Hooke lab - lab write-up PDF

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1 The Hooke’s Law and SHM SPH 4U Tuesday, October 31, 2017 Introduction There are many forms of energy such as gravitational potential energy and kinetic energy, two forms that have been discussed thoroughly throughout previous lessons. Another type of energy is elastic potential energy which refers to the energy of springs or other elastic objects. To understand the energy that a spring can contribute, different components of the spring must first be studied such as the force constant. Different springs have different force constants which determines how much force is required to stretch or compress a spring by a certain amount. Much like a coefficient of friction, a force constant is unique to different springs as it depends on the type of material, the size of the spring (length, diameter of wire, diameter of spring), and the shape of the spring. The force applied to the spring, the amount of stretch of the spring, and the force constant are all related to one another through Hooke’s law: F =kx . When a spring is stretched, the increased length or extension of the spring is directly proportional to the force applied to it. However, it is important to note that if the force applied by the spring is being sought, it would be proportional and opposite to the force applied to the spring and hence the formula, F =-kx , would be used. If the relation between the force applied and the stretch of the spring was graphed, it would be a linear line much like the one seen in Fig.1. Now that the basic components of a spring are clear, finding the actual energy the spring contributes is easy. The elastic potential energy can also be found as the work the spring has done. To find the work that the spring has done, we would just simply find the area under the

 x  would be the formula line as, by definition, W=FΔd . By consulting the graph above, W=½ F Max  ” can be used to find the area. To generalize this formula for any amount of work, “F Max

2 substituted with “kx” according to Hooke’s law. With this substitution, the final formula for the work done by the spring, or the elastic potential energy is: E E =½ kx2 .  Moving on to springs in vibration or in other words, simple harmonic motion, the energy of any vibrating object can be found at different points of motion. Assuming that there is no dampening of the amplitude of the vibration, the system conserves energy with no loss. Ignoring gravitational energy, when the vibration of the spring is completely to one end, meaning that the stretch of the spring equals the amplitude of the motion, all the energy in the system is elastic kA 2 , where “A” stands for amplitude, is the equation that can be potential energy. Thus, E E =½  used to find the total elastic energy in simple harmonic motion. This energy is constantly converted to kinetic energy and back as the spring moves. Due to this direct conversion, kinetic  2 k(A 2 -x  ). This equation shows that at the point where energy can easily be found by using E k =½ 

the springs is not stretched at all and “x ” equals zero, all the elastic energy has become kinetic energy and the object on the spring is at maximum velocity. If the vibration is up and down, potential gravitational energy also contributes to the total energy of the system. Furthermore, the period and frequency of the vibrating spring can also be used to find components of the simple harmonic motion through the equations: T = 1f =2 𝝿

√

m k

.

By understanding the different features of springs, simple harmonic motion, and using the derived formulas, the behaviour of springs can be studied and understood. The effect of different values of applied forces and the differences in the period of a vibrating spring can be compared to see how the spring reacts in each situation due to its value of force constant. By understanding how the force constant of a spring affects its behaviour and total elastic energy, springs can be manipulated to benefit many fields of work and studies. Purpose Determination of force constant for extension springs by static and dynamic methods and verification of the Hooke’s law. Hypothesis I believe that the force constant for a spring should decrease the longer the spring is. This is because, with three springs made of the same material, the longest one should have the most give. If identical weights are put on each spring, the longest one should have the most stretch

3 while the shortest has the least. This can also be seen in the equation F =kx . If “F ” remains constant, then “k ” and “x ” are proportional in that if “k ” decreases, “x ” must increase. The longest spring can manage the longest stretch, hence it should have the lowest force constant. Additionally, due to Hooke’s Law, the applied force and the stretch of the spring have a linear relationship. Hence, the graphs produced from these variables should be linear as well. Observations Part A. Static Method (3.1 and 3.2) Spring

Short

Medium

Long

Initial Length L0 (m) 0.020

0.049

0.057

Mass suspended (kg)

Applied Force F=mg (N)

Final Length L1 (m)

Stretch L1-L0 (m)

0.020

0.196

0.030

0.010

0.050

0.491

0.061

0.041

0.070

0.687

0.082

0.062

0.100

0.981

0.116

0.096

0.120

1.18

0.134

0.114

0.020

0.196

0.090

0.041

0.050

0.491

0.152

0.103

0.070

0.687

0.194

0.145

0.100

0.981

0.259

0.210

0.120

1.18

0.301

0.252

0.020

0.196

0.113

0.056

0.050

0.491

0.208

0.151

0.070

0.687

0.266

0.209

0.100

0.981

0.364

0.307

0.120

1.18

0.427

0.370

Final Length

Stretch L1-L0

Part A. Static Method (3.3) Spring

Initial Length

Mass

Applied

4 L0 (m) Long +

suspended (kg)

0.155

Medium

Force F=mg (N)

L1 (m)

(m)

0.020

0.196

0.256

0.101

0.050

0.491

0.414

0.259

0.100

0.981

0.678

0.523

Part B. Dynamic Method (4.1) Spring

Initial Length L0 (m)

Mass suspended (kg)

Time of 10 cycles, t1 (s)

Time of 10 cycles, t2 (s)

Time of 10 cycles, t3 (s)

Long

0.057

0.150

14.03

13.86

14.00

1.396

0.170

14.30

14.33

14.28

1.430

0.200

15.72

16.02

15.53

1.576

0.100

9.32

9.22

9.35

0.930

0.120

9.85

10.07

10.18

1.003

0.150

10.89

10.82

10.91

1.087

Medium

0.049

Average Period, T (s)

5 Analysis

This graph shows the relation between the force applied to a spring and the stretch of the spring. Multiple tests on four different springs were conducted and lines of best fit were found for each set of data. The slope of each line represents the force constant for each spring as the slope, be definition, is N/m which is the value of k. The blue line is the small spring, yielding a force constant of 9.87 N/m. The red line is the medium spring with a force constant of 4.67 N/m and the yellow line is the long spring with a value of 3.17 N/m as its force constant. The green line is a combination of both the medium and long spring, showing a force constant of 1.87 N/m. Clearly, the longer the spring got, the lower the force constant became. For the fourth test in which the medium and long spring were connected, the resultant force constant should be related to the force constant of each individual spring. To find this relation, the equation, k =

F x

, should be looked at: kTotal  =

F x total

 = kTotal

F x1+x2

6 (x1 =

F k1

F and x2 = k2 )

k Total  =

F F /k1 +F /k2

k Total  =

(k1) (k2) k1 + k2

Thus, the equation above is what the total force constant should equal in terms of k1 and k2. If the variables are plugged in, where k1 equals the force constant of the medium spring, 4.67 N/m, and k2 is the long spring, 3.17 N/m, kTotal should be: k Total  = k Total  =

(k1) (k2) k1 + k2

(4.67) (3.17) 4.67 + 3.17

  =1.89 N/m k Total The calculated value of kTotal is 1.89 N/m while the experimental value on the graph was 1.87 N/m. Since the two values are very similar, the derived equation should be correct and the experimental data has proven to have very minimal errors. In the dynamic method where ten cycles were measured for each spring, the average period was found for each calculation. Using the equation for the period of simple harmonic motion, T =2 𝛑 Spring

√

m k

, the force constant can be found: Mass Suspended (kg)

Average Period, T (s)

Force Constant (N/m)  2 k = 4m𝛑 2 /T

Average Force Constant (N/m)

Short

N/A

N/A

N/A

N/A

Long

0.150

1.396

3.04

3.17

0.170

1.430

3.28

0.200

1.576

3.18

0.100

0.930

4.56

0.120

1.003

4.71

0.150

1.087

5.01

Medium

4.76

7 Through this method and the calculations in the chart above, the force constant of the medium spring came to be 4.76 N/m, while the force constant in the long spring came to be 3.17 N/m. There are no calculations for the short spring as there were no tests conducted for it. The two methods used to obtain the force constant of each spring was the static method (stretching of the spring) and the dynamic method (vibrations of the spring). Through the static method, the short spring had a force constant of 9.87 N/m. Since there is no other test to compare this value to, it is assumed that this force constant is accurate if the static method deems to be accurate overall. For the medium spring, the static method resulted in a force constant of 4.67 N/m while the dynamic method gave a value of 4.76 N/m. Both these values are extremely close to one another, with only a deviation of 0.09 N/m. Due to this small difference, both methods seem to be agreeable and reliable. For the long spring, both the static and dynamic method gave a result of 3.17 N/m. Since the two methods are identical in their answer, they have proven to be dependable ways to find the force constant of a spring. Based on these answers, there is no better method that should be used over the other as the results are quite similar anyways. Results In the chart below are the force constant values of each spring obtained through both the static and dynamic method: Spring

Static Method Force Constant (N/m)

Dynamic Method Force Constant (N/m)

Short

9.87

N/A

Medium

4.67

4.76

Long

3.17

3.17

Discussion The force constant of each spring decreased as the spring got longer. This agreed with the hypothesis as the shortest spring was estimated to have the highest force constant and the longest spring the lowest force constant. Additionally, the graphs turned out to be linear, showing a linear relationship between the force applied on the spring and the stretch of it. This confirmed relationship ensured the reliability of Hooke’s law, F =kx . The resultant force constant from the

8 static method and the dynamic method was only slightly different or exactly the same. The slight difference in values can be easily explained when reflecting back on the procedure and materials. A major factor affecting the collected data in the static method would be how the stretch of the spring was measured. When using a metre stick to measure centimetres, it is impossible to get an accurate reading. There will always be a discrepancy of ±0.1 cm, altering the final result accordingly. Additionally, most of the recordings were done while the spring was still vibrating by a few millimetres as it was difficult to get it to stop moving completely. Due to this movement, the recorded stretch became even more altered, further changing the final result. Furthermore, a major factor affecting the collected data in the dynamic method was how the time was kept. As this section relied heavily on human reaction to count ten vibrations and instantly hit the stopwatch, there is a large area for human error. This method of using a stopwatch can never be accurate as it will always result in time recordings that are slightly off. The time recording was the main factor needed to get the period of the vibration for each trial, making it a huge influencer on how the period changed. Additionally, when setting the spring into vibration, my group started with large amplitudes, overstretching the spring to some degree. We did fix this mistake halfway throughout the tests but the experiments that were already conducted in this manner had altered recordings. It is also possible that the springs used where also overstretched in previous labs, adding on to the errors made in our own lab. Due to these numerous systemic and random errors, the data recorded was altered in some way. An improvement for future use is to consider using different steps in the procedure to minimize the effect of the errors present. For example, in the static method, when measuring the stretch of the spring, it would be best to wait for the spring to settle at its equilibrium point. This could be done by stabilizing the weight as much as possible by hand and then letting the small millimetres of vibration dampen. Then, when the spring is still, the recording can be taken with the most accuracy possible given the measuring tools. Additionally, a step that was taken to improve the measurements was that one person held the metre stick at the right position while another read the actual length. In the beginning, one person was doing both tasks and the metre stick would move from its original position, changing the recording. This method was quickly discarded and split into two different jobs for maximal accuracy. In the dynamic method, the

9 time recordings where the most vital data in getting the results. For most accuracy, the group member taking the time measurements could have practiced with a trial run before the actual tests to make sure that their reflexes were fast enough. Additionally, since recording the time with a stopwatch held many chances for human error, the procedure had extra steps and required three separate recordings. This way, the final time used in the calculations for the period was an average of three trials, making it a little more accurate to the actual time. These extra steps that were already in the procedure and the suggested improvements are all valuable next steps that can be considered to get more accurate results in future activities. Conclusion In conclusion, the purpose of this lab was to find the force constant of several springs through a static and dynamic method and verify Hooke’s law. This purpose was achieved as the force constant was found to decrease as the length of the spring increased according to both methods. When the relationship between the force applied to each spring and the stretch of the spring was graphed in the static method, the lines of best fit showed a linear relationship. This verified Hooke’s law as the force applied was directly proportional to the stretch of the spring as seen in the equation F =kx. Therefore, the purpose of this lab was successfully achieved and many lessons on proper data recordings were learned. Application Throughout this activity and the calculations, the springs used were treated to be ideal springs. However, they were actually real springs that differ greatly from the characteristics of an ideal spring which is frictionless, massless, and linear. For example, a real spring would have a stretch even if there is no applied force. The spring would stretch under its own weight unlike an ideal spring that should have no displacement when no extra force is applied. Additionally, when a spring is put in simple harmonic motion, an ideal spring would continue to vibrate as energy is conserved. However, a real spring losses energy to internal friction forces and dampens over time, eventually settling in an equilibrium position. Furthermore, the force constant of a real spring can change over time as it gets worn out from use, unlike an ideal spring. An ideal spring is also able to stretch and compress without boundaries whereas a real spring can break if it is

10 stretched too far and it has a maximum compression point. Due to these many reasons, it can be seen that an ideal spring is different from a real spring. The knowledge of springs and force constant is extremely important when looking at cars. Steel springs in vehicles must hold the weight of the vehicle, passengers, and any added weight without excessive sagging. They also must help compensate for irregularities in the road surface, making the ride smooth for passengers (Tire Review, 2014). They must extend when the wheel moves down and compress when the wheel moves up so that the vehicle body remains level (How A Car Works, n.d.). However, due to this violent compression and extension, the vehicle would essentially lose control on a rough road surface. Thus, shock absorbers must be used to dampen the spring’s frequency. It is crucial to know the spring’s force constant as that will determine how much weight the spring can hold while still being able to manage some extent of compression and extension to compensate for a rough surface. Additionally, as a vehicle moves faster, a stiffer spring is required to keep the axles and wheels in contact with the road surface. This is why higher performance cars have stiffer suspension springs than regular vehicles (Tire Review, 2014). A normal passenger car usually has springs with a frequency of 0.5-1.5 Hz while a sports or track car has springs with a frequency ranging from 1.5-2.0 Hz. A lower frequency means that the force constant is lower and that the spring is able to absorb irregularities better. This is more important in passenger vehicles as a normal driver wants a smooth drive over the need of speed. A higher frequency means that the spring constant is higher and that the springs do not absorb irregularities as well. However, this is not crucial as speed is valued more in this type of vehicle (WSCC, 2012). This information on springs and force constant is important to car manufacturers as they would want the car to run according to its purpose. A passenger car does not need extreme speed and a sports car does not need as much comfortability on rough roads.
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