Lab report 1 adjusted - Motion under variable acceleration PDF

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Motion under variable acceleration ...

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Experiment AM1.1: Motion under variable acceleration

Executive Summary: The main aims of this experiment was to find the coefficient of friction of the system, as well as to investigate the difference between the theoretical and experimental results. A dynamic cart was attached to a pulley system using a light, inextensible string. The time for the dynamic cart to travel a constant 0.9m was measured using a Leybold Zählgerät Counter. It was concluded that as the normal force of the cart increased, so did the frictional force.

Introduction: Sir Isaac Newton first presented his three laws of motion in the “Principia Mathematica Philosophiae Naturalis” in 1686 [1]. This experiment is based on his second law, which states that the acceleration of an object is equal to the force upon the object divided by the mass of the object. This equation is then applied to this experiment, via a pulley system, on a both the horizontal axis (dynamic cart) and the vertical axis (hanging mass). A hanging mass (hanging vertically) with masses 10g, 20g, 30g and 40g is attached via an inextensible string (which is resting on a pulley) to a dynamic cart (on a horizontal track) of masses 500g, 1000g and 1500g. The system is released and the time is taken, for the cart to travel 0.9m, via a Leybold Zählgerät Counter. The experimental value of acceleration is then calculated as well as the theoretical value, using Newton’s second law. The importance of this experiment is to prove Newton’s second law, any differences between the two values for acceleration are noted and explained. This experiment shows the conversion of gravitational potential energy (of the hanging mass) when the system is taut, to kinetic energy when the cart and hanging mass are moving. Aims and Objectives: 

Determine the Dynamic Coefficient of Friction. o Calculate the reaction force of the dynamic cart. o Use equation (Eq. 5) to calculate the Dynamic Coefficient of Friction. Investigate the differences between the theoretical values of acceleration and the experimental values of acceleration. o Measure the experimental values. o Calculate the theoretical values using Newton’s Second Law equation. o Calculate the difference between the two values as a percentage. o Consider possible reasons why there is a difference between the values. o



Methodology: The layout of the experiment consists of a dynamic cart at rest, on a level table. It is attached via an inextensible string, resting on a pulley, to mass hanging over the edge of the table. Initially, the dynamic cart is held at rest with only 10g (mass mH), attached to the hanging mass. The system is then released and the dynamic cart was allowed to accelerate over a distance of 0.9m (distance d). The time taken was recorded using a sensor and trigger which is inbuilt into the system, two repeats were done for every 10g mass added to the hanging mass. The average time (t) was then calculated by taking both of the recorded times and dividing them by two. After the times for 10g, 20g, 30g and 40g were taken, the average velocity (v) was calculated using the equation:

v=

d t

The experimental acceleration ( aexp . ) was then calculated using the formula:

aexperimental =

2d (Eq . 4) 2 t

This result was then compared to the theoretical acceleration (

atheoretical =

athe . ), obtained using:

mH g ( Eq .3) m Tot

where m Tot is the total of the two masses (hanging mass + dynamic cart mass) and g is acceleration due to gravity. After this the percentage difference between the values of acceleration was calculated by using:

Percentage difference=

(athe . −aexp .) × 100(Eq .6) athe .

Finally, the frictional force ( F K ) was obtained using the equation:

F K =m H g−m C aexp . ( Eq. 5) Once all these values were calculated for all 4 hanging masses (mentioned above), the experiment was repeated but this time adding 500g to the dynamic cart and then repeated a final time by adding another 500g on top of that. The dynamic cart had an initial mass of 500g therefore, the masses used in the calculations for the dynamic cart were 500g, 1kg and 1.5kg respectively. The same masses for the hanging mass were used for all three masses of the dynamic cart. Results: Mass (g)

10 20 30 40

Test 1 – Without added mass Theoretical Average Experimental Acceleratio Velocity Acceleration n 2 (m/s) (m/s ) (m/s2) 0.256 0.145 0.192

Differences (%)

Frictional Forces (N)

24.5

0.024

0.377

12.5

0.025

0.493

0.555

10.4

0.033

0.661

0.726

9.0

0.035

Time 1 (s)

Time 2 (s)

Average Time (s)

3.43 9 2.33 8 1.90 1 1.65 0

3.605

3.522

2.335

2.337

0.385

0.330

1.922

1.912

0.471

1.650

1.650

0.545

Table 1 - No mass added to the Dynamic Cart

Normal Forces (N) 4.904 9.807 14.711

10g Hanged Mass 0.024 0.046 

Calculated Friction Forces (N) 20g Hanged 30g Hanged Mass Mass 0.025 0.033 0.070 0.057 0.077 0.039

Table 2 - Calculated Frictional Forces against Normal Forces

40g Hanged Mass 0.035 0.048 0.077

Ca l c u l a te d f r i c tio n a l f o rc e a ga i n st n o r m a l F o rc e 0.08 Calculated frictional force (N)

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 4

6

8

10

12

14

16

Normal Forces (N) 10g Hanged Mass Linear (20g Hanged Mass) 40g Hanged Mass

Linear (10g Hanged Mass) 30g Hanged Mass Linear (40g Hanged Mass)

20g Hanged Mass Linear (30g Hanged Mass)

Figure 1 - Calculated Frictional Force against Normal Force

Discussion: Generally, as the normal force increases, so will the frictional force as shown in Figure 1. So in theory the gradients of the lines should be equal as the coefficient of friction has a constant value. However, this was not the case and this is due to experimental errors from -the system. These errors are also the reason for differences between the experimental and theoretical results. In this experiment we calculated the frictional force between the cart and the tracks, in order to make this experiment more accurate, the friction between the string and the wheel will need to be taken into account as well as the friction of the wheel on the axis it was rotating about. Other reasons for these experimental errors could be that the track was not completely level therefore either increasing or decreasing the time the cart takes to travel the 0.9m. There is also the possibility that the mass attached to the end of the string was swinging when the cart was released, this would increase the time taken for the cart to travel the given distance. Another source of error could be that the string was extensible which would have increased the time taken for the cart to pass the sensor. The timing of the experiment was done using an in-built sensor, therefore I don’t believe there was any error there. There was an error with one of the values for the 30g hanging mass, it is an anomaly and was not included in the calculation of the coefficient of friction. Conclusion: As the normal forces increase, the frictional forces increase. The coefficient of friction for the system was calculated to be 0.0048. The gradient of the lines should be the same for all four added masses as the coefficient of friction is the same throughout the system, however due to errors within the experiment they were different.

Appendix:

Mass (g)

10 csc2 0 30 40

Mass (g)

10 20 30 40

Time 1 (s)

Time 2 (s)

5.72 5 4.119

6.051

2.74 3 2.34 5

Test 2 – With 500g added mass Theoretical Average Average Experimental Acceleratio Time Velocity Acceleration n 2 (m/s ) (s) (m/s) (m/s2) 5.888 0.153 0.052 0.097

Differences (%)

Frictional Forces (N)

46.4

0.046

3.499

3.809

0.236

0.124

0.192

35.4

0.070

2.806

2.800

0.321

0.230

0.286

14.6

0.057

2.318

2.332

0.386

0.331

0.377

12.2

0.048

Differences (%)

Frictional Forces (N)

 38.8

 0.077

Time 1 (s)

Time 2 (s)

 4.69 0 3.53 5 2.96 4

 4.593

Test 3 – With 1kg added mass Theoretical Average Average Experimental Acceleratio Time Velocity Acceleration n (s) (m/s) (m/s2) (m/s2)    0.065 4.777 0.188 0.079 0.129

3.027

3.281

0.274

0.167

0.192

13.0

0.039

2.961

2.9625

0.304

0.205

0.255

19.6

0.077

Assuming the table top is level, then the force on the hanging mass (FH) is:

∑ F=ma → ∑ F H =mg−T =m H a H (Eq . 1) The force on the cart (FC) is:

∑ F C=T −F K =M C a(Eq . 2) where T is the tension in the string. When calculating the theoretical value of acceleration, we assume that there is no friction. Therefore:

m H g=a ( mH +m C ) ( where ( m H +m C ) =mTot ) atheoretical =

mH g (Eq .3) mTot

The theoretical value for acceleration can be calculated using:

1 2d d= a t 2 → a experimental= 2 (Eq . 4) 2 t where d is the maximum distance travelled by the cart (0.9m). Using the experimental acceleration, the frictional force can be calculated using:

Fnet =M C a →T −F K =m C aexp . → F K =T − mC a exp . (where T =m H g)

F K =m H g−m C aexp . ( Eq. 5) To calculate the percentage difference between the theoretical and experimental acceleration:

Percentage difference=

(athe .−aexp . ) × 100(Eq .6) athe .

References: [1 ]- Sir Isaac Newton (1686) Principia Mathematica Philosophiae Naturalis, England: Londini...