April 01, 2015 Centripetal Force with a Motor Lab

Objective: Find the relationship between the angle and the angular velocity.

The Set-up: 

The apparatus is a motor mounted on a surveying tripod. A long bar is connected vertically to the motor.  And a meter stick is mounted on the long bar. At the end of the meter stick is a string that has a weight attached. 

Data Collection:

Here are a few things that we measure:

The length of the string (L) =165.4cm ± 1 cm

The radius of the circle made by string/mass (R) =97.5 cm ± 1 cm

The height of the horizontal rod (H) =200cm ± 1 cm

The angle and height (h) are the variables that can be obtained through experiment. After collecting the data, we can use them to find angular velocity.

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Next,  we come up with a model of the angle and the angular velocity.

After finding the model of the angle and the angular velocity, we begin our data collection. We find angular velocity by divide a number of rounds that the weight made with the time. As for the angle, we obtain by recording the height that the weight reaches. Then we can calculate the angle using inverse cosine. 

Here is our data.

Table 1

Trial	Height of the mass, h (cm)	Period, T (sec)
1	47.3± 0.5	3.70
2	62.4 ± 0.5	3.26
3	85.0 ± 0.5	2.84
4	118.5 ± 0.5	2.20
5	140.8 ± 0.5	1.93
6	161± 1	1.56

Sample Calculation

Table 2

Trial	Angle	Theoretical Angular Velocity (rad/sec)	Experimental Angular Velocity (rad/sec)
1	22.6	1.59	1.70
2	33.7	1.84	1.93
3	46.0	2.17	2.21
4	60.5	2.68	2.86
5	69.0	3.19	3.26
6	76.4	3.96	4.03

The sample calculation shows the theoretical angular velocity. Theoretical angular velocity is the angular velocity that the mass should have under ideal situation.

Since the apparatus is not under ideal situation, there is the experimental angular velocity. The experimental angular velocity can be found by:

Angular velocity = 2pi/T.

T is period, it is the time it takes for one revolution. T is recorded in table 1.

For example, the experimental angular velocity of trial 1 would be 2pi/3.70 = 1.697≈1.70 rad/sec.

Finally, we input both the theoretical and experimental angular velocity into loggerpro, we get a graph that looks like the following. 

Experimental angular velocity vs theoretical angular velocity graph

Here we see that our experimental value, the red line, did not quite match the theoretical value, the black line. The reason is because there are some uncertainties involved in this apparatus. First of all, when we measure the length of the string, the radius and the height of the horizontal rod, there is an uncertainty value of ±1cm since we measure with the meter stick. Secondly when measuring the height of the mass (h), there is an uncertainty value of ±0.5 cm. When we measure the period, the time it takes for one revolution, we count the revolution with our eyes. There is some uncertainty involves since our counting are not exact. Also, there are some system error involves. When the apparatus turns, the horizontal rod, which is a meter stick, does not turn in a horizontal manner, sometimes it can wobble a little. Also, there is air resistance. The air resistance can slow down the angular velocity.

Summary

We begin this experiment with an apparatus that involves the rotation of the mass hanging on a string. Our goal is to find the relationship between the angle and the angular velocity. Firstly, we measure a few things that are always constant; they are the length of the string, the radius and the height of the horizontal rod. Then we find a formula for the angle and a formula for the angular velocity. To find the angle, we need the height of the mass. We get the height of the mass through experiment; it is recorded in table 1. After finding the angle, we can calculate the theoretical angular velocity using the formula we derived. We also take down the period, or the time that the mass takes to make one revolution. The period is for finding the experimental angular velocity. To find the experimental angular velocity, we simply divide 2pi by the period. The angle, theoretical angular velocity and experimental angular velocity are recorded in table 2. Then, we compare the theoretical value with the experimental value. We make a graph showing the difference. There is a small discrepancy between the two, it is due to the uncertainties involve in this lab.

April 6, 2015 Work-Kinetic Energy Theorem Lab

Objective: To show that work equals to the change in kinetic energy.

Exp 1

The Set-Up:

The Apparatus

We place a metal track on the table. At one end of the table is a force sensor stabled by the table with a table clamp. On the other side of the metal track is a motion sensor. Both force sensor and motion sensor is connected to loggerpro. We calibrate the force sensor with a force of 4.9 N applied. Then we set the motion sensor so that toward the sensor is positive.  A cart is place on the metal track with a spring connecting from the cart to the force sensor, so that when the cart is pulled to a certain distance, the force will also be recorded.

Data Collection: 

Force vs Position Graph

We pull the cart and obtain a graph of force vs position. We linearized the force vs position graph. The slope of is the spring constant, k, of the spring in this apparatus. 

The spring constant is 2.904 N/m.

To find the work done by the spring, simply use the integration option in loggerpro. Loggerpro will calculate the area under the curve, that is the work.

Position vs force graph. Area under the curve

In this experiment, the work that the spring have done is 0.08410 m*N.

Exp 2

The Set-Up:

The set-up in part 2 is the same as part 1.

We measure the mass of the cart plus the addition weight on top of it, the total mass is 0.574 kg.

Data Collection:

We use the same force vs position graph from part 1 and integrate it at three different positions. The area under the curve is the work done by the spring.

We want to see how work connects to kinetic energy.  So we analyze the area under the curve and kinetic energy to see the relationship between them. 

Force(purple) and Kinetic energy (blue) vs Position Graph ---1st position

The purple area represents the work that the spring did at that position. The blue line represents kinetic energy. The boxes show that the kinetic energy at this position is 0.071 J, the work is -0.07432 m*N.  Note that m*N= J, from this we see that they are representing the same thing, that is, work is the change in kinetic energy. We see more evidence in the second graph. There is a negative sign in work might because work is a vector, direction matters in work, the negative sign might come from the force is opposite the displacement. 

Force and kinetic energy vs Position Graph--2nd position

Here we have the information of both lines at the same position. Here the kinetic energy is 0.123 J, the area under the purple curve, or work, is -0.1232 m*N. They are very close together, from this we see that work is kinetic energy.

Force and kinetic energy vs Position Graph--3rd position

 This is the third position we choose to analysis the area under the curve and the kinetic energy. Once again we see that kinetic energy is 0.146 J and work is -0.1458 m*N.

 Work is the change of kinetic energy.

Exp 3

The Set-Up:

We open a file called “Work KE theorem cart and machine for Phys L.” It is a video about a professor uses a machine to pull back on a large rubber band. The force is recorded by a force transducer onto a graph. This rubber band is connected to a cart. Once released, the cart passes through two photogates. We want to find the final speed and thus calculate the final kinetic energy of the cart. 

Data Collection:

We first begin by finding the force vs position graph for the robber band. We trace the the force pattern made by the force transducer onto the white board. Then we make a graph using the data.

Stretch of the Rubber Band(m) [x-axis] vs Force(N) [y-axis] Graph

Work is the curve under the force vs position graph. If we want to find the work done by the rubber band, we have to we separate the curve into four sections and find their area one by one.

Area A: ½ (0.27 m) (68 N) = 9.18 m* N

Area B: (0.38-0.27 m)(68 N) = 7.48 m*N

Area C: ½ * (32+68N)(0.42-0.38m) = 2 m*N

Area D: ½ * (32+37)(0.65-0.42m)= 7.935 m*N

Work = total area under the curve= 9.18 + 7.48 + 2+ 7.935=26.6 m*N

Next, we examine the final kinetic energy of the cart attached to the machine.

Here are some basic information given from the video:

mass of the cart = 4.3 kg

Distance between the two photogates =15 cm= 0.15 m

Time that the cart passes the two photogates = 0.045 sec.

KE= ½ m v^2

v=d/t = 0.15 m/0.045 sec= 3.33 m/sec

KE=1/2 (4.3 kg)(3.33)^2= 23.8 J

Since the is initially at rest, its initial kinetic energy is zero, therefore, the charge in kinetic energy of the cart would just be the final kinetic energy. 

Notice that the total work done by the professor is 26.6 m*N while the final kinetic energy of the cart is 23.8 J. Since work= the change in KE, why the value is different? It is different because there is some uncertainty involve, such as friction in the system, or calculation rounding error. 

Summary: 

This lab is separate into three experiments. 

The first experiment seeks to find the spring constant, k, in a spring. We find the spring constant by using the force vs position graph. We obtain the graph by pulling the cart. Then we linearized the graph to give us a slope, the slope of the graph is k. k= 2.904 N/m.

The second experiment deals with the work-energy theorem. Using the force vs position graph from experiment 1, we find the area under the curve. The area under the curve is the work done by the spring. Then we combine kinetic energy graph into the force vs position graph. We chose three positions to analyze them. We find that work is the change in kinetic energy. 

The third experiment involves a video of a professor pulling a rubber band through a machine. The force of the rubber band is recorded. We look at the force curve and find the total work done by the professor. Then the same rubber band is connected to a cart. When the cart is released, it passes through two photogates. The mass of the cart, the distance between the two photogates and the time is given. We use these information to find the change in kinetic energy of the cart. Since work is the change in kinetic energy, they should be the same value. However, we find that they are not the same value, the difference might due to friction in the rubber band. 

April 08, 2015 Energy Conservation Lab

Objective: To prove energy is conserved in a vertically- oscillating mass-spring system.

The apparatus

In this lab we are trying to find the total energy of this vertically oscillating spring- mass system. We firstly derive the formula for each type of energy, then we use force sensor and motion detector and with the help of loggerpro we come up with the numerical value of this spring mass system.

First of all, there are five types of energy present in this system

1.       Elastic potential energy (EPE)

2.       Gravitational Potential Energy (GPE)

3.       Kinetic energy (KE)

4.       Gravitational Potential Energy of the spring (GPE_spring)

5.       Kinetic energy of the spring (KE_spring)

The first three types of energy are common. EPE is there because we have a spring and the energy from spring is elastic potential energy. GPE is present because the spring is hanging at a certain height; there is GPE since we set the ground to be the origin. KE is there because at the end of the spring connects the mass, velocity is created from the oscillation of the spring. 

The last two types of energy are not so easily to identify. In order to identify them, we need to see the spring itself as the system. 

GPE_spring is one of the energy in this system because different part of the spring has different height, as the spring oscillates, the height of the different part of the spring changes. When the height is changing, we have different potential energy from different part of the spring. In order to calculate the total potential energy of the spring, we need to use integration to come up with a function with a variable, y₀ . After we have the function, we can then use loggerpro to help complete the process.

The above picture is the integration to find the function of energy. We begin by finding dm.

a)      dm is a piece of mass we chosen to represent one part of the spring. Since the spring is constantly changing, we need to choose one part to integrate. This piece of mass, dm, has a height, we called it dy. The piece of mass, dm, is divided by the height of the piece of mass, dy. This is equal to the total mass of the spring, M, divide by its changing length, h-y₀. 

b)      We write out the function of potential energy using dm. Note that the height is y₀ not h.

c)       We integrate from y₀ to h because that is the length that the spring is changing its position. y₀ is referring to the end position of the spring at each position. It is a constantly changing variable since the end of the spring is constantly changing its position as it oscillates. 

d)      Eventually, our result involves the sum of two fractions. One fraction is constant and the other one includes a change in position, y₀. We are interested only on the one with y₀ because we are looking for the function that would describe the energy at different position during oscillation. The constant fraction just gives us the energy at a constant height.

KE_spring is part of the total energy of this system because as the spring oscillates, it creates different downward velocity at different part of the spring. The top of the spring is not going to move a lot, it would have less velocity than the part of the spring that is at the end , which would have more velocity. This change of velocity at different part of the spring also requires the use of integration. Our goal would be to come up with a function describing the energy of the spring with a variable, v_end. From then on, we can use loggerpro to help find the kinetic energy of the spring.

Note that the origin is at the top so that the downward velocity would be positive.

a)      In this calculation we begin by finding dm. To find the mass of a piece of spring dm, we divide the mass of the spring, M, by the length of the spring, L. Just as in the last calculation, dm is divided by dy, its height.

b)      Next, we find the function of kinetic energy. The formula for kinetic energy is ½ mv². We start by substituting dm into m  and v_piece into v. The reason it is v_piece is because we are integrating the change of velocity of a piece of spring. Then, we replace v_piece by v_end.  This v_end is multiplied by y/L. Since velocity is not constant throughout the spring, we cannot just take the velocity at the end of the spring as our velocity.   Therefore, we need the ratio of the spring to find the velocity at certain position. y/L is a good ratio because y is the change of length, and it is divided by the total length of the spring. 

c)       We separate the constant from variable. Then we integrate y² from 0 to L. This covers the whole spring. We want to find the velocity of each pieces of the spring.

d)      Finally, we rearrange our function so that it looks like the formula of kinetic energy.

Now, we have our five functions of energy written out in a format that we may plug into loggerpro.

1.       Elastic potential energy (EPE )=  ½ *k* “∆y”²

ü  We may obtain the spring constant, k, through loggerpro or measures by meterstick.

ü  ∆y is the unstretched position of the spring

2.       Gravitational Potential Energy (GPE) =  m*g* “position”

3.       Kinetic energy (KE) = ½ *m* “velocity”²

4.       Gravitational Potential Energy of the spring (GPE_spring) = (M_spring /2) * g * "position"

5.       Kinetic energy of the spring (KE_spring ) =( ½) * (M_spring/3) * “velocity”²

Note that m is the mass connected to the end of the spring; it is not the mass of the spring!
The mass is 0.3 kg and the mass of the spring, M_sping, is 0.087 kg.

The Set-Up:

We secure a long rod vertically by the rim of the table using a table clamp. On the long vertical rod we secure a shorter rod horizontally. Then we put a force sensor at the end of the horizontal rod. The force sensor is zeroed and calibrated so downward motion is positive. A spring is hanged on the force sensor. On the ground, below the spring is a motion detector. Both the force sensor and motion detector is connected to the loggerpro.

 Data Collection:

Force vs position graph

We begin by finding the spring constant, k, of the spring in the apparatus. We keep the spring still then measure it using the force sensor. The force sensor gives us a force vs time graph, we linearized the graph.  Here the slope is the spring constant. The spring constant is 8.487 N/m.

Position vs time graph (Top) & Velocity vs time graph (Bottom)

We find the position of the mass using motion detector. We hang a mass at the end of the spring and let it oscillates, through loggerpro, the motion detector records a sinusoidal graph. With the position data, we manage the loggerpro to give us the velocity graph as well.

Table of all five energy and total energy

 Now, we have the data of the velocity and position of the mass at every time increment as the spring oscillates, we add five columns on loggerpro for the five energy functions. We input the functions, we get all five energy at different time increments.

 Notice that the far right corner is the total energy. The total energy is not constant, it is changing but the difference is just a little, it is this way because there is some uncertainty to this spring- mass system, such as air resistance, that could alter our result by a little.

Also, all five energy, doesn’t matter what time increment we pick, add up to be around the same value , thus, we can say that energy is conserved in this vertically oscillating spring-mass system.

Total energy vs time graph

Finally, we put the total energy data into a graph, we get a sinusoidal graph, almost straight line. The mean is 1.699 J, which the total energy of this system. 

Summary:

In this lab, we explore the topic of energy conservation. We try to find the sum of all energy in a mass-spring system, and we prove that the total energy is the same at every time increment.

We start the lab by finding the function of all the energy that exists in the mass-spring system. We come up with five energy, EPE, GPE, KE, GPE_spring, and KE_spring. We use integration to find the function for GPE_spring and KE_spring. After having all five functions for energy, we begin finding the velocity and position at every time increment. We also find spring constant. Then, we input the functions into loggerpro. We obtain a table that has the amount energy for all five energy types at each time increment. We look at the total energy, it all revolves around 1.699 J. We have successfully prove that energy is conserved in this vertically- oscillating spring- mass system!