Physics Ia

Oscillations on a curved path Investigate factors which affect the motion of a sphere on a curved track

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Topic- Oscillations on a curved path Research Question: Investigate factors which affect the motion of a sphere on a curved track Personal Engagement From tranquil and unruffled desserts to raging storms with winds of up to 2400 kilometres an hour; from scorching-planetary surface temperatures of 450 degrees Celsius to planetary satellites covered in liquid methane in outer space – our solar system has it all. I was fascinated by our solar system from the moment I learnt about it. Being the curious learner that I was, I soon started to read more and learnt about a force that seemed rather mysterious at the time: Gravity – The force that held the earth and the other planets together and the reason why all the plants revolve around the sun. On researching, further, I found something fascinating but strange, the path of most planets wasn’t circular as one would intuitively assume, but in fact were elliptical. This is one of the questions that has “bugged” me a lot as a kid which is why when I started to think of a “physics IA” topic I realised that it would be an interesting opportunity to study more about the nature of the centripetal action on bodies with a non-circular path. Until now, in IB I learnt about and know that the centripetal force can be calculated using the equation,

where m is the mass of the object, v is the velocity of the object

and r is the radius of the path or distance from the centre. Now I will try to attempt to form a relation between the centripetal force and the mass, radius and the velocity for a non-circular path.

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Hypothesis I had decided to find the relationship between the time period of oscillation of a sphere on a curved and its radius, but after performing the experiment while I was researching about the possible relationship between them, I came across this website1 whose link is provided below where this formula was given, √

……..Equation-1

This helped me in the hypothesis, so the hypothesis is that the relationship between the period (T) of oscillation of a sphere along the curved track and the radius (r) of the sphere is ……..Equation-2 Variables Independent variable- Radius (r) of the spheres. 6 spheres of different radii are taken (the spheres used are of different materials like the metal ball, marble ball, golf ball, crazy ball and 2 plastic balls) and as the radii are different their masses are not the same) Dependent variable- Time-period (T) of oscillation of the spheres over the curved track. All the spheres were made to oscillate on the curved track from different heights on the curved track to check the total number of complete oscillation the spheres performed without considerable loss of energy, it was seen that on average the spheres could perform atleast 6 such Oscillation, so time taken for 6 oscillation was first found and then it was divided by 6 to find the time period. Controlled variables: 1. Radius (R) of curvature of the track-This was kept fixed randomly and will be found while analysis of data will be done. 2. Length of the curved track (AB, see Diagram-1) about which the spheres oscillate = 60 cm 1

https://www.physicsforums.com/threads/spherical-ball-rolling-on-a-concave-surface.683493/

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3. Height of the track above the table=3.1 cm 4. Height of the starting point of the sphere from where it is released=8.4 cm 5. Number of oscillations counted for each sphere=6 6. Draught-The experiment was performed in a closed lab, all fans and windows were kept closed so that wind does not affect the oscillation of the spheres 7. Rolling friction between the surface of the spheres and the curved track is assumed to be minimum as there is ideally a single contact point between the 2 surfaces Apparatus Curved track, Stand, boss and Clamp-2 sets, 6 spheres of different radii, stopwatch (0.01s), meter rule (0.1cm), vernier calliper (0.01cm), marker Diagram-1

Curved track B

A 3.1cm Schematic diagram

Actual Setup

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8.4cm

Procedure: 1. Measure the diameter of each sphere using a vernier calliper by holding the sphere between the jaws of the calliper. 2. Take 3 readings for the diameter by placing one of the spheres between the jaws of the calliper in 3 different positions so that any non-uniformity in the sphere is considered. 3. Take average diameter from the 3 readings and then half it to find the radius. 4. Repeat steps 1 to 3 for the remaining 5 spheres and record all data in a table. 5. Take the 2 boss, stand and clamps and use it to fix the curved track firmly on the top of the table as shown in diagram-1 6. Use a marker a mark on the track such that the Length of the curved track about which the sphere will oscillate is 60 cm. 7. Now, Take the sphere with radius r = cm. 8. Place it on the left hand side from the centre where a mark is made and leave it so that it performs Oscillatory motion. Measure time t for 6 oscillations. 9. Repeat step-8, 4 more times for the same sphere and then take average t. Find 10. Repeat steps 7 to 9 for the remaining 5 spheres with different radii. T2 11. Record all the data in a suitable table. Find

.

12. Plot a suitable graph to find the relationship between Time period and radius of the sphere.

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Raw data-1 Table-1: For radius ‘r’ of the spheres

MetalBall

1

Mass/g 35.2

Marble

2

30.3

Plasticball

3

28.3

Plasticball

4

30.5

golf ball

5

45.6

crazy ball

6

55.4

M.S.R /a/cm 1.0 1.0 1.0 1.5 1.6 1.6 2.8 2.8 2.7 3.2 3.2 3.2 4.1 4.1 4.1 4.9 4.9 4.9

V.S.R/ /cm

V.S.D/b 1 2 1 9 1 2 1 1 8 1 1 2 7 8 8 8 7 6

0.01 0.02 0.01 0.09 0.01 0.02 0.01 0.01 0.08 0.01 0.01 0.02 0.07 0.08 0.08 0.08 0.07 0.06

T.R /a+c/cm 1.01 1.02 1.01 1.59 1.61 1.62 2.81 2.81 2.78 3.21 3.21 3.22 4.17 4.18 4.18 4.98 4.97 4.96

Mean /D/cm 1.01

/cm 0.51

r/cm 0.01

1.61

0.80

0.01

2.80

1.40

0.01

3.21

1.61

0.01

4.18

2.09

0.01

4.97

2.49

0.01

Calculation for Table-1 The limit of reading of the vernier calliper used to measure the diameter of the sphere =0.01 cm ∴ The uncertainty

=

For sphere 1 i.e the metal ball, Absolute uncertainty in radius ∴ as the absolute uncertainty is lesser than the instrument uncertainty the radius of the metal ball is 0.51

0.01 cm (here the instrument uncertainty which is 0.005 is rounded off to the

second decimal place as the radius is in 2 decimal places). Similar calculation are done for the remaining spheres and recorded in Table-1 6|Page

Raw data Table-2

Sphere

Sr.No

Mass/g

radius /r/cm

MetalBall Marble Plasticball Plasticball golf ball crazy ball

1 2 3 4 5 6

35.2 30.3 28.3 30.5 45.6 55.4

0.51 0.80 1.40 1.61 2.09 2.49

Time taken for 6 Oscillations /s

/s

13.17 13.31 11.82 11.73 11.75 9.87

13.78 13.23 11.9 11.85 12.06 9.88

/s

/s

13.17 12.71 11.97 11.88 11.86 9.78

13.78 12.94 11.82 11.78 11.44 10.07

/s 13.48 12.98 11.85 11.87 11.28 9.46

Processed data for Table-2

Sphere

Sr. No

Mass /g

MetalBall Marble Plasticball Plasticball golf ball crazy ball

1 2 3 4 5 6

35.2 30.3 28.3 30.5 45.6 55.4

Sphere

Sr. No

T/s

MetalBall Marble Plasticball Plasticball golf ball crazy ball

1 2 3 4 5 6

0.05 0.05 0.01 0.01 0.06 0.05

Time taken for 6 Oscillations

radius /r/cm 0.51 0.80 1.40 1.61 2.09 2.49

/s 0.01 0.01 0.01 0.01 0.01 0.01

13.17 13.31 11.82 11.73 11.75 9.87

/s 13.78 13.23 11.90 11.85 12.06 9.88

/s 13.17 12.71 11.97 11.88 11.86 9.78

/s 13.78 12.94 11.82 11.78 11.44 10.07

/s 13.48 12.98 11.85 11.87 11.28 9.46

Avg t/s 13.48 13.03 11.87 11.82 11.68 9.81

t/s 0..31 0.30 0.08 0.07 0.39 0.31

2.25 2.17 1.98 1.97 1.95 1.64

/ 5.04 4.72 3.92 3.88 3.79 2.67

0.23 0.22 0.05 0.05 0.25 0.17

Sample Calculation for sphere-1

Since a stopwatch was used, to measure the time taken for 6 Oscillations the uncertainty in each value is

∴Uncertainty in t =

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… as the stop watch is an digital device.

Now, ∴Uncertainty in T =

∴ Uncertainty in All above calculation are performed similarly for other spheres and they are recorded in Processed data for Table-2 Analysis of data As the hypothesis made is that the square of the Time-period ( the radius (r) of the oscillating sphere, let’s plot a graph of

) should is proportional to v/s r to check the validity of

the hypothesis.

5.30

"T^2 v/s r" best fit line y = -1.0568x + 5.5715 R² = 0.9175

T2 v/s r 4.80

T^2 v/s r Worst fit line y = -1.3964x + 5.985 R² = 1

4.30

T2/s2 3.80

3.30

2.80

2.30 0.40

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0.90

1.40

r/cm 1.90

2.40

2.90

From the graph for the line of best fit we get,

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∴ Also, |

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∴

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∴ Now let’s consider the equation provided in the hypothesis √

Squaring both sides we get, ∴ Now, comparing above formula with

i.e.

The gradient,

And the y-intercept, As we have plotted a graph of

against r, and also found its gradient and y-intercept we

can use these values and compare it with the above equations.

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∴

From

∴ Conclusion and Evaluation The value of

(the correlation coefficient) of the graph is 0.915, though it is not very closer

to 1 but even at this value we can assume that there exists a linear relationship between and r.

which

is very large indicating large errors in the readings. If we compare the values of the gradient i.e. m from experimental value is

and

, the

and the theoretical value is

we see that there is a large discrepancy in the values which implies that the curved track is non-circular even though a linear relationship holds between

and r .

Further if is possible to determine the radius of curvature if the track provided it is circular without actually measuring, this is the strength of physics. As I pretty sure that the curved track is not circular, I would definitely like to go ahead and use further math and physics to

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find a proper equation of the form given in Equation-1for a non-circular track or in general any curved track. Safety Precaution 1. While I was performing the lab, the spheres had a tendency to roll down the table so I had kept them in a container to avoid this. 2. I ensured that the surface of the track is clean and without any dents so that it was almost frictionless. 3. The lab was performed in a closed environment with no disturbance whatsoever and the windows and fans were kept closed.

Limitations and Improvements 

I have used excel to plot the graph and as excel uses least square fitting and its plots graph without considering the error bars that I have put and I have studied that the best fit line should be drawn so that it should pass through all the error bars, this could have affected my results. I can improve this by plotting the graph manually or check some other graph plotting software which will help resolve this issue.



I have measured the time for 6 oscillations using a stopwatch so this could have led to uncertainty in time measurements due to response time, as an improvement I could have used light gates to measure time automatically which could have given precise readings thereby reducing the large difference between the experimental and theoretical value for the gradient and also as I had squared the time-period the uncertainty got further doubled.



The spheres used had different radii, but I had not used spheres made of the same material as I was not able to find them, I could have got spheres made of same material if I had some more time.

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

As the spheres were of different materials their masses were not constant, they were very different and I did not take into account the effect of mass on the time-period though I had measured the mass and recorded it in table-2. I could have also seen the effect of mass on the time period.

Bibliography https://www.physicsforums.com/threads/spherical-ball-rolling-on-a-concave-surface.683493/

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