Effect of height on the oscillation of a pendulum

Effect of height on the oscillation of a pendulum

Abstract

The work aims at exploring the impact of the length of a pendulum’s string on the ideal opportunity for the time of that pendulum. Given the past information, it was realized that a pendulum carries on in a swaying way, implying that the quickening is consistently corresponding to the negative separation. In any case, it is likewise known through kinematics that an opportunity to travel a specific separation is relative to the increasing speed, speed, and separation of movement. Through controlling the length of the string, it is acceptable that it’s controllable for both the separation it voyages, as the circular segment length would be more noteworthy, as well as the speed of an item. The explanation will be expressing the subsequent one because when the string length is more outstanding, the article would have an absolute bottom that is lower than past models, and consequently mean a higher dynamic vitality. This commonly involves a higher speed at that example. Even though there is all accounts, two contending factors here, it would be anticipated that the expanded separation would end up being more grounded than the expanded speed since it is known for a fact that hotshot keeping pendulums in holy places or pendulum tickers have truly long strings and truly sluggish development. Accordingly, it can be estimated that the string length increases the time for the period, which is directly proportional to the time of the object.

Introduction

Mesmerized by the isochoric movement of the pendulum and the law of conservation of energy, I stumbled upon Galileo’s interrupted pendulum, a way of showing how energy is conserved with a very different movement. Something I was shocked to see at first and was fascinated by. An interrupted pendulum is one in which the string strikes a rod directly below the pivot point, causing the pendulum bob to deviate from its previous circular trajectory into a trajectory of a smaller radius. Depending on the original angle of displacement (θ) and where the rod is placed relative to the total length of the string, the Bob may or may not wind itself entirely around the rod. So, essentially an interrupted pendulum is a pendulum that has two halves, one with a long string and one with a shorter one. Therefore, the experiment was to be done using different lengths of the string as well as recording the results for each length. I use this same idea to understand and derive a formula for the period of the pendulum. The interrupted swing is also a way of showing how the law of conservation of energy is obeyed and conserved. In this experiment, we will be stimulating the motion of a pendulum, which has a fixed obstacle in its trajectory. For simplicity, I assumed that the object vertically aligned. As shown in the figure below.

Background

Law of conservation of energy this principle states that energy cannot be destroyed, nor can it be created; it can only be changed from one form to another. Therefore, the energy in a closed system will be the same. There is a simple way to confirm this fact, and I would like to prove it using a derived proof.

Let the schematic above be the motion of a ball in the air, where A is the highest point it reaches, B is any point midway, and C is the point when the ball makes initial contact with the ground.

At Point A

K.E = 0 (Since A is the highest point reached, the ball is stationary in the air)

P.E = mgh = mg(X+Y)

• Total Energy = 0 + mgX + Y = mg(X + Y)

At Point B

K.E = ⁶₇ ×m×(v⁷)

(v⁷) = u⁷ + 2as = 0 + 2g(X)

1

∴ K. E = ₂ ×m×2gX = mg(X)

P.E = mgh = mg(Y)

∴ Total Energy = mg(X) + mgY = mg(X + Y)

At Point C

P.E = 0 (Since C is the lowest point reached, the ball has h=0, making P.E (mgh) = 0)

K.E = ^@_A ×B×(C^A)

(C^A)=D^A+AEF=G+AH(I+J)

∴ K. L = _A ×B×AH(I + J) = BH(I + J)

∴ MNOEP LQRSHT = G + BH I + J = BH(I + J)

Thus, it is proved, energy will always stay the same in all points of motion.

The period of a pendulum is the time taken for the pendulum to complete one full oscillation, which means the time taken for it to complete one full to and fro movement. The length of a simple pendulum affects the period of the pendulum, but how does the change in the obstruction height affect the period and how?

The amplitude of a pendulum is the angle that it makes between the mean position, the position at rest, and the releasing position. The pendulum works in the gravitational field of the Earth, which is also an excellent example of a conservative field. Gravitational potential energy and kinetic energy are forms of mechanical energy. The gravitational potential energy can be seen as stored energy due to the conservative earth field. On the other hand, kinetic energy is the energy due to the motion of a body at a particular velocity. The mechanical energy is conserved throughout the movement of the pendulum.

The total of both the kinetic energy and gravitational energy, is invariably constant, which equals to the total mechanical energy. The highest point of the motion of the simple pendulum is also called the endpoints of the swing. Since there is no motion at these points, the total energy is equal to the gravitational potential energy. And at the lowest point of the pendulum as ∆h is 0 the total energy is the same as the kinetic energy. This point is called the equilibrium point, as when not in motion, the pendulum bob stays at that position. As the pendulum reaches one end of its movement, the value of ∆h keeps increasing till the end. And this value of ∆h is calculated but subtracting the height at this point minus the height at equilibrium.

Theory

In this experiment, a simple pendulum was used to measure the relationship between the period of oscillation and the mass of a pendulum bob, the length of a pendulum’s suspension string, and the oscillation amplitude. As seen in the experimental set up in Figure 1, a pendulum is a system consisting of weight, known as the Bob, suspended from a pivot point by a length of string around which Bob can swing back and forth in a single plane.

Oscillation is the repetitive movement of an object back and forth at a specific amplitude and period. The period of Oscillation (T) is the number of oscillations of an object in a one-second time frame. For small oscillations (𝜃₀ “60°), the period can be calculated using Equation 1 below, where L represents the length of the pendulum string, and g is the acceleration due to gravity.

Equation 1

For large oscillations (𝜃₀” 60°), the period can be calculated using Equation 2 below, where 𝜃₀ is the amplitude of oscillation.

Equation 2

A Vernier Photogate and the Logger Pro software were used to record and analyze the data captured while the pendulum oscillated during the numerous trials conducted during the experiment. A photogate is an instrument used to accurately measure time intervals, which was imperative to calculate the period of oscillation in this experiment accurately. Each trial consisted of releasing the Bob and allowing it to swing for a specified number of oscillations, depending on the trial, while the time was measured with the photogate. The collected data was then loaded into the Logger Pro software for analysis.

To evaluate the relationship between the length and the period of oscillation of the pendulum the Bob was suspended with varying lengths of string and evaluated for a period of ten oscillations each. The Logger Pro software was then used to produce a plot of the period of oscillation versus the length of the pendulum string to visualize the relationship.

To investigate the effect of Bob’s mass on the period of oscillation, the weight of the two bobs used in the experiment, an aluminum and a steel bob, were measured. This was done using an electronic balance, while the diameter of the bows was measured using a Vernier Caliper. Each mass was then suspended using the same length of string and evaluated for the same length of time. The period of oscillation for each trial was then calculated using Equation 1.

The relationship between the amplitude angle of release and the period of oscillation was explored by using five different angles of release between 30° and 90° while using the same Bob, pendulum length, and several oscillations per trial. The Logger Pro software was then used to produce a plot of the period of oscillation versus the amplitude of oscillation. The amplitude oscillation angle was then calculated using Equation 3 below, where x is the horizontal displacement of the pendulum as it swung back and forth.

Equation 3

The uncertainty of the amplitude of oscillation was calculated using Equation 4, where σ_L and σ_x are the uncertainties on the metre stick used to measure the lengths of string and the horizontal displacement.

Equation 4

Finally, the system’s conservation of energy was evaluated. The law of conservation of energy states that the overall energy of an isolated system, such as the pendulum set up, should remain constant over time. This means that no external forces are exerting an effect on the system’s activity. As a result, for the pendulum system to be conservative, Equation 5 must be found to be accurate, where ∆K is the kinetic energy of the system, and ∆U is the potential energy. In a pendulum, the kinetic energy is at a maximum when Bob is in the equilibrium position, and the potential energy is at a minimum. The ratio of the two energies changes with the position of the Bob as the pendulum oscillates.

∆K + ∆U = 0               Equation 5

Kinetic energy is the magnitude of the energy due to motion that a system holds. The value can be calculated using Equation 6, where m is the mass of the object, and v is the velocity of the object’s motion.

Equation 6

Potential energy is the magnitude of the energy stored in a system due to its position in space. Potential energy can be calculated using Equation 7, where h is the height at which the object is placed.

Equation 7

Variables

Dependent

Time period – The time was taken for five complete oscillations and then the time for one is found this is done to reduce the percentage error in the reading.

Independent

Length of string – The length of the string is measured and is kept the same for all trials for that length. And then is changed for the trials of the other lengths.

Obstruction Height (z) – The obstruction height was measured from the bottom of the pendulum bob and string and is kept the same for one set of readings and then is changed for the new collection of data

The independent variable would be the length of the string, as that would be the establishment of this lab. The needy variable would be the timeframe. Be that as it may, I am a little unsteady on this suspicion because, for a fact, I have seen pendulums slowly decline inadequacy/development, and don’t know how that would influence the timekeeping. Along these lines, to moderate any potential impacts, I timed the initial three oscillations and located the standard time for that period, and afterward, time the second 3 oscillations and the third three oscillations in a similar way. Along these lines, I would have the option to have an increasingly exact time estimation. As I will utilize the same rope for every preliminary, I ought not to have any issues with that length evolving. The hour of the period will be estimated when it takes for the bounce to travel three full periods, and afterward separated by three. To get a precise length estimation, I will gauge the length after the string is tied onto the ring stand. Different things I would keep steady would be the mass of the item on the rope and the stature of the underlying drop. The explanation I am keeping these consistent is that I realize that the object has some measure of potential vitality when it is at first dropped. That inherent vitality is reliant on the two statures of the item, just as the mass of the article. Even though I don’t have the foggiest idea about the points of interest, I would guess that changing both of these variables would bring about varying values for the time of wavering. Maybe those variables would be appropriate for another experiment.

Controlled

Drop angle – The drop angle was kept the same for all the sets of data and trials did ensure that the Bob in all the trials will have the same amount of energy. This was done by putting a pin on the other side, which would ensure that the pendulum bob is dropped from the same angle.

Digital Stopwatch used – The same stopwatch is used for all the trials and sets of data in the experiment, ensuring that, if any, systematic error is the same throughout.

The Bob used – The same pendulum bob was used so that the mass remains the same throughout the trials.

String used – The same string was used for the trials for the same length, ensuring that the errors remain the same for all the trials of that length if any.

Type of string used – The same type of string was used, ensuring that the string would stretch by the same length for all trials. It was also made sure that the string does not slack at the starting point.

Other variables that were controlled are:

• Length of twine (75.0 cm
• Type of twine (100% cotton kitchen twine)
• Mass of stopper (15.0 g)
• Rubber type (Styrene-butadiene)
• Timer (Timex watch)
• Room conditions (22˚C)
• Ceiling suspension point (light above the dining room table)

Risk Assessment

• A safe distance was kept from the oscillating pendulum bob.
• The stand should be kept on a stable surface ensuring that it does not fall

Apparatus

• Inelastic string

• Stand

• G clamp

• Pendulum bob

• Obstructing Pin

• Digital Stopwatch (±0.01s)

• Metre Rule (±0.05cm)

• Digital Scale (±0.001g)

Figure 1. Experimental setup for the simple pendulum experiment

Piece of string

Two bobs (cylinder)

The meter stick range is 0.0 – 100.0 cm and a precision of 0.1 cm

The caliper range is 0.000 – 15.000 cm and the accuracy of

The photogate range is

The simple pendulum, pictured in Figure 1 above, is a system composed of a weighted bob that is suspended by a length of string to a pivot point. The Bob swings freely back and forth around this pivot point, allowing the period of oscillation to be measured using a photogate. The photogate measures the time in which it takes for an object to block and then unblock an infrared beam across the device.

Method

1. The desired string length is measured and cut by using a metre rule.
2. The mass of Bob is taken using a three-decimal digital scale, and it is recorded for further data.
3. A measured length string is attached to the pendulum bob on one end and the other to the stand.
4. The stand is fastened to the table using the G-clamp.
5. The pin is attached to the stand, at a measured height from the bottom.

6. The pendulum bob is then pulled to one side, and the angle (q) between the mean position and the pendulum bob is kept small so that Bob does not have enough kinetic energy to go around and wind itself ultimately.
7. The pendulum bob is then left, and the time taken for five oscillations is taken. As the time for one oscillation will be very small, and there would be a lot of random error in it while recording the data

8. The same thing is done for different obstruction heights and different lengths of string.

Observation

1. The oscillations for the looked shorter length of string looked faster, but surprisingly it was not.
2. The pendulum bob would spin.
3. The string would slack at one end when the kinetic energy of the Bob was high.
4. When the obstruction height was small, the angular displacement seemed larger than when the obstruction height was greater.
5. The string was sometimes at an angle with the pin. We assume that it is perpendicular to the pin for more straightforward calculations.

Data Collection and Analysis

Table 1: Properties of the Weighted Bobs

Part A: Length Effect on the Period of Oscillation

Table 2: Measures of Period of Oscillation and Lengths of Aluminum Bob

Part B: Mass Effect on the Period of Oscillation

Table 3: Measured Period of Oscillation of Copper Bob

Part C: Amplitude Effect on the Period of Oscillation

Table 4: Oscillation Periods for Oscillation Amplitudes of Copper Bob at a Length of (0.463 ± 0.01) m

Table 5: Consistency Tests for Coefficients of Equation 2

Part D: Conservation of Energy

Calculated ∆K = 0.06 J

Calculated ∆U = 0.20 J

Discussion

In this experiment, the relationships between the period of oscillation of a simple pendulum and the mass of the bob, the length, and the oscillation amplitude were examined. From the experiment, it was determined that the weight did not affect the period of oscillation. Conversely, when the length of the pendulum’s string was increased, the period of oscillation increased. It was also determined that increasing the oscillation amplitude of the pendulum caused the period of oscillation to increase. Finally, it was determined that the system does not obey the law of conservation of energy as the calculated values of kinetic and potential energy were ∆K = 0.06 J and ∆U = 0.20 J. Since these two values do not sum to zero the system is not considered conservative.

The most dominant source of error in this experiment was the effectiveness of the photogate laser sensor’s detection. If the sensitivity of the sensor were increased to ensure that Bob was detected with greater accuracy, the data collected would be of a higher level of precision. A force sensor placed at the equilibrium position of the pendulum could also be used in this experiment. Placing the zeroed sensor at the equilibrium point would allow the force to be measured to determine the centripetal force of the pendulum. This would enable the centripetal velocity and acceleration of the swing to be calculated while also providing another method by which to calculate the period of oscillation. An experiment to determine the value of gravitational acceleration using a simple pendulum can be carried out using Equation 8 below. In the analysis, an individual would need to complete oscillation trials at varying lengths and substitute their determined period values and their corresponding lengths into Equation 8 to calculate the value of gravitational acceleration.

Equation 8

In this experiment, it was determined that mass did not impact the period of oscillation of the pendulum. A swing, such as those found in a children’s playground, work in much the same way as a simple pendulum; however, it is known that the period of oscillation of a swing when an adult versus a child is on the swing is slightly different. One possible explanation for this is due to the fact that most adults have a larger weight than children. Increased weight means that the gravitational forces exerted on an adult are larger than those on a child. As a result, the adult will likely have a smaller period than the child as more kinetic energy is required to overcome the gravitational forces and air resistance, attempting to bring the swing back to its equilibrium position so they would have a reduced angular velocity which directly impacts the period of oscillation.

Conclusion

Additionally, it was determined that an increased length of the pendulum string causes the period of oscillation to increase as well. In Part A of the experiment varying lengths of string were used for the trials. All of the lengths, however, were larger than 30 cm. The decision to use lengths greater than 30 cm was made to avoid having too small of a period of oscillation. If the length used was less than 30 cm, the period was likely to drop below a value of 1 Hz or one oscillation per second.

It was also determined that the oscillation amplitude has a directly proportional effect on the period of oscillation of a simple pendulum system. In this experiment, the coefficients of Equation 2 were calculated up to the third order, and consistency tests were completed. As seen in Table 5, the first two coefficients referred to as ‘a’ and ‘b’ were found to be inconsistent and, therefore, insignificant. In contrast, the third-order coefficient ‘c’ was found to be consistent and, consequently, significant. Over time the oscillation amplitude decreases as a direct result of friction experienced in the pivot point; as a result, the Bob’s velocity slows, and the height to which it reaches during its oscillations also decreases. This phenomenon is a direct indication that the total system energy is dissipated, not conserved, over time.