds, but it put a lot of pressure on
our reaction times, which may not have always been as correct as they
could be. Reaction times were always going to be a key factor anyway,
because it leads to a likelihood of human error, and the fact that we
were sometimes dealing with less than half a second didn't help.

We could've used a taller desk, and perhaps had some magnetic sensor
that could tell when the trolley(s) had passed a particular point.

There is also a probability that some of the trolleys' energy was lost
as heat, due to friction between the wheels and the axles and between
the wheels and the slightly uneven, wooden desk. The wheels might have
seemed loose enough, but there was no way they could be 100% efficient
against the axles.

We could've used an air bed under the trolley, instead of wheels, to
remove the effects of friction.

Altogether though, I was happy with the outcome of the experiment, as
it made analysis fairly easy, and I think the method was good, if the
above problems could just be solved.


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