Lab report(shm) - lab report of simple harmonic motion PDF

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Lab report -Simple Harmonic Motion

Introduction A long pendulum swings in the grandfather clock, the wooden bridge shakes when people walk on it, a swing with a child on it … We can see that simple harmonic motion is very close to our lives. (See Fig.1, Fig.2, Fig.3) Actually, all the things around us are vibrating at a certain frequency all the time, the frequency is called their natural frequency, and this kind of vibration is called free vibration. The natural frequency of water is 2.45GHz, that means if water is given a driving frequency of 2.45GHz, the water molecular will oscillate at a rate of 2.45 billion times per second and strongly collide with each other because of the driving frequency, and the water is heated up on the macro. People used this resonance theory to create microwave oven that can create a driving frequency of 2.45GHz and heat the food up. However, some criminal individuals or organizations may use the theory to produce weapons that can create a driving frequency the same as our bodies’ frequency. This could be very harmful to the society and should be strictly banned.

Figure 1

Figure 2

Figure 3

The first microwave oven was invented by an American radar engineer Percy Spencer in 1945. The chocolate in his pocket started to melt when was doing a radar experiment, he was inspired and discovered the thermal effect of microwave, he then used the theory to invent the first microwave oven, radar oven. (See Fig.4)

Figure 4

Simple harmonic motion refers to a periodic motion that when an object is vibrating, its displacement has a sinusoidal relationship with time. In simple harmonic motion, there is always a force trying to ‘drag’ the object back to its equilibrium position where the object stops moving, which is called restoring force. The restoring force is directly proportional to the displacement and has an opposite direction as the displacement in simple harmonic motion. Time period refers to time taken for one oscillation. The time period of simple harmonic motion is really an interesting thing that is worth to investigate. For example, when a pendulum oscillates, it seems that many factors could affect its time period: its angle to the vertical line, the mass of the pendulum bob, the length of the string and so on. But, do they all really affect the time period? Simple pendulum is one of the models of simple harmonic motion, this system consists of a string and a bob. (See Fig.5) When the bob is pulled back a certain amount and let it go, then the bob starts to swing back and forth periodically. Mass-spring system is another model of simple harmonic motion, which is a spring with a mass connected to it. (See Fig.6) When the mass is pulled and released, it also starts to oscillate periodically.

Figure 5

Figure 6

In this experiment, strings with pendulum bobs, springs with hanging masses, a stopwatch, a protractor and rulers were used to investigate the factors that could affect time period of simple harmonic motion. Measured results were compared with theoretical results to see whether the hypotheses were correct. The simple pendulum was also used to investigate damping and resonance in simple harmonic motion. Hypothesis: In the string with pendulum bob experiment, the time period was only determined by the length of string, whatever its angle to the vertical line was, whatever how heavy the pendulum bob was, the longer the string, the bigger the time period. In the spring with hanging mass experiment, if the hanging mass was big and the stiffness constant of the spring was small, the tension would be difficult to ‘drag’ the mass to its equilibrium position, and the time taken would be longer. That means the bigger hanging mass or the smaller spring constant, the bigger the time period. For damping and resonance, the damping force could only affect the amplitude of the oscillation but can not affect the time period.

Theory Simple harmonic motion is the motion in which the restoring force directly proportional to the displacement, but their direction is opposite. The equation is as shown below:

𝐹 = −𝑘𝑥

(1)

where F represents the restoring force, x represents the displacement, k is a constant which varies in different situations, ‘ -’ means that the direction of restoring force is always opposite to that of displacement. To take an example of the motion of the x-axis projection of an object in uniform circular motion. To judge whether this motion is simple harmonic motion, the only thing that needs to do is to judge whether its restoring force and displacement obey Eq.1

Figure 7 Form the Fig.7, the relevant equations can be written as follows: 𝐹 = 𝑚𝜔2 𝑟 = 𝑚𝜔2 𝐴

(2)

where F is the centripetal force of the object, m is the mass of the object, ω is the angular velocity of the object, r is the radius of the circle which is the same as the amplitude A of the motion of the projection. 𝐹𝑥 = 𝐹𝑐𝑜𝑠𝛼

(3)

where 𝐹𝑥 is the x-component of F, α is the angle between F and 𝐹𝑥 , and also is the angle between the line of F and x-axis. 𝑐𝑜𝑠𝛼 =

𝑥 𝐴

(4)

where x represents the displacement of the projection. Combining Eq.2, Eq.3, Eq.4, the following equation can be derived: 𝐹𝑥 = −𝑚𝜔2 𝐴 •

𝑥

𝐴

= −𝑚𝜔2 𝑥

where ‘-’ means that the direction of 𝐹𝑥 is opposite to the direction of x.

(5)

Comparing Eq.1 with Eq.5, it’s clear that 𝑚𝜔2 is the constant number k. Therefore, the motion of the x-axis projection of an object in uniform circular motion is simple harmonic motion. Let 𝑘 = 𝑚𝜔2 , so ω can be written as:

𝜔=√

𝑘

(6)

𝑚

As the equation for time period of the motion of the object is 𝑇 =

2𝜋

𝜔

, substitute ω into the

equation, the general equation for time period of a simple harmonic motion can be written as: 𝑚

𝑇 = 2𝜋√

(7)

𝑘

where T is time period, m is the mass of the object, k is the constant number that builds up the relationship between restoring force and displacement. The equation for displacement, acceleration, velocity in simple harmonic motion can all be derived from Fig.7. Displacement: 𝑥 = 𝐴𝑐𝑜𝑠𝛼 Because 𝛼 = 𝜔𝑡 , 𝜔 = 2𝜋𝑓 , where 𝑓 represents frequency, the general equation for displacement in simple harmonic motion can be written as: 𝑥 = 𝐴𝑐𝑜𝑠2𝜋𝑓𝑡

(8)

Acceleration: 𝑎 = −𝑎0 𝑐𝑜𝑠𝛼, where 𝑎0 is the centripetal acceleration of the object, 𝑎 is the xcomponent of 𝑎0 , ‘-’ means that the direction of 𝑎 is always opposite to the direction of x-axis. (See Fig.8) Because 𝑎0 = 𝜔2 𝐴, 𝜔 = 2𝜋𝑓 , the general equation for acceleration in simple harmonic motion can be written as: 𝑎 = −𝐴(2𝜋𝑓)2 𝑐𝑜𝑠2𝜋𝑓𝑡

(9)

Velocity: 𝑣 = ±𝑣0 𝑠𝑖𝑛𝛼, where 𝑣0 is the linear velocity of the object, 𝑣 is the x-component of 𝑣0 , ‘±’ means that the direction of 𝑣 can be the same or opposite to the direction of x-axis. (See Fig.9) Because 𝑣0 = 𝜔𝐴, 𝜔 = 2𝜋𝑓, the general equation for velocity in simple harmonic motion can be written as: 𝑣 = ±2𝜋𝑓𝐴𝑠𝑖𝑛2𝜋𝑓𝑡

Figure 8

(10)

Figure 9

The 𝑠𝑖𝑛 and 𝑐𝑜𝑠 in Eq.8, Eq.9, Eq.10 can be changed to 𝑐𝑜𝑠 and 𝑠𝑖𝑛 , because when evaluating the motion of the y-axis projection of the object, the displacement x has a 𝑥 relationship with A which is 𝑠𝑖𝑛𝛼 = 𝐴. That means 𝑠𝑖𝑛 and 𝑐𝑜𝑠 in these equations should be changed into 𝑐𝑜𝑠 and 𝑠𝑖𝑛 in that situation.

Combining Eq.8 and Eq.9, the relationship between displacement and acceleration can be written as the equation: 𝑎 = −(2𝜋𝑓)2 𝑥

(11)

Combining Eq.8 and Eq.10, the relationship between displacement and velocity can be written as the equation: 𝑣 = ±2𝜋𝑓√𝐴2 − 𝑥 2

(12)

Time period of simple pendulum For simple pendulum, (See Fig.10) the following analyses are to derive the equation for its time period.

Figure 10

Figure 11

As can be seen in Fig.10, the weight of the pendulum mg is decomposed into two directions. Therefore, 𝐹 = 𝑚𝑔𝑠𝑖𝑛𝛼, where F is the restoring force. When the angle α is very small (usually 𝑥 considered as lower than 5˚), x’ is approximately the same as x (See Fig.11), so 𝑠𝑖𝑛𝛼 = 𝐿 . The equation for restoring force in simple pendulum can be written as: 𝐹=−

𝑚𝑔

𝐿

𝑥

(13)

where F is the restoring force of the pendulum, L is the length of the string, x represents the displacement of the bob, ‘-’ means that the direction of F is opposite to the direction of x. Comparing Eq.1 and Eq.13,

𝑚𝑔 𝐿

can be seen as k. Therefore, 𝑘 =

𝑚𝑔 𝐿

. Substituting k into Eq.7,

the general equation for time period of simple pendulum can be derived as: 𝑇 = 2𝜋√

𝐿

𝑔

(14)

where T is the time period, L represents the length of the string because the pendulum bob is seen as a particle and its radius is neglected. It shows that the time period of simple pendulum is only determined by the length of the string and gravitational acceleration, and will not change when the mass of the pendulum or the angle to the vertical line is changed.

Time period of spring with hanging masses For the simple harmonic motion of spring with hanging masses, the tension is the force that wants to ‘drag’ the hanging masses back to its equilibrium position, so the restoring force of

the hanging object is tension, according to Hooke’s Law, the equation for restoring force of the object can be written as: 𝐹 = −𝑘𝑥

(15)

where F is restoring force, k is the spring constant, x represents the displacement of the object, ‘-’ means that the direction of F is opposite to the direction of x. Therefore, the k in Eq.1 refers to the stiffness constant of the spring in this situation. The equation for time period of spring with hanging masses can be written as: 𝑚

𝑇 = 2𝜋√

𝑘

(16)

where T refers to the time period in the spring with masses system, m represents the mass of hanging masses, k is the stiffness constant of the spring. This equation shows that, in the system of spring with hanging masses, the time period is only determined by the mass of hanging masses and the stiffness constant of the spring.

Energy in simple harmonic motion In simple harmonic motion, the total energy consists of two parts: kinetic energy and potential energy. To take the example of mass-spring system. (See Fig.12)

Figure 12 The equation for kinetic energy is: 𝐸𝑘 =

1

2

𝑚𝑣 2

(17)

where 𝐸𝑘 represents kinetic energy, m represents the mass of the object, v is the velocity of the block. The equation for potential energy is: 𝐸𝑝 =

1 𝑘𝑥 2 2

(18)

where 𝐸𝑝 represents potential energy, k refers to spring constant, x represents the displacement of the block. Combing Eq.6, Eq.8, Eq.10, Eq.17, Eq.18, the general equation for total energy of simple harmonic motion can be written as: 𝐸=

1

2

𝑘𝐴2

(19)

where E is the total energy in a system of simple harmonic motion, k is a constant number, A is the amplitude of the oscillation. This equation indicates that the total energy in

simple harmonic motion is only determined by the amplitude of the oscillation. The total energy remains the same as the object oscillating. That is because, when an object is undergoing simple harmonic motion, the object is not affect by any external forces and there is no energy input or energy loss, so there is only kinetic energy and potential energy converting to each other, and the total energy remains the same as the energy when the object starts moving.

Damping Simple harmonic motion is an idealized motion of harmonic motion where internal friction and air resistance are neglected. However, in real harmonic motion, those factors can affect the amplitude of the oscillation because some of the mechanical energy is transformed into thermal energy. The damping in simple harmonic motion means that due to external forces such as air resistance, the mechanical energy of the system dissipates over time, so the amplitude of the oscillation decreases over time. There are two factors that can affect the degree of damping: the nature of the surrounding medium and the velocity of the oscillator. If a mass-spring system is immersed in the water, the damping force will be much bigger than that of the system which is set surrounded by air, so the bigger the friction, the higher the damping force. If the oscillator moves very fast, the damping force will be even higher, and more energy will dissipate. Therefore, the equation for damping force can be written as: 𝐹𝑑 = −𝑏𝑣

(20)

where 𝐹𝑑 is the damping force, b represents the natures of the surrounding medium such as viscosity, 𝑣 is the velocity of the oscillator, ‘-’ means that direction of damping force is always opposite to the direction of the velocity. Thus, the total force that wants to ‘drag’ the oscillator back to equilibrium position is the sum of restoring force and damping force. The equation for this total force is: 𝐹 = −𝑘𝑥 − 𝑏𝑣

(21)

𝑚𝑎 = −𝑘𝑥 − 𝑏𝑣

(22)

Supposing the mass of the oscillator is 𝑚, the acceleration of the oscillator is 𝑎 at the time 𝑡, and the oscillator starts to move at the maximum point. According to Newton’s second Law of Motion, Eq.21 can be written as: To get the equation for displacement 𝑥 in damped simple harmonic motion, the solution to Eq.22 is as shown below: 𝑥 = 𝐴𝑒−

𝑏𝑡 2𝑚

cos (𝜔′𝑡)

(23)

where 𝐴 is the amplitude of the oscillation, 𝜔′ is the angular frequency of the damped simple harmonic motion. This equation shows that in a damped simple harmonic motion, the amplitude of the oscillation is not a constant but decrease over time. The condition under which that Eq.23 works is given by the following equation: 𝜔′ = √

𝑘

− 𝑚

𝑏2

4𝑚2

(24)

There are 3 types of damping: under damped, critically damped and over damped. (See Fig.13)

Figure 13 Under damped is the situation that the object oscillates several times before it stops. In this situation, 𝑏 2 < 4𝑚𝑘, that means damped obey the rule of Eq.23.

𝑘 𝑚

−

𝑏2

4𝑚2

> 0, so Eq.24 works and the displacement of under

Over damped is the situation that the damping force is so big that object takes a long time to achieve its equilibrium position. In this situation, 𝑏 2 ≫ 4𝑚𝑘, that means

𝑘

𝑚

−

𝑏2 4𝑚2

< 0, so Eq.24

no longer works, and the object will not oscillate. Instead, it will gradually stop at its equilibrium position due to the damping force. Critically damped is the situation that the object takes the shortest time to come to the equilibrium position, and all the energy is gone before its first oscillation. In this situation, 𝑏 2 = 4𝑚𝑘, that means 𝑏𝑡

𝑘 𝑚

−

𝑏2

4𝑚2

= 0, so 𝜔′ = 0 and the equation for the displacement becomes 𝑥 =

𝐴𝑒 −2𝑚 . Therefore, the object will not oscillate and will stop at the equilibrium position in a very short time.

Resonance If an object is given a driving force periodically, it will oscillate at the frequency the same as the driving frequency. If the driving frequency is the same as the natural frequency of the object, the energy input will be higher than the energy lost in its harmonic motion, so the amplitude will increase. To take an example of resonance coupled pendulums. The 5 pendulum bobs are connected to one string by 4 kinds of strings. (See Fig.14)

Figure 14 The length of strings has the relationship: 𝑙𝐸 < 𝑙𝐷 < 𝑙𝐶 < 𝑙𝐵 = 𝑙𝐴 . When pendulum A starts to oscillate acting as the driving force, it will drive the public string to oscillate, then other pendulums then start to oscillate as well. However, although the natural frequencies of

pendulum B, C, D, E are different, they will oscillate at the frequency the same as pendulum A which provides the driving frequency. In addition, when B, C, D, E is oscillating under the driving frequency provided by pendulum A, the amplitude of the oscillations of the 5 pendulums has the relationship: 𝐴𝐸 < 𝐴𝐷 < 𝐴𝐶 < 𝐴𝐵 = 𝐴𝐴 . That means when the length of string is close to the length of string A, or in other words, the difference between natural frequency and driving frequency is close to 0, the amplitude becomes higher. That can explain why the amplitude becomes higher when driving frequency is close to natural frequency. (See Fig.15)

Figure 15

In this experiment, the focus was the time period in simple harmonic motion, several factors were evaluated to see whether they could affect time period or whether they obey the theories, such as the length of string, the mass of pendulum, the angle of the string to the vertical line, the stiffness constant of spring and so on. Damping and resonance were also investigated to see whether they agree with the theories.

Methods Simple pendulum Equipment (See Fig.16): • • • • • • •

3 strings with the length of 0.265m, 0.445m, 0.59m 3 pendulum bobs with radius and mass of (1,1cm, 9.44g), (1.5cm, 29.21g), (2cm, 71.18g) 2 rulers 1 protractor 1 stopwatch (SATZ) 1 balance (MKL622) 1 clamp stand

Figure 16

Procedure: 1. Set up the equipment with the clamp stand, string 2, bob 3 and protractor. 2. Pull the pendulum bob to an angle of 5˚ to the vertical line. (See Fig.17)

Figure 17

3. Release the bob, make sure it started moving from stationary state. 4. Measure the time taken for 10 oscillations. Because the bob moved so fast that it was difficult to catch the point where the bob reached the equilibrium position, a video was taken to record the movement of the bob as well as the time shown on the stopwatch. Then use the time when the bob passed equilibrium position the 20th time to subtract the time when it passed equilibrium the first time, and then let this total time divided by 10 to get the time period. 5. Reset the equipment, change the angle to 15˚ and repeat step 3-4. 6. Reset the equipment, change the angle to 30˚ and repeat step 3-4. 7. Reset the equipment, change the string to string 1 and repeat step 2- 4. 8. Reset the equipment, change the string to string 3 and repeat step 2-4. 9. Reset the equipment, change the bob to bob 1 and repeat step 2-4. 10. Reset the equipment, change the bob to bob 2 and repeat step 2-4. 11. Use formula to calculate the time period in each situation, compare the measured results to the theoretical results. 12. Draw line graphs to demonstrate the difference between the measured results and theoretical results in order to evaluate whether the results obey the theories. The main method in this experiment was control variables method. There were 3 variables evaluated in this experiment: angle, length of the string, mass of the bob. Step 1-2 were the preparations before releasing the pendulum bob, the angle of the string to the vertical line should be kept stable in order to reduce the error. Step 3-4 were the procedure of the experiment, release the bob and take notes of the measured time period. Step 5-6 were to evaluate angle, during which the string and the bob were not changed, only angle was changed. Step 7-8 were to evaluate the length of string, during which the angle and the bob were not changed, only the string was changed. Step 9-10 were to evaluate the mass of bob, during which the angle and the string were not changed, only the bob was c...