1 June 2015: Physical Pendulum Lab

Purpose:

The purpose of this experiment was to find the period of physical pendulums of different shapes that oscillate at small angles.

Apparatus:

Figure 1: Apparatus of the first portion of the experiment

The apparatus of the first part of the experiment involves two ring stands, with a knife edge attached horizontally. The knife edge is used to hold up the ring that is concaved to half its thickness at a part of the inner side. There is another ring stand, with a photogate attached. Tape is attached to the ring so that the photogate can detect the position of the ring. The photogate measures how much time it takes for the physical pendulum to pass the photogate twice. This relates to the period of the pendulum, where it reaches the amplitude in the positive and negative direction from equilibrium and reaches equilibrium. A computer with LoggerPro was also needed to collect the data from the photogate and provide the period of the system.

Figure 2: Apparatus of the second portion of the experiment

The second apparatus of this experiment involves the same setup, except that different physical pendulums are attached. A semicircle by the center of its diameter was pivoted, and it was also pivoted for a second trial by the highest tip of the curve. 

Procedure (Part 1): 

First,  the moment of inertia of the ring was obtained using the following calculation:

Figure 3: Finding moment of inertia of the ring

The ring was treated as a cylindrical shell, which has a moment of inertia of 1/2MR^2 rotating about its center. Then, the parallel axis theorem was used to find the moment of inertia halfway through its thickness. Using the calculated moment of inertia, torque was used to find the angular acceleration of the pendulum while oscillating at small angles. The form of alpha = omega^2 * theta was found, and the angular velocity of the system of obtained using this form. Then, the period was found by dividing 2pi by the calculated angular velocity; the theoretical period was found to be 0.717 seconds.

Next, the actual experiment was performed where the ring was oscillated at small angles, and the experimental period was measured by the photogate. Once the equipment was set up as shown in the apparatus, the experiment was performed, and the result is shown below:

Figure 4: Experimental period of the physical pendulum involving the ring

Discussion (Part 1):

The experimental period of the physical pendulum with the ring was found to be 0.720 seconds. The percent error of the period of this physical pendulum was calculated below:

1.

As can be seen, the percent error in this part of the experiment is very small, indicating that the theoretical period is the true period. It also shows that the assumption that when the angle is small, sin theta is almost equal to theta is also a good assumption. Lastly, it also shows that the method used to obtain the period is also accurate and correct. 

Conclusion (Part 1):

One reason that the experimental period value is slightly larger than the theoretical is due to uncertainties in the measurements of the dimensions of the ring. In addition, there is also uncertainty in the measurement of the period by the photogate. Also, some friction is present in the pivot where the ring is in contact with the knife edge, lowering the angular frequency by a little and resulting in a higher frequency. Lastly, air resistance as the pendulum is oscillating can result in a slower angular velocity and a larger period. As can be seen though, because the percent error is very small, these errors and uncertainties have little effect on the system.

Procedure (Part 2):

First, the center of mass of the semicircle pendulum was obtained, using the procedure shown below:

Figure 5: Deriving the center of mass of a semicircle

Picking dm as a strip of the semicircle, the center of mass in the x direction was zero, based on an origin in the center of the diameter. The center of mass in the y direction was found to be 4R/3pi.

Next, the moments of inertia of the semicircle about the center of the diameter, the center of mass, and the tip of the curve were obtained, using the same procedure as in part 1.

                       Figure 6: Deriving the moment of inertia of a semicircle by the middle of the diameter

Using semicircular shells, the moment of inertia of the semicircle was found to be 1/2MR^2. Next, the moment of inertia of the semicircle at a pivot at the center of mass was found using the parallel axis theorem, and its derivation is found below:

Figure 7: Deriving the moment of inertia of the semicircle about its center of mass

Lastly, the moment of inertia at the tip of the curve of the semicircle was found using the parallel axis theorem again.

Figure 8: Deriving the moment of inertia of a semicircle about the highest tip of the rim

Then, using the derived moments of inertia, the periods for the semicircle pendulum oscillating about the center of the diameter and the tip of the rim were calculated, as shown below:

Figure 9: Deriving period of pendulum oscillating about the center of diameter

The same was performed for the semicircle oscillating about the highest tip of the rim.

Figure 10: Deriving period of pendulum oscillating about the tip of the rim

Next, the shapes were measured and cut out from foam, and the actual experiment was performed, as in part 1, and the actual periods were obtained. The period found for the pendulum about its center of diameter is shown below:

Figure 11: Actual period of oscillation about center of diameter

Likewise, the period is shown below for the pendulum about the tip of the rim:

Figure 12: Actual period of oscillation about tip of rim

Discussion (Part 2):

The percent error between the actual and theoretical values for the period of oscillation about the center of diameter was found to be 4.8%. Likewise, the percent error for the the period of oscillation about the tip of the rim was calculated as -4.7%. The percent error is high for a system such as a physical pendulum. These high percent errors, along with the percent error of the first part, show that the errors in this experiment are resulting from human error, and not error from the path taken to calculate the theoretical values  and the procedure used.

Conclusion (Part 2):  

The reason for such a high percent error is that, due to the lack of time, the physical pendulums were not pivoted correctly, resulting in the wobbling of the pendulums and movement back and forth, instead of only side to side movement like a regular pendulum. In addition, there was some noticeable friction at the pivots, resulting in the physical pendulums slowing down over time during the experimental gathering of data. To make this experiment better, better pivots should be used  for the semicircle pendulums that does not result in the wobbling of the pendulums, and more time would be needed to perfect the experiment. Lastly, more dense material should be used for the pendulums instead of foam), giving the pendulums more control during the oscillations.

20 May 2015: Conservation of Energy/Conservation of angular momentum

Purpose:

The purpose of this experiment was to use the principles of conservation of energy and conservation of angular momentum to determine the final height a system reaches. The system used involves a meter stick pivoted near one end that is released from a horizontal position and is involved in an inelastic collision with clay at the vertical position (bottom of the swing). Determining a theoretical final height similar to an experimental also proves the principles of conservation of energy and angular momentum.

Apparatus:

Figure 1: Apparatus used in this experiment

The apparatus used in this experiment involves a meter stick pivoted near one of its ends, with tape on its other end. A clay piece, which was stood on three paper clips (used as legs), was also used and also had tape wrapped around it. The tape allows for the clay to attach to the meter stick upon collision. The meter stick was pivoted on a rotational sensor, which was used only for its minimal friction during rotation. A metal stand and a horizontal rod clamped to it were used to hold the rotational sensor and the meter stick. 

A camera was also needed to record the collision between the end of the meter stick and the clay as well as the final height it reached. This experimental value for final height was used to prove that the method used to find the theoretical final height is correct. Lastly, a computer with LoggerPro software was also needed to obtain the recording of the experiment and to use video analysis in LoggerPro in order to determine the final height the system in the experiment reached.

Procedure:

First, some masses and dimensions of the system were measured. The mass of the meter stick was found to be 147 g, and the mass of the clay was found to be 28 g. The meter stick was found to be pivoted at 10 cm away from one end.

Once all of the equipment was set up, the actual experiment was performed and recorded, where the meter stick was released from the horizontal position and it collided with the clay when it reached the bottom of the swing. After the inelastic collision, both the clay and meter stick swung up to a final height. After recording, the video analysis software in LoggerPro was used to find the final height the system reached before falling back down, as shown below:

Figure 2: Video analysis of the recording of the experiment

The origin was placed at the point of collision between the meter stick and clay. The final point plotted right before the system starts falling back down to the vertical provides the final height of the system. 

Figure 3: Data obtained from video analysis

Looking at the column of "y (m)" in the data obtained from the video analysis at the last data point (row 14), it can be seen that the final height the system reached was 0.3651 m. This experimental final height will be used to determine if the method used to find the theoretical final height is correct.

Next, the theoretical final height the system would reach, using the dimensions obtained, was found. The method used to obtain this height involves splitting the experiment into three phases, which includes:

1. The use of conservation of energy from when the meter stick starts from rest at the horizontal position and reaches the bottom of the swing, but right before the collision with the clay. This method was used during this phase of the experiment was used to find the angular velocity of the meter stick at the bottom right before the collision.
2. The use of conservation of angular momentum during the inelastic collision between the meter stick and the clay piece. This method was used for this phase in order to find the angular velocity of the clay and meter stick system after the collision.
3. The use of conservation of energy from the point after collision to the final height the system reaches before falling back down to the vertical position. This method was used for this phase to find the theoretical final height the system (clay and meter stick) reaches.

Discussion:

The derivations used to final the theoretical final height are displayed below:

Figure 4: Derivation (Phase 1 and 2) of the theoretical final height the system reaches

First, the moment of inertia of the meter stick was derived since it is used throughout all three phases of the derivation for the final theoretical height the clay and meter stick system reaches. The parallel axis theorem was used to find the moment of inertia of the meter stick when the axis of rotation is 10 cm from one of the edges. This was derived by taking the moment of inertia of a thin rod rotating about its center, and then adding to it the mass of the meter stick multiplied by the distance the pivot moved from the center of mass squared. The moment of inertia about the pivot was found to be 0.03577 kg m^2.

Second, the conservation of energy principle in phase 1 was used. At the beginning of phase 1, there is only gravitational potential energy of the meter stick. At the end of phase 1, there is both rotational kinetic energy of the meter stick and gravitational potential energy of the center of mass of the meter stick. The angular velocity of the meter stick found right before the collision was 5.68 rad/s.

Third, the conservation of angular momentum principle was applied in phase 2 to find the angular velocity of the clay and meter stick system after the collision. Before the collision, there is only angular momentum of the meter stick. After the collision, there is angular momentum of the meter stick and clay piece. The inertia of the meter stick and clay system was found by treating the clay as a point mass, and then adding the moment of inertia of the meter stick to the moment of inertia of the clay. The moment of inertia of a point mass in mr^2, where r is the distance the mass is from the axis of rotation. In this case, it would be the length of the meter stick from the pivot to the end that collides with the clay, or 90 cm. The angular velocity of the meter stick and clay system was calculated to be 3.47 rad/s.

 Figure 5: Derivation (Phase 3) of the theoretical final height of the system

Finally, the conservation of energy principle was used again in phase 3 of the experiment to find the final height the mass and end of the meter stick reaches. The point where gravitational potential energy is zero was placed at the pivot. At the beginning of phase 3, there is rotational kinetic energy of the system and gravitational potential energy of the meter stick. At the end of phase three, there is gravitational potential energy of the clay and the meter stick. The gravitational potential energy is negative in this phase since the center of mass of the meter stick and the clay are found under the point of zero gravitational potential energy. At the beginning, the height from the point of zero GPE of the meter stick is -0.4 m (the center of mass is in the middle of the meter stick, but it is pivoted 10 cm away from the edge). The height for the clay is -0.9 m. At the end, the height for the center of mass of the meter stick is -0.4cos(theta) since the meter stick moves an angle theta above the vertical, and the vertical distance of that movement is cosine of the angle times the hypotenuse, which is -0.4 in this case. The same reasoning is used for the height of the clay at the final height. 

The relationship between theta and h was found using the following method:

Figure 6: Determining the relationship between h and theta

Substituting (h-L)/L for cos(theta) in the expression for phase 3, h becomes the only unknown variable and it can be solved for using algebra. It was found that the theoretical final height of the system was 0.386 m. 

Conclusion:

Comparing the theoretical value (0.386 m) to the experimental value (0.3651 m), it can be seen that they are similar to each other. The percent error of the experimental value is -5.41% (it is negative since the experimental value is less than the theoretical, which is expected), which is small for such an experiment with many sources of uncertainty/error and is satisfactory. Therefore, it can be concluded that the conservation of energy and angular momentum principles are proven to be correct and the method used to find the theoretical final height of the meter stick and clay system is also correct.

The sources of uncertainty/error in this experiment are:

• Uncertainty in the plotting of data points during the video analysis
• Uncertainty in the measurement of the masses of the meter stick and clay as well as the length of the meter stick.
• Error in the assumption that the clay's center of mass is exactly at the end of the meter stick upon collision.
• Friction in the pivot of the meter stick.
• Sound and heat generated upon collision between the meter stick and clay.
• Friction between the paper clips acting as the stands of the clay and the ground upon collision.
• The meter stick used was not perfectly straight; there was some curving in it, affecting the center of mass of the meter stick and the position of the center of mass and the edge of the meter stick not being same in the horizontal position upon collision.

13 May 2015: Finding the moment of inertia of a uniform triangle about its center of mass

Purpose:

The purpose of this experiment was to find the moment of inertia of a right uniform triangular plate about the center of mass in the following orientations:

Apparatus:

Figure 1: Apparatus used in this experiment

The apparatus consisted of the same rotational stand used in the angular acceleration lab. However, in this experiment only one disk was rotated (by allowing air to run through the end), and the same disk was used throughout. The same torque pulley (large) was also used throughout the  experiment. The counts per revolution was also set at 200 as in the angular acceleration lab. In this lab, however, the triangle holder was used instead of a drop pin. The holder was used to hold the uniform triangle used in this experiment from its center of mass, as shown in the figure above. Lastly, the hanging mass was also used to accelerate the disk, as in the angular acceleration lab.

A computer with LoggerPro was also used to collect the data recorded by the rotational sensor on the apparatus for analysis. Lastly, analytical balances and calipers were used to measure the masses of the hanging mass, the uniform triangle, the disk, and the pulley / triangle holder system as well as to measure the dimensions of the pulley, the triangle, and the disk.

Procedure:

First, the moment of inertia of a uniform triangle about its center of mass was derived. This derivation was used to determine the accuracy of the moments of inertia obtained experimentally in this lab. However, the parallel axis theorem was used in this derivation, because it turns out that the calculus involved in finding the moment of inertia directly at the center of mass is complex. Therefore, it was easier to find the moment of inertia of the triangle about its edge and then use the parallel axis theorem to find it about the center of mass. 

Figure 2: Parallel-Axis Theorem for a uniform right triangle

Using the parallel axis theorem, the moment of inertia of the triangle at the edge can be found and then subtracted by the mass of the triangle multiplied by the distance from the edge to the center of mass squared. That distance d, based off of center of mass in the x direction, is b/3, where b is the length of the base of the triangle.

Figure 3: Derivation of the moment of inertia of the right triangle about its center of mass

The moment of inertia about the edge was found to be (1/6)(m)(b^2), and the moment of inertia about the center was found to be a third of that, or (1/18)(m)(b^2). 

Next, the masses and dimensions of the parts of the apparatus were measured. The mass of the triangle was recorded as 456 g. The shorter leg was found to be 9.80 cm, and the length of the longer leg was found to be 14.85 cm. The mass of the large torque pulley used to wind and unwind the string attached to the hanging mass was found to be 36.3 g, and it diameter is 5.36 cm. The mass of the rotating disk used in the apparatus (the large steel disk) was found to be 1.360 kg, and it diameter was 12.630 cm. Lastly, the mass of the triangle holder was found to be 26 g.

Then, once the equipment was all set up, the air was turned on to the apparatus with the hose clamp open so that the bottom and top disk rotated independently of each other. Then, once the string was wrapped around the torque pulley and the hanging mass was at its highest point, the data measurements were started and the mass and disks were released. Using the angular velocity graphs obtained from the experiments, the angular accelerations of the rotating disks as the mass was going upward and as the mass was going downward were found by looking at the slopes of the angular velocity graphs. The angular velocity graphs obtained as well as data obtained for each experiment is found below. In the first experiment, the triangle was not attached; the only attached object was the triangle holder. In the second experiment, the triangle was attached and oriented such that the shorter leg was parallel to the floor. In the third experiment, the triangle was also attached and oriented such that the longer leg is parallel to the floor.

Figure 4: Angular velocity graph of the first experiment

Figure 5: Angular velocity graph of the second experiment

Figure 6: Angular velocity graph of the third experiment

As can be seen, the angular acceleration as the hanging mass was moving up is not the same as when it was going down, due to the fact that there is frictional torque. To lower the effect of the frictional torque onto the results, the same approach will be taken as in the angular acceleration lab, which is to take the average of the magnitudes of the angular accelerations.

Discussion:

The angular acceleration of the system without the triangle attached was found so that the moment of inertia of the triangle in each orientation can be found. Using the angular accelerations, the moment of inertia of all three systems can be found. Because of this, the moments of inertia of the triangle in the two orientations alone can be found by subtracting the moments of inertia of the systems with the triangles by the moment of inertia of the system with no triangle. 

The theoretical moments of inertia of the triangle in each orientation can be found using the derived equation and the dimensions measured. However, to determine the accuracy of this value, the moments of inertia of the triangle in those orientation are found experimentally using the procedure explained above. To find the experimental moments of inertia, the same derived equation was used as in the angular acceleration lab, displayed below,:

1.

where r is the radius of the torque pulley, and m is the mass of the hanging weight. Using the experimental angular acceleration data and the measurements of the dimensions. the actual moment of inertia of the system with no triangle attached was 0.00294 kg m^2, obtained as follows:

2.

To find the moment of inertia of the triangle when the base was parallel to ground, the moment of inertia of the system with the triangle attached was found.Then, the experimental moment of inertia of the triangle was obtained by subtracting the moment of inertia of the system with the triangle and the moment of inertia with the triangle. Using the same method used above from the moment of inertia without the triangle, the moment of inertia of the systems in this experiment are tabulated below.

Figure 7: Experimental moments of inertia of the three systems

Subtracting the moments of inertia of trial 2 and 3 by trial 1, the experimental moments of inertia of the triangle in the two orientations was found:

Figure 8: Experimental moments of inertia of the triangle in the two orientations about its center of mass

It was expected to find that the moment of inertia of the triangle when the longer leg is parallel to the ground (its base) is higher than the moment of inertia when the shorter leg is parallel to the ground. This is because the moment of inertia of the triangle depends on its mass (which stays constant) and its base; the triangle with the longer base, by the equation, should have a larger moment of inertia than the triangle with the shorter base. a sample calculation for finding the experimental moment of inertia of the triangle is found below:

3.

To measure the accuracy of the experimental moments of inertia of the triangle, the theoretical moments of inertia were computed using the derived equation for the moment of inertia of a triangle about its center of mass, and were compared to the experimental values. The theoretical values are displayed below, along with a sample calculation for finding the theoretical moment of inertia.

Figure 8: Theoretical moments of inertia of the triangle in the two orientations about its center of mass

4.

Conclusion:

The theoretical and experimental moments of inertia of the triangle about its center of mass when its shorter leg is parallel to the ground are very similar; there is only a 4.12% percent error of the experimental value, showing that the method used to find the moment of inertia of the triangle in that orientation was accurate and a useful method. This also proves that the moment of inertia derived for a triangle rotating about its center of mass is the correct formula. 

On the other hand, the difference between the experimental and theoretical moments of inertia of the triangle in the orientation where the longer leg is parallel to the ground is fairly large (10.2% percent error). This is mainly due to error in the angular acceleration values obtained during the experiment. In addition, because the values used in this experiment are small, any small change in the angular acceleration would have a large effect on the accuracy of the results and would cause the difference to be larger. The uncertainty in the measurement of the length of the longer leg is very small, having little effect on the error in the experimental moment of inertia. 

Because the experimental value for the moment of inertia of the triangle with its longer side parallel to the ground is larger than the theoretical value, the angular acceleration obtained was smaller than actual. This error could have resulted from friction in the disks due to not having the air turned on as much as needed. Another source of error could be that the triangle was not perfectly oriented such that the longer leg was not parallel to the ground, resulting in a different value range obtained for the moment of inertia. In addition, as seen in the angular acceleration lab, the rotational sensor and the LoggerPro software already contain uncertainties in their measurements of the angular displacement and angular velocity of the rotating disks.

11 May 2015: Moment of Inertia and Frictional Torque

Purpose:

The purpose of this experiment was to experimentally determine the moment of inertia and frictional torque of a large metal disk rotating about a shaft in its center.

Apparatus:

Figure 1: Apparatus of this experiment

The apparatus for this experiment consisted of a large metal disk that rotates about a metal shaft that goes through its center. This The disk and shaft are held up by metal stands attached to a metal plate. A large caliper was also needed to accurately measure the diameter of the large metal disk, and a small caliper was used to measure to the diameter of the central shaft. In addition, a camera was needed to capture the spinning of the disk, and  a computer with LoggerPro was used to collect the video using movie capture and to obtain data from it. Lastly, a 500 gram cart and a 1 meter track were needed to verify the accuracy of the calculated moment of inertia and frictional torque of the system.

Procedure (Part 1):

The dimensions of the large metal disk and the central shaft (the system) were measured using the calipers. Using the dimensions, the moment of inertia of the system was calculated. The derivation is displayed in figure 2 below:

Figure 2: Derivation of the moment of inertia using the measured dimensions

The mass of the system was provided as a stamp onto the side of the rotating metal disk. 

Discussion (Part 1):

Since the system is composed of three different cylinders of different diameters and lengths (the large metal disk, and the two cylinders of the central shaft on each side), the moment of inertia of the system was found by using the equation for the moment of inertia of a cylinder: 

1.

The moments of inertia of the three "cylinders" in this system were calculated separately, then added together. Because this system was composed of the same metal or metal alloy, the mass of each of the three cylinders of the system was obtained by finding how much of the total volume the cylinder consists of, then multiplying it by the total mass to get the mass of that cylinder.

2.

The dimensions measured of each cylinder were used to find each volume of the cylinder, and the total volume of the system was found by summing the volumes of all the three cylinders. The large metal disk, whose volume was pi((0.1000275 m)^2)(.0157 m) = 4.96 x 10^-4 cubic meters, was found to contain around (4.96 x 10^-4 m)/(5.75 x 10^-4 m) = 86.3% of the volume of the whole system, and so its mass was found to be (0.863)(4.808 kg) = 4.15 kg. Using the same approach, the mass of the cylinder of the central shaft that contained the length of 5.00 cm was found to be 0.324 kg (6.74 % of the volume/mass), and the mass of the other cylinder was found to be 0.332 kg (6.96 % of the volume/mass).

Summing the moments of inertia of all three cylinders, the moment of inertia of the system was found to be 0.0209 kg squared meters.

Procedure (Part 2):

Once the moment of inertia of the system was obtained, the frictional torque between the large disk and the central shaft of the system was obtained. The data needed to obtain the frictional torque was taken from a video capture that was performed. The video capture consisted of the recording of the rotating large disk decelerating by the frictional torque in the central shaft. A point on the rim of the disk was used for the analysis of the video capture to obtain the change in the angle that the disk rotates during a set time period. This point was marked by a large dot on a piece of tape attached to the rim of the disk. Once the video capture of the rotating disk was analyzed by plotting points of the position of the dot on the rim using LoggerPro, the data obtained consisted of the x and y positions of the dot as well as the velocity of the dot in the x and y position.

Discussion (Part 2):

Theta, the angle that the dot on the rim moved through during each time period, was found by taking the inverse tangent of the quotient of the y position and the x position, as shown below:

Figure 3: Finding frictional torque in the system

The purple dot is the dot on the rim that was monitored during the video capture. Once theta at each time period was found, a graph of theta versus time was obtained, shown below:

Figure 4: Graph of theta vs time

A quadratic fit was performed for both slopes on the graph. The reason that the graph had two different slope is that each slope counts for one rotation, and two rotations were recorded in the video capture. A quadratic fit was performed because the slope of the graph is related to a rotational kinematics equation, shown below:

3. 

Using the average value of A, the angular deceleration of the disk was found as follows:

Figure 5: Derivation for the angular deceleration of the rotating metal disk

Knowing that torque is the product of inertia and angular acceleration, the frictional torque was obtained using the derivation below:

Figure 8: Derivation of frictional torque

Procedure (Part 3):

To determine the accuracy of the derived frictional torque and angular deceleration resulting from the frictional torque, a 500 gram cart was attached to one of the cylinders of the central shaft using a long string, and a meter metal track was placed 60 degrees above the horizontal from the ground. Calculations, shown below, were performed in order to determine a theoretical time it takes for the cart to roll down the meter track from rest. Then, three trials were performed where the cart, initially at rest, was rolled down the track while attached to the system; the time it took for the cart to travel down the track was measured. It took 6.9 seconds for the cart to reach the bottom of the track for the first and third trial, and it took 6.8 seconds for the second trial.

 Figure 9: Derivation of the theoretical time needed for the cart to travel down the track from rest

The above calculation is performed when the angle was at a theoretical 40 degrees.

Discussion (Part 3):

 First, the major external forces acting on the cart were found. These forces include its weight, the tension from the string, and the normal force. Summing forces in the direction parallel to the track, an expression for tension was found in terms of theta, the carts weight, and acceleration (eq. 1). Note that the acceleration in this experiment is different from the angular deceleration caused by the frictional torque.

Then, the net torque on the system was recorded. The torques acting on the system were the frictional torques and the torque resulting from the tension in the string, which are going in opposite directions. Then, the net torque was equaled to the inertia of the system and the angular acceleration, using that relationship. Then, the torque from the string was put in terms of the tension and the radius it is pulling at from the center, and the angular acceleration was replaced with a/r. Next, the tension of the string was solved for in terms of acceleration, inertia, frictional torque, and radius (eq. 2).

Next, equations 1 and 2 were set equal to each other, and the acceleration was solved for in terms of inertia, radius, frictional torque, theta, and the mass of the cart. Then, using the kinematic equation used for the slope of the theta versus time graph in part 2, the theoretical time for the cart to roll down the track was found.  The calculation shown above for acceleration was done for a theoretical angle of 40 degrees. Using the true angle of 60 degrees, the following calculation is performed:

4.

The acceleration of the cart was found to be 0.0409 m/s/s, and the time it would take for the cart to roll down the track was 6.99 seconds.

As can be seen, the theoretical time and actual time was very close. The difference was 0.09 seconds (1.3%) between trials 1 and 3, and the difference was 0.19 seconds (2.7%) between trial 2.

Conclusion:

The sources of uncertainty in this experiment were very small, making the difference between the theoretical time and actual time it took for the cart to roll down the track from rest very small. The major source of uncertainty that caused the actual time to be slightly less than the theoretical was the slight slipping of the string from around the central shaft at around the end of the trial, where the cart was almost about to get to the bottom of the track. To reduce this uncertainty/error, the central shaft at the edges where the disk does not come into contact can be made a little rougher. Other than that error, all the other uncertainties in this experiment were so small that their error had very little to no effect on the experimental results.

6 May 2015: Angular Acceleration

Purpose:

The purpose of this experiment was to determine the components of applying a known torque on a rotational object that influence the angular acceleration of that object. Using this information, the experimental moment of inertia of the rotating object can be found.

Apparatus:

Figure 1: Apparatus of this experiment

The apparatus of this experiment consisted of a rotational stand, where metal disks are placed on the stand for rotation. Only one disk or multiple disks can be rotated together on the stand through the use of running air through a hose clamp; when the hose clamp is open, the bottom and top disks rotate independently of each other. However, when the hose clamp is closed, both the bottom and top disk will rotate together. In order to allow the disk(s) to rotate with the air on, a drop pin had to be placed in the center of the disks where the rotational stand is. Air is also used to decrease greatly any friction between the disks. Next to the rotational stand is a rotational sensor. On the sides of the disks there are 200 marks, which the rotational sensor reads and uses to measure the change in the angle of the disks (theta) and the angular velocity of the disks (omega). The rotational sensor reads 200 counts each rotation of the disks. On the end of the apparatus is a nearly frictionless pulley, where a hanging weight's string is placed to allow for the smallest energy loss possible in the system. 

A computer with LoggerPro software was also used to obtain and store the data measured by the rotational sensor. An analytical balance was also used to measure the mass of the disks and the torque pulleys (the torque pulleys are placed between the disks and drop pins). Lastly, to measure the diameters of the disks, the following caliper was used:

Figure 2: Large caliper used to measure the diameters of the disks

A smaller caliper was used to measure the diameters of the torque pulleys.

Procedure (Part 1):

The diameters and masses of all the disks and pulleys used in this experiment were measured. The data is shown below:

• diameter and mass of top steel disk: d = 12.630 cm ; m = 1360 g
• diameter and mass of bottom steel disk: d = 12,630 cm ; m = 1348 g
• diameter and mass of top aluminum disk: d = 12.630 cm ; m = 465 g
• diameter and mass of smaller torque pulley: d = 2.81 cm ; m = 10.0 g
• diameter and mass of larger torque pulley: d = 5.36 cm ; m = 36.3 g

Then, LoggerPro was set up and the rotational sensor was hooked up to the computer. Once the equipment was all set up, the air was turned on to the apparatus with the hose clamp open so that the bottom and top disk rotated independently of each other. Then, once the string was wrapped around the torque pulley and the hanging mass was at its highest point, the data measurements was started and the mass and disks were released. Six experiments were performed, with one component changed in each experiment. Experiments 1, 2 and 3 involved changing the weight of the hanging mass, experiments 1 and 4 involved changing the radius of the torque pulley, and experiments 4,5 and 6 involved changing the mass of the rotating disks. Using the angular velocity graphs obtained from the experiments, the angular accelerations of the rotating disks as the mass was going upward and as the mass was going downward were found by looking at the slopes of the angular velocity graphs. The angular velocity graphs obtained as well as data obtained for each experiment is found below:

 Figure 3: Angular velocity graph of experiment 1

α up = 1.080 rad/s/s

α down = 1.189 rad/s/s

  Figure 4: Angular velocity graph of experiment 2

α up = 2.366 rad/s/s

α down = 2.194 rad/s/s

  Figure 5: Angular velocity graph of experiment 3

α up = 4.069 rad/s/s

α down = 3.051 rad/s/s

 Figure 6: Angular velocity graph of experiment 4

α up = 2.312 rad/s/s

α down = 2.080 rad/s/s

Figure 7: Angular velocity graph of experiment 5

α up = 6.938 rad/s/s

α down = 6.238 rad/s/s

Figure 8: Angular velocity graph of experiment 6

α up = 1.157 rad/s/s

α down = 1.056 rad/s/s

  Figure 9: Data obtained for experiments 1 through 6

To ensure that the angular acceleration obtained by the rotational sensor is the actual acceleration value, the acceleration of the hanging mass was obtained using the velocity graph obtained from a motion detector through its slope. This test was performed for experiment 5. The velocity graph is displayed below:

Figure 10: Velocity graph of the hanging mass in experiment 5

a = 0.1652 m/s/s

Discussion (Part 1):

It was expected that the angular acceleration when the hanging mass was accelerating upward was larger than the angular acceleration when the hanging mass was accelerating downward. This is because the net torque on the rotating disk is larger when the hanging mass is accelerating upward than when accelerating downward. When the hanging mass is accelerating upward, the torque applied on the rotating mass by the tension in the string of the hanging mass is in the same direction (clockwise) as the torque from friction (since both are slowing down the rotating disk), resulting in a larger net torque. A larger net torque results in a larger angular acceleration (torque = moment of inertia x angular acceleration) since moment of inertia is constant in this experiment. On the other hand, when the mass is accelerating downward, the torque from the hanging mass is in the opposite direction of the torque from friction; the torque from the mass is speeding up the rotating disk while the torque from friction is slowing it down. Therefore, a smaller net torque is obtained. Friction in the system is seen from the graphs; the slopes are not perfectly straight lines in that there is some small zagging up and down in the graphs. Therefore, in order to help cancel the effect of friction on the system, the average angular acceleration for each experiment was taken by taking the average of the upward and downward angular acceleration.

Looking at experiment 5, in order to determine if the acceleration measured is the true acceleration, the relationship between angular acceleration and linear acceleration can be used to find the "experimental" radius, and see if that radius agrees with the measured radius of the pulley. Doing so, 

1. 

The radius of the large torque pulley used for that experiment was 2.68 cm. The values are fairly close, proving that the acceleration data gathered is the true acceleration of the rotating mass. The reason for the small difference between the two radii is uncertainties in measurements as well as the friction in the system. 

Taking a look at experiments 1 through 3, it can be seen that the hanging mass does affect the rotational acceleration of the disks. when the mass is nearly doubled (from 24.6 g to 49.6 g), the angular acceleration nearly doubles (from 1.135 rad/s/s to 2.280 rad/s/s), and when it triples (from 24.6 g to 74.6 g), the angular acceleration nearly triples (from 1.135 rad/s/s to 3.560 rad/s/s) as well. Therefore, the hanging mass is directly proportional to the angular acceleration of the rotating disks.

Looking at experiments 1 and 4, it can be seen that the radius of the torque pulley also has an effect on the angular acceleration. When the radius of the pulley was nearly doubled (from 1.40 cm to 2.68 cm), the angular acceleration nearly doubled (from 1.135 rad/s/s to 2.196 rad/s/s). Therefore, the radius of the torque pulley is proportional to the angular acceleration of the rotating disks. 

Lastly, looking at experiments 4 through 6, it can also be seen that the mass of the rotating disks affects the angular acceleration. when the mass was made about three times lighter (from 1360 g to 465 g), the angular acceleration was nearly tripled (from 2.196 rad/s/s to 6.588 rad/s/s). In addition, when the mass was nearly doubled (from 1360 g to 2708 g), the angular acceleration was about twice as slow (from 2.196 rad/s/s to 1.107 rad/s/s). Therefore, the rotating mass is inversely proportional to the angular acceleration. 

Procedure (Part 2):

Using the data obtained from part 1, the moments of inertia of the disks or disk combinations in each experiment were found. The experimental values for moment of inertia were found by using the derived equation shown below:

2. 

For comparison of the experimental moment of inertia to see its accuracy, the actual moment of inertia of the rotating mass was found using the following formula:

3.

The calculations are shown below:

 Figure 11: Experimental and actual moments of inertia of exps 1/2

  Figure 12: Experimental and actual moments of inertia of exps 3/4

 Figure 13: Experimental and actual moments of inertia of exps 5/6

Discussion (Part 2):

Looking at experiment 1, the percent error between the experimental value of moment of inertia (0.00298 kg m^2) and the actual value (0.00271 kg m^2) is 9.96%. Looking at experiment 2, the percent error between the experimental value of moment of inertia (0.00299 kg m^2) and the actual value (0.00272 kg m^2) is 9.93%.  Looking at experiment 3, the percent error between the experimental value of moment of inertia (0.00287 kg m^2) and the actual value (0.00271 kg m^2) is 5.90%. Looking at experiment 4, the percent error between the experimental value of moment of inertia (0.00292 kg m^2) and the actual value (0.00271 kg m^2) is 7.75%. Looking at experiment 5, the percent error between the experimental value of moment of inertia (0.000964 kg m^2) and the actual value (0.000927 kg m^2) is 3.99%. Lastly, looking at experiment 6, the percent error between the experimental value of moment of inertia (0.00583 kg m^2) and the actual value (0.00540 kg m^2) is 7.96%. 

Although the percent error varies between the experiments, the percent error was less than 10%, indicating that the values are accurate and that the error is due to uncertainty. There is more percent error in some experiments than others, showing that there is more uncertainty in some experiments than others. These uncertainties are found in measurements of dimensions (different equipment used to measure dimensions of different parts of the apparatus) as well as varying friction (due to the use of different disks or disk combinations and the use of different torque pulleys).

Conclusion:

There are some sources of uncertainty/error in this experiment. These uncertainties/errors include:

• uncertainty in the measurement of the diameters of the disks and torque pulleys
• uncertainty in the measurement of the mass of the disks and torque pulleys
• the use of different equipment with different disks and torque pulleys depending on the size and mass (using different analytical balances and calipers between the larger disks and the smaller disks and torque pulleys)
• uncertainty in the measurement of the angular acceleration of the rotating mass by the rotational sensor
• uncertainty in the measurement of the linear acceleration of the hanging mass by the motion detector
• friction between the disks, and different friction between the disk and the rotational stand

In order to lower the uncertainty, better equipment would be needed. In addition, higher experimental values would also be needed to lower the effect of error and uncertainty on the results.