Dynamics Lab Report Assignment final PDF

Preview

Summary

simple pendulum and springs...

Description

Dynamics Lab Report Assignment: Simple Pendulum 1913340 Group 4a Lab date – 11th February 2020 Abstract A simple pendulum apparatus can be set up to investigate how the time period may vary due to a change in the weight of the sphere or the length of the nylon cord attached to the sphere. In this experiment 2 spheres of different masses were tested, measuring the time period for an oscillation at different cord lengths. It was found that the mass does not affect the time period of any pendulum oscillation. The experiment can be used to find an accurate measurement of g by drawing a graph of

𝑇2 𝐿

4𝜋2

where g is the 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 .

Introduction Aim of the experimentThis experiment was conducted to identify whether the pendulum follows the theory, finding an accurate value of g using our values, as well as pinpoint what factors affect a pendulum. Background information and theoryTheory suggest that the time period of a simple pendulum is Where: T = Time period (s) L = Length of pendulum (m) Figure 1- equation for time period of a simple pendulum

g = 9.81 m/s^2

If a pendulum is set in motion so that is swings back and forth, its motion will be periodic. The time that it takes to make one complete oscillation is defined as the period T. There are 2 types of solid body vibrations: free vibrations and forced vibrations. Free vibrations occur when the system, such as a pendulum, moves without restraint or a driving force. When it is in free vibration, a pendulum can be an example of Simple Harmonic Motion (SMH), this is where the external forces are countered by a restoring force causing the pendulum to accelerate towards the centre or slow down as it moves away from its mean rest position. Forced vibrations occur when the system has a driving force. Forced vibrations can cause resonance to occur, this is when the driving frequency equals or approaches the natural frequency (the frequency the body oscillates at without any driving force) causing the vibration to reach its amplitude. An example of resonance is the millennium bridge where the footsteps of the people walking across the bridge matched the natural frequency of the bridge, causing it to resonate. The

bridge was closed on 12th June 2000 so that engineers could apply dampeners to the bridge to reduce the resonant response. This experiment could be relevant and easily applied to an engineering situation, for example engineers can use the concept a pendulum swinging and relate it to an amusement park ride or a clock. The way pendulums behave can also help engineers further their knowledge on centripetal forces, inertia and Newtons first law of motion. Pendulums can help us accurately measure time; this is because pendulums continue to swing without changing speed unless they are acted upon by an external force such as air resistance (therefore they are used in grandfather clocks). A theoretical graph of displacement against time for the pendulum can be made by assuming that the theoretical cord length is 0.55m, this means the value for the theoretical time period calculated for length 0.55m can be used (in appendix). It is also assumed that the amplitude is 30mm as in the experiment the sphere was moved 30mm horizontally to one side which was its starting position, implying this is the amplitude. With this information the graph below can be produced where the time period is 1.49s.

Displacement - Time Graph 0.04

Figure 2 - Displacement as a function of time for one full oscillation

Displacement (m)

0.03 0.02 0.01 0 -0.01

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-0.02 -0.03 -0.04

Time (s)

The Experimental Problem Experimental setupThe problem that was investigated is whether the mass of the sphere/bob affects the time period of a simple pendulum oscillation. In order to carry experiment out, the equipment needed were2 bobs/spheres of different masses, Mass scale, Metre ruler, Stopwatch, TM161 pendulum experiments simple pendulum module MethodologyIn this experiment the length of string is the independent variable and we measure the time period which is the dependent variable. Measure the mass of each sphere to identify which is the lighter sphere. Fit the sphere to the string and set the string length to 550mm by moving the knob up (the knob can also be turned to keep the string in place as well as releasing the string to change the distance). Push the sphere to one side horizontally, about 30mm which can be measured by a ruler. Allow the ball to swing freely and start stopwatch. Count 50 oscillation and stop stopwatch (one oscillation would be from the point of release to the other side and back to the point of release). Record time for 50 swings. Change the length to 500mm by moving releasing and moving the knob

downwards. Repeat steps 3-6 for new length. Change length every 50mm interval until 200mm, recording the time for 50 oscillations. Repeat experiment for the heavier sphere. Record data in Table 1 found in appendix, one table for lighter sphere and one for heavier sphere. Once all the data is collected and put in a table, multiple graphs can be plotted using Microsoft Excel or any online graph creator. A graph of time period against length can be plotted where there are 3 lines, 2 for measured period of the lighter and heavier sphere and the other for theoretical. In order to identify a value for g, a graph of

𝑇2 𝐿

can be drawn where the gradient is equal 𝑙

was found by squaring both sides of the equation 𝑇 = 2𝜋 ( 𝑔)

0.5

4𝜋2 𝑔

. This

and dividing by L.

Results and Discussion Experimental resultsEach sphere was weighed using a mass balanceLighter sphere = 67.95g Heavier sphere = 206.43g The raw data for the lighter sphere (Table 2) and for the heavier sphere (Table 3) are in the 𝑙

appendix. The “Theoretical Period” column was calculated using the formula 𝑇𝑇 = 2𝜋 ( ) 𝑔

0.5

. An

example using the formula can be shown below When length = 550mm 0.55 ) 𝑇 = 2𝜋 ( 9.81

0.5

T = 1.49s The theory suggests the mass of the pendulum does not affect the time period of the equation, this is because in the equation for the time period of a pendulum, mass is not one of the variables. The experiment does not fully conclude that the theory is 100% correct as the values that I received for the time period does not match the theoretical period exactly. However, they may not match exactly because of the human reaction time causing them to be some degree higher or lower, we have determined this to be 0.2 seconds. The experiment was conducted indoors therefore the wind was negligible/ there was a constant airflow and Time-Length graph with light and heavy sphere wouldn’t affect the 2 pendulum or data. Time (s)

1.5 1

Figure 3 – Comparison of the light and heavy sphere to the theoretical period

0.5 0 0

0.1

0.2

0.3

0.4

0.5

Length (m) time period for lighter sphere theoretical time period

time period for heavier sphere

0.6

From both the lighter and heavier sphere line (Figure 2), it clearly shows that the mass does not affect the time period of the pendulum as both graphs have lines which are like the theoretical period line. It is also clear that the length of the nylon cord/ string which holds the sphere has a major impact, the longer the string the longer the time period is. The graph shows the error margin of time which is ±0.2, the theoretical period lies in all the error margins. This shows that there was no abnormal behaviour in the equipment or any major experimental behaviour. In certain values in the experiment the theory is supported. For example, at 200mm string length theory suggests that the period of the pendulum would be 0.9 seconds and from our readings we found that the value of the time period is 0.9 seconds as well.

Calculations Applying the equation

𝑇2 𝐿

to find the gradient of the line for the lighter sphere –

Time Period Squared- Length for lighter sphere Time Period Squared (s^2)

3 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Length (m)

Figure 4 - graph showing positive relationship between 𝑇 2 𝑎𝑛𝑑 𝐿, light sphere, used to calculate g Gradient for lighter sphere= g=

4𝜋2

3.77

2.13−0.81 0.55−0.2

= 3.77

therefore 3.77=

4𝜋2

therefore 3.77=

4𝜋2

𝑔

= 10.47

Same process is applied for the heavier sphere – Gradient for heavier sphere= g=

4𝜋2

3.77

= 10.47

2.13−0.81 0.55−0.2

= 3.77

𝑔

Time Period Squared- Length for heavier sphere Time Period Squared (s^2)

3 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Length (m)

Figure 5 – graph showing positive relationship between 𝑇 2 𝑎𝑛𝑑 𝐿, heavy sphere. They both give the same value of g since the gradient of each line were almost identical. The error bars on the graph were determined by doubling the value of the error bars for T. We do this for 𝑇 2 because the general rule to find the new error bar is to multiply by the by you are raising it to. Since T is raised to the power of 2, we multiply by 2, therefore the new error bar value will be 0.2 × 2 = 0.4. The theoretical value of g can be calculated using the same process but with the theoretical values for time period. I calculated the value to be 9.8, this value can be compared to the experimental value and a percentage error for the experiment can be calculated using – 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒−𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒 𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒

× 100

Using this formula, the percentage error for the experiment was 6.84%

Conclusion In conclusion the experiment would be considered a success as we were able to clearly identify if the mass of the sphere affects the time period as well calculating a value for g which was accurate to 93.16%. Figure 1 also shows that the length of the cord and the time period are proportional.

References https://www.teachengineering.org/lessons/view/cub_pend_lesson01 http://www.phys.utk.edu/labs/simplependulum.pdf

https://www.britannica.com/science/vibration

Free vibrations of a mass spring - Static and Dynamic analysis Abstract A mass spring system can be used to investigate Hooke’s law as well as establishing relationships between variables such as spring stiffness and frequency. In this experiment we tested springs of 2 different lengths, measuring the displacement at certain forces which allowed us to find the spring constant. Using the spring constant and VDAS, the frequency at different masses can be found. It was found that 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ∞

1

√𝑚

and springs obey Hooke’s law.

Introduction Aim of the experimentThis experiment was conducted to identify whether springs follow the theory of Hooke’s law. Hooke’s law states that the force needed to extend or compress a spring by some distance is proportional to that distance, providing external factors such as temperature remain constant and the solid is in its elastic limit. From experiment 2, dynamic experiment, we can also identify a relationship between frequency mass and spring stiffness. Background information and theoryHooke’s law states that the force needed to extend or compress a spring by some distance is proportional to that distance, providing external factors such as temperature remain constant and the solid is in its elastic limit (𝐹 ∞ 𝑋). A constant of proportionality, K, was added to the proportionality to produce this equation Where: F = Force (N) K = Spring constant (N/m) Figure 1 – Hooke’s law equation

E = Extension/Displacement (m)

This experiment can be related to real world engineering problems, learning about springs can improve our knowledge on cars which use the concept of a mass spring system in its shock absorber assembly. The mass-spring system in a car helps absorb bumps in the road, as a car goes over a bump, the spring in its system becomes compressed, but the elastic potential energy of the spring immediately pushes the it back to a state of equilibrium. Cars also have dampers which inhibit the springs oscillations. Experiment 2 is an example of a damped free vibration. When the spring is displaced from its rest position and released it causes the mass to vibrate. It is as its maximum velocity as it passes the equilibrium point. Air resistance would be the damping force causing the spring to come to rest 1 eventually. For experiment 2 the frequency (Hz) is found by doing 𝑇 , where T is the time period. Time 𝑚

period for spring can also be found using 𝑇 = 2𝜋√

spring constant is unknown to us.

𝑘

however we cannot use this formula as the

The Experimental Problem Experimental setupThe problem that was investigated is whether the springs follow the theory of Hooke’s law. In order to carry experiment out, the equipment needed were2 springs of different length Masses ranging from 200g-1000g TM164 free vibrations of a mass spring system Methodology (Experiment 1)1) 2) 3) 4) 5)

Fit the short spring into the machine and measure its initial length Add 200g and measure the springs new length, record total change in displacement Repeat step 2 until total mass added is 1000g Repeat steps 1-3 for the longer spring Record data in a table such as table 1 found in the appendix

The data can be put into two graphs of force against displacement, one graph for the shorter spring and one graph for the longer spring. In order to recognise if the theory of Hooke’s law has been followed, the graphs produced should have a straight line and a positive correlation. From the gradient ∆𝐹 of the graph the spring constant can be calculated, which is ∆𝐷. The spring constant of a spring can

help us determine the spring stiffness, this would therefore help us identify a relationship between stiffness and frequency. The higher the spring constant the stiffer the spring is. Each spring was measured using a ruler to find its initial lengthShorter spring = 71mm Longer spring = 101mm The graphs below show a straight line with a positive correlation which shows that springs obey Hooke’s law.

Force- Extension Graph For Long Spring 12 10

12 10 8 6 4 2 0

Force (N)

Force (N)

Force-Extension Graph For Short Spring

8 6 4 2 0

0

10

20

30

40

Extension (mm)

Figure 2 - Force-Extension Graph for Short Spring, evidence for Hooke’s law

0

20

40

60

80

Extension (mm)

Figure 3 - Force-Extension Graph for Long Spring, evidence for Hooke’s law

Spring constant calculationsThe raw data for the Shorter spring (Table 2) and for the Longer spring (Table 3) are in the appendix. The spring constant for the short spring was calculated as 9.81−0 = 33−0

297 N/m

The spring constant for the longer spring was calculated as 9.81−0 68−0

= 144 N/m

Methodology (Experiment 2)Experiment 2 was analysing springs dynamically; the same equipment was used however we used an extra piece of software called VDAS Versatile Data Analysis Software to analyse the time period. The method used changed1) Fit the spring and adjust the position of sensor 2) Open VDAS and enter Length of spring and spring constant 3) Enter platform mass (0.594kg), enter added mass and set time base to 100ms and displacement as 5mm 4) Initiate communication 5) Push platform down until the top of the activator level lines up with the bottom of the dotted line sensor 6) Release platform and wait 5 seconds before terminating communication 7) Measure the period at 3 points on the trace and record ∆𝑇 𝑎𝑠 𝑤𝑒𝑙𝑙 𝑎𝑠

1 ∆𝑇

8) Repeat experiment for 200g mass intervals until the added mass is 1000g 9) Record all the data in a table such as table 3 in appendix

Results and Discussion

Frequency - Mass Graph For Short Spring

Experimental results-

4

Average Frequncy (Hz)

All the raw data for the experiment can 3 be found in the appendix via table 5 and 6. In order to identify if there is any 2.5 relationship between the frequency of 2 oscillation and mass, a graph can be 1.5 drawn between the 2 variables. Both 1 graphs show a negative correlation, the 0.5 line is not straight and begins to bend, 0 0 0.5 1 1.5 with more data points the curvature of Mass (Kg) the line would be more visible. The negative curvature shows that the not only is the relationship inversely proportional, but Figure 4 - Frequency-Mass Graph Short Spring the frequency is proportional to the inverse square root of mass, 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ∞

1

.

√𝑚

3.5

2

3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

Using VDAS the timebase was set to 100ms, this allowed each increment along the x-axis to change by 0.02. Therefore, the error margin for the time period is ±0.01s. The vertical error bars are present on the graph, but they are very small therefore not visible. The masses of 200g did not have any uncertainty as it was the exact mass.

Mass (Kg)

Figure 5 - Frequency-Mass Graph Long Spring The stiffer the spring is the higher the natural frequency of vibration is, this statement can be proven by looking at the starting frequency of both springs in tables 5 and 6 in the appendix. The stiffer spring, which has a spring constant of 297 N/m, had an average starting frequency of 3.39s when no masses where added (just the platform mass). On the other hand, the other spring which has a spring constant of 144 N/m, had an average frequency of 2.65s.

Theoretical graphs of displacement, velocity and acceleration as a function of time Now that all the required information has been collected, three theoretical graphs can be produced for one of the springs. In this case short spring has been chosen which has a spring constant of 297 𝑚 N/m, with this the theoretical time period of the spring can be calculated using 𝑇 = 2𝜋√ 𝑘 . For mass it is assumed that there is no added mass therefore there is just the platform mass which is 0.594kg. Using the equation, a time period of 0.281 seconds. To calculate the amplitude the equation 𝑦(𝑡) =

V1 𝜔

sin(𝜔𝑡) + y1 cos(𝜔𝑡)

Where -

Displacement - Time Graph For Short Spring

V1 is the velocity which is 0 at the start

0.006

y1 is the displacement which is 5mm 𝜔 is the natural angular frequency To work out the natural angular

𝑘

frequency the equation 𝜔 = √ 𝑚. Using this gives us √

297

0.594

= 22.36.

All the information can be inputted into the first equation giving 𝑦(𝑡) = 0.005 cos(22.36 × 0.281) = 0.00497m

0.004

Displacement (m)

Average Frequency (Hz)

Frequency - Mass Graph For Long Spring

0.002 0 -0.002

0

0.05

0.1

0.15

0.2

0.25

-0.004 -0.006

Time (s)

Figure 6 - Displacement as a function time for one full oscillation