k = spring constant of the spring, x = compressed spring length,
m = mass of metal ball, y = projection height,
g = gravity.
The factors of projection height, from the equation derived above,
Range α√y
The range should increases as y (projection height) increase, and we can predict that the range is directly proportional to the square root of projection height with a variation constant which equals to (√((2k)/(mg)))x.
Range α x
The range should be increases with x (compressed length of spring), and the range is directly proportional to the compressed length of spring with a variation constant which equals to √((2ky)/(mg)) should always hold true.
Experimental Setup Description
I have designed an experiment to study the effects of projection height and compressed length of spring on the range of a metal ball. The metal ball we used is made of steel, and its weight is 17.0 g ± 0.05g.
In this experiment, the projection height y and the compressed length x are varied separately and the projection range is determined. First of all, the spring constant of the spring is measured and the experiment is carried out on the elevating platform with a hole filling with sand and the cylindrical container with a movable wooden block connected to its base with a spring is set into place. The position of the container can be adjusted to certain height measuring by the meter ruler. A metal ball is pushed into the container and the compressed length x is measured by the 25cm-rule. The metal ball is then released and moves along the cylinder until it leaves it and undergoes projectile motion afterwards. The projection range can be obtained by measuring the distance between the point directly under the muzzle of the cylindrical container and the center of crater formed when the metal ball hits the sand. It is difficult to measure the compressed spring length using the whole cylinder and hence the cylidrical container is cut into half with the base remains entired. (See Diagram 2)However, the measuring error of the projection range using this method is large due to the uncertainty of the position of the centre of the crater. Therefore, the sand platform is replaced by a carbon paper(with carbon side up and a white paper taped on top of it) which makes a mark when the metal ball hits it. The range of the metal ball now is measured from the point directly under the muzzle of the container and the mark on the recording paper. The Diagram 3 and Diagram 4 illustrate the experimental setup used initially and finally:
Diagram2: Experiment apparatus
Diagram 3: The experimental setup initially
Diagram 3: The experimental setup finally
Experimental Methods
Procedure - Determining the spring constant of spring by putting different weight on it
- Measure the natural length of the spring by using the ruler without compressing or extending the spring.
- Place the slotted mass on the spring and measure the compression of the spring.
- Take away the metal weight from the spring and check whether the spring is deformed.
- Repeat step 2 to 3 by placing different slotted masses on the spring, say 20g,50g,100g,200g.
- Tabulate the data and plot a graph of reading of the spring balance against the extension using Mircsoft Excel.
Natural length of the spring = 15cm±0.1cm
From the excel, the slope of the curve = 0.050 m/N
Since F=kx (where F is the force applied to the spring, k is the spring constant and x is the compression of the spring) , the slope of the graph=1/k.
Therefore, spring constant k=1/0.050=20.0 N/m
Procedure - Determining range of metal ball by varying projection height
- Fix the container onto the stand clamp using steel clips. Make sure the stand clamp is fixed tightly using clamps and screws.
- Tape several recording paper (A4 paper and carbon paper) onto the table using cello tape.
- Fix a meter rule to the table using a stand clamp and steel clips. Place it behind the muzzle of simple spring gun. Make sure the meter rule is in contact with the table.
- Attach a 25cm-rule onto the upper wooden board of simple spring gun using cello tapes.
- Make sure the simple spring is parallel to the surface of the table using a spirit level.
- Mark a point onto a paper vertically below the muzzle of the spring gun. This will be the starting point of the projectile.
- Adjust distance height y between the bottom of metal ball and the table to 10cm using the meter rule as reference.
- Compress the spring and movable wooden block until they are compressed to 5cm of length using the 25cm-rule as reference.
- Place the metal ball in front of the movable wooden block. Make sure it is touching the movable wooden block.
- Release the movable wooden block to launch the metal ball.
- When the metal ball lands, mark the point of impact. It should be a black spot.
- Measure the distance between the black spot and the starting point using a zigzag ruler.
- Repeat steps 6 to 11 but vary the height each time. The heights are: 20cm, 30cm, 40cm, 50cm, 60cm, 70cm and 80cm.
- Tabulate the data and plot a graph of Change of range against the square root of the height.
Now having collected the data we can determine the square root range of the metal ball. The data are listed below along with its corresponding theoretical values. (gravity = 9.81ms-2). Note that the square root values are rounded to two decimal places:
Table 1:
Graph 1:
Procedure - Determining range of metal ball by varying compressed spring lengths
Similar to the procedure of determining range of metal ball by varying projection height, the steps 1 to 6 are the same as the previous procedure, we can begin to determine the range of metal ball by varying compressed spring length.
1-6. Repeat the steps in previous procedure
7. Compress the spring and movable wooden block until they are compressed by a length of 3cm using the 25cm-rule as reference.
8. Adjust distance height y between the bottom of metal ball and the table to 40cm using the meter rule as reference.
9. Place the metal ball in front of the movable wooden block. Make sure it is touching the movable wooden block.
10. Release the movable wooden block to launch the metal ball.
11. When the metal ball lands, mark the point of impact. It should be a black spot.
12. Record the distance between the black spot and the starting point using a zigzag ruler.
13. Repeat steps 7 to 11 but vary the compressed length spring length. The lengths are: 3cm, 4m, 5cm, 6cm, 7cm, 8cm, 9cm and 10cm.
14. Tabulate the data and plot a graph of Change of range against the compression of the spring..
The following data was collected after having carried out this experiment and the corresponding theoretical values are also stated:
Table 2:
Graph 2:
Graphing the experimental data and the theoretical data
Both the experimental range values and theoretical range values are plotted in the same graph in order for us to compare the differences between them.
From the Graph 1 and Graph 2, all experimental range values obtained are less than the theoretical ones although their differences vary. The experimental ranges almost resemble a straight line.
Interpretation of the graphs
It is clear from the above graphs that the data collected via experimental method does not match the theoretical hypothesis very accurately. The following provides an explanation:
- There is friction between the metal ball and the wooden board, when launching the metal ball, it rotates along the wooden board, so part of the elastic potential energy in the spring is converted into work done against friction and not all of the elastic potential energy in the spring is converted into linear kinetic energy of metal ball.
Moreover, as the metal ball rotates during the launch, part of the elastic potential energy stored in the spring is converted into the rotational kinetic energy of the metal ball.
Also, during the release of metal ball, some elastic potential energy is lost due to work done against air resistance as the metal ball has traveled a certain distance in this process.
The container may not be clamped tight enough to prevent movement during the launching of the metal ball, suggesting that there was recoil during the process. As a result, some of the elastic potential energy stored in the spring is converted into the linear kinetic energy of container.
However, in hypothesizing the range of metal ball using theoretical method, we assumed that all elastic potential energy stored in the spring before the launching of metal ball is converted into the linear kinetic energy of the metal ball upon release. Hence, the projection velocity of metal ball in the experiment is less than we predicted, resulting in a smaller horizontal range.
- After the metal ball is launched, it is not only subject to the influence of gravity, but also to the air resistance. Therefore, gravity is not the only factor causing the acceleration of metal ball, but also air resistance which is acting on the metal ball against its motion. As a result, the horizontal velocity of the metal ball decreases during its flight in air due to the horizontal component of the air resistant and the vertical acceleration of the metal ball decreases due to the vertical component of the air resistant on the metal. However, the resistant force is extremely small compared to the gravitational force and hence the time of flight of the metal ball is more or less constant. Consequentially the range of the metal ball is smaller than the theoretical one.
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In addition, when carrying out the experiment, some trials are observed to have side way motion. This may have been a result of air current.
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The graph of “metal ball range against compressed spring length” showed the difference between experimental range value and theoretical range value. The theoretical range value increase progressively as the compressed spring length increases/decrease. A greater compressed spring length results in a longer metal ball trajectory. Thus, the work done against friction would be greater as
Work done against friction = friction x distance traveled
Moreover, a longer range will lead to a larger rotational kinetic energy of the metal ball and work done against air resistance. Hence, more elastic potential energy stored in the spring will be converted into work done against friction, work done against air resistance and rotational kinetic energy of the metal ball. Therefore, if a smaller portion of elastic potential energy stored in the spring is converted into linear kinetic energy of the metal ball, the range obtained during the experiment will be more diverged to the theoretical values as the compressed spring length increases.
Comparing experimental results to theoretical hypothesis
According to the theoretical hypothesis, the range of metal ball should be directly proportional to the square root of the projection height. Thus, the quadrupling of the projection height would lead to twice the range of metal ball. Also, the range of the metal ball should be directly proportional to the compressed spring length. Thus, by doubling the compressed spring length, it would as double of the range of metal ball. Let's determine whether the hypothesis above applies to the experimental data.
Experimental results: Relationship between the projection height and the range of metal ball
- Projection height = 10.0cm,
corresponding range = 21.9cm
Projection height = 40.0cm,
corresponding range = 46.0cm ~ 2.10 x 21.9cm
- Projection height = 20.0cm,
corresponding range = 30.8cm
Projection height = 80.0cm,
corresponding range = 63.7cm ~ 2.07 x 30.8cm
From the comparison above, quadrupling the projection height is roughly twice the metal ball range. The trend of the growth of the experimental ranges as increasing the projection height is acceptable.
Experimental results: Relationship between the compressed spring length and the range of metal ball
- Compressed spring length = 3.0cm,
corresponding range = 27.3cm
Compressed spring length = 6.0cm,
corresponding range = 53.0cm ~ 1.94 x 27.3cm
- Compressed spring length = 5.0cm,
corresponding range = 43.0cm
Compressed spring length = 10.0cm,
corresponding range = 91.8cm ~ 2.13 x 43.0cm
From the comparison above, doubling the compressed spring length leads to nearly twice the range of the metal ball. The trend of the growth of the experimental ranges as increasing the compressed spring length is acceptable.
Evaluation and Conclusion
Through conducting the experiments in this investigation, there are several important observations I’d like to point out.
Firstly, there was a larger difference between the experimental values of range and the corresponding theoretical values of range for a greater spring length generally. The reason is the greater the distance travelled by the metal ball the greater the amount of work done against friction, work done against air resistance and rotational kinetic energy of metal ball.
Secondly, as expected, the projection range of metal ball increases with both increases of compressed spring length and the projection height, satisfying our predictions of such a trend to occur. There are differences between the experimental values of range and the corresponding theoretical values of range. The experimental values of range obtained are less than the corresponding theoretical values of range. This is mainly due to the air resistance.
Thirdly, the trajectory of the projected metal ball clearly resembled a parabolic path, which matched with the description and characteristic of a typical projectile motion. This effectively indicates that the ejected metal ball had underwent a parabolic path despite the effects of air resistance.
The projected metal ball had shown side way motion in some of the trials during the experiment although the fan and air conditioner are not in operation. This indicates that the metal ball is easily influenced by air current and this kind of random error is difficult to unavoidable.
The following are a series of improvements that can be made to the experimental methods:
- Using a spring with a larger spring constant such as 80N/cm or 100N/cm will allow a larger portion of elastic potential energy stored in the spring to be converted into the linear kinetic energy of the metal ball. Furthermore, lubricating the runway of the container can reduce the friction between the container and the metal ball. This will lead to more accurate results.
- It is known that the smaller size of object, the smaller the air resistance. Based on this principal, we can use a smaller sized metal ball in order to reduce the effects of air resistance on the metal ball. Previously mentioned in the interpretation of graphs, I noticed there was side way motion. Therefore, using a heavier metal ball will reduce the effect of air current on the metal ball. In addition, I should make sure that all fans and air conditioners are not in operation throughout the entire experimental procedure to reduce air current.
- A motion detector can be used to measure the projection velocity of the metal ball with a higher degree of accuracy and precision. Nevertheless, uncertainties in the measurements will still remain. However, we can still use the data to estimate how much elastic potential energy stored in the spring has been converted into linear kinetic energy of the metal ball. Then, we can roughly determine whether a greater displacement range would lead to an increase in work done against friction, work done against air resistance and rotational kinetic energy of the metal ball. Moreover, by using the measurements of the motion detector, we can calculate the approximate standard value of the range, and also estimate the effect of air resistance on the metal ball during flight. In addition, we can investigate the effects of air resistance on metal ball of different size and mass during flight.
- More accurate experimental results can be obtained if the experiment is performed in a vacuum since there will be no air resistance acting on the metal ball during flight. However, it requires a high level of technology and advanced apparatuses to help carry out the experiment which cannot be done in a school laboratory.
An investigation on how range is affected when launching the metal ball at different angles is worthy of scholarly inquisition. However, stating a hypothesis proves to be a complicating task as different projection angles leads to different amount of gain in gravitational potential energy during the trajectory of the metal ball. An experiment setup in which the metal ball does not gain gravitational potential energy during its trajectory can be used to investigate this experimental variable- different launch angles, simplifying the investigation. Also, to expand on this investigation, we could introduce another variable, which in this case is surface area and shape. For instance rather than using just metal balls, I could use cylinder shape and bullet shaped projectiles.
Appendix 1
By conservation of energy, during the ejection of the metal ball,
Gain in linear kinetic energy of the metal ball = loss in elastic potential energy of the spring
After the metal ball is projected
Consider the vertical motion of the metal ball,
By
Consider the horizontal motion of the metal ball,
By
Works and Program Cited:
Program:
Microsoft Office Excel 2007
Copyright © Mricosoft
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Microsoft Office Excel 2007 is the basic graphing software from
Microsoft Corporation. It combines graphing, TI calculator data import, curve fitting and other analytical tools into one easy-to-use program.