11-May-2015: Moment of Inertia and Frictional Torque

Purpose: Use appropriate measurements to calculate the total moment of inertia and frictional torque in the system,  and use their relationship to calculate the time it takes a cart to travel 1 meter.

Procedure: Given a large metal disk on a central shaft, we were to measure the mass, length, and radius of the disk and shaft in order to find the total moment of inertia. The mass of the entire disk and shaft was labeled in grams on the side but we needed the mass of each individual piece. Given that both parts were of the same density, we solved for the volumes of each part to figure out the percentage of the total mass each part was. Once we found the mass of each piece, we were able to solve for the total moment of inertia by adding up each individual moment of inertia. The inertia of each cylinder is given as  I = 1/2*M*R^2

Calculations for total moment of inertia

Calculating for total moment of inertia, we got I = 0.021 Nm.

Then, we were to solve for the frictional torque acting the apparatus. The trick to this part was finding an easy way to measure the angular acceleration of the disk needed for the calculation. By using video capture to record a specific amount of rotations the disk made over a period of time, we were able to plot a graph of theta over time. 

graph of theta vs time from video capture.

Solving for the 2nd derivative of the function of theta vs. time gives us the angular acceleration of the apparatus.

Calculations for angular acceleration of disk

Our calculations gave us Alpha = -1.1158 rad/sec^2.

Next, we connected the apparatus to a 500 gram dynamic cart by winding it with a string. The cart is placed on an inclined track so that it can roll down a distance of 1 meter. We recorded the angle of the incline of the track to be 58.1 degrees from the horizontal. Our objective from here was to use our measurements of the total moments of inertia and angular acceleration to calculate how long it should take for the cart to travel 1 meter from rest, and compare our result to the actual experimental trails.

Cart on inclined track attached to disk/shaft apparatus

Below is our calculations for finding the amount of time it should take, theoretically, for the cart to travel 1 meter. In order to find out how long it takes, we first had to find out the cart's acceleration, and before finding that, we had to create a set of equations using Newton's second law. We have our calculation for frictional torque using T(frictional) = I*Alpha. We drew our rigid body diagram of the system in order to write an equation for our forces in the x and y direction. Having 2 unknowns, T(radial) and the acceleration of the cart, we had to solve for T(radial) using the equation for net torque, T(radial) - T(frictional) = I*Alpha .  Using the relationship between angular acceleration and linear acceleration Alpha = a/r , we can substitute to get a singular equation solving for the acceleration of the cart. Finally, using kinematics we can use the equation distance = 1/2*a*t^2 to solve for time.

Calculations for time (t) it should take for cart to travel 1 meter down incline.

Results: The calculated result for the amount of time it should take the cart to travel 1 meter down the track came out to 8.58 seconds. We then used a stop watch to measure how long it actually took for the cart to travel 1 meter in three trials. The average of the three trials came out to 8.04 seconds.  The average of our measured results was 6.3% off of the calculated result.

Conclusion/Uncertainty: By measuring specific characteristics of the disk/shaft apparatus, we were able to solve for its total moment of inertia. Then, by using a video capture method to solve for angular acceleration, we were able to solve for the frictional torque of the apparatus. Finally, by using those values and understanding of Newton's second law, we were able to solve for the acceleration of the cart and eventually the time it took the cart to travel 1 meter. Although we were not within the 4% mark, it was still reasonably close. There were a few aspects that could have skewed our final results including the human error on the stopwatch, friction between the cart and the track, and the propagated uncertainty compounded from our measuring utensils.

13-May-2015: Finding the Moment of Inertia of a Uniform Triangle About its Center of Mass

Purpose: To determine the moment of inertia of a right triangular thin plate around its center of mass for two perpendicular orientations of the triangle. Then compare the experimental and theoretical results for the moment of inertia of the triangle in each of the two orientations.

Procedure: The apparatus consisted of two rotating disks adjacent to a Pasco rotational sensor. We attached a torque pulley on top of the two disks with a string wrapped around it. The other end of the string goes over another pulley and hangs off the edge with a hanging mass. A triangular thin plate was attached on top of the torque pulley with a mounted holder. Turning on the compressed air allowed the disk(s) to rotate freely without friction, which caused the hanging mass to oscillate up and down.

Lab apparatus, disk/holder/triangle system with hanging mass

To approach this experiment, we had use our understanding of the parallel axis theorem, which states that

I(parallel axis) = I(around cm) + M*d^2, where d is the displacement from the original cm and M is the mass.

Because the limits of integration are simpler if we calculate the moment of inertia around a vertical end of the triangle, you can calculate that moment of inertia and then get the moment of inertia around the center of mass with

I(around cm) = I(around one vertical end of the triangle) + M*d^2, where d is the displacement from the original cm and M is the mass.

We had to derive the moment of inertia of the triangle around its center of mass and use the parallel axis theorem to find the moment of inertia around its new axis. The derivation for this is below.

Derivation of moment of inertia of triangle around its new axis

The new moment of inertia of the triangle came out to I = 1/18*M*R^2. In order to solve for the moment of inertia of the right triangular thin plate, we had to first measure the the triangle's mass M, horizontal length R1, and vertical length R2. These came out to M=0.455 kg, R1=0.09853 m, R2=0.14950. R would depend on the orientation of the triangle during the experiment.

Next was the experimental portion of the lab. The moment of inertia of the entire disk/holder/triangle system can be solved for by knowing the torque exerted by the tension from the hanging mass and the angular acceleration of the system produced by the torque. Newton's 2nd law relates these values with

Torque = Tension*radius = I*alpha  

So, by finding the total moment of inertia of the entire system and subtracting it by the moment of inertia of just the disk/holder, we can get the moment of inertia of just the triangle. This moment of inertia will be our experimental value of I. We first recorded the angular acceleration and the moment of inertia of just the disk/holder, then added the triangle to the system and measured the angular acceleration and moment of inertia for the entire system. 

Note: Because there is some frictional torque in the system (the disk is not completely frictionless and the frictionless pulley is not massless), the angular acceleration of the system is not the same when it is descending as when it is ascending. For this reason we had to derive an equation which took into account the frictional torque. The derivation is below

derivation of experimental moment of inertia

We began recording the data of the angular acceleration of just the disk/holder system. Angular acceleration was found by finding the slope of the angular velocity vs. time graph below.

graph of angular velocity vs time for disk/holder system

We then took the average angular acceleration going up and going down, then found the average of the two. Then using our derived equation, and our measured mass and radiuses, we were able to calculate the experimental moment of inertia of the disk/holder system.

experimental moment of inertia of disk/holder system

Next, we added the triangle to the system in the vertical orientation and began recording the angular acceleration of the disk/holder/triangle system. Once again, we found the angular acceleration using the slope of the angular velocity vs time graph. 

Graph of angular velocity vs time for disk/holder/triangle system (vertical)

Taking the average acceleration, we were able to calculate the moment of inertia for the disk/holder/triangle system (vertical).

experimental moment of inertia of triangle (vertical) and theoretical moment of inertia

By subtracting our disk/holder/triangle system inertia with the disk/holder system inertia, we were able to get the experimental moment of inertia of just the triangle which came out to 2.09x10^-4 kgm^2. Then by using the theoretical equation we derived earlier of I = 1/18*M*R^2, we were able to find the theoretical moment of inertia which was 2.44x10^-4 kgm^2. Our percent error came out to 14.3%

Lastly, we turned the triangle so that it was horizontal and again measured the system's angular acceleration using the slope of the angular velocity vs time graph below.

graph of angular velocity vs time for disk/holder/triangle system (horizontal)

Then, using that angular acceleration and our measured values of M and the disk's r, we were able to calculate the triangles experimental moment of inertia (horizontal).

experimental moment of inertia of triangle (horizontal) and theoretical moment of inertia

Comparing the experimental value for moment of inertia of the triangle (horizontal) I =5.32x10^-4 with the theoretical value I=5.65x10^-4 gave us a percent error of 5.84%.

Conclusion/Uncertainty: Through this equation, we were able to derive an equation for the triangle's moment of inertia on using the parallel axis theorem. Using our data from measuring the disk's angular acceleration and dimensions and properties, we were able to calculate the experimental value for the triangles moment of inertia. Then, by comparing our experimental results for moment of inertia and the theoretical value using our derived equation, we were able to see how accurate we were able to get using our understanding of newton's second law and the parallel axis theorem. Our results, even though the percent error was a little larger than we had hoped, was still close enough to verify our experiment.

A few errors occurred during our experiment beginning with the fact that our vertical triangle was not perfectly vertical. We did not catch that until after we recorded our data and may be why our percent error for the vertical was large compared to that of our horizontal results. Other things such as the friction from not having a perfectly clean disk and slight error in measurements of the triangle may have skewed our results a bit.

 

04-May-2015: Angular Acceleration

Purpose: To apply a known torque to a disk (or disk combination) and measure its angular acceleration in order to calculate the object's moment of inertia. Then compare the results to the theoretical values for the objects' moment of inertia.

Procedure: The apparatus consists of an axis where 2 disks stack and rotate. Adjacent to it is a Pasco rotational sensor. On top of the disks, we attach a torque pulley with a string wrapped around it. The other end of the string goes over another pulley stationed at the edge and hangs a mass off of the table. Turning on the compressed air allows either just the top, or both disks to rotate, which causes the hanging mass to oscillate up and down. We had to clean the disks with alcohol so that we prevent friction from causing any erratic results.

Lab apparatus

Part 1: We wanted to perform 6 trials, using different sized pulleys and disks (or disk combinations), and record the angular acceleration produced by each set of circumstances. In addition we wanted to use one of our trials and look at v (hanging mass) vs. ω (disk) and a (hanging mass) vs. α (disk).

To begin the experiment, we had to first measure the various masses and diameters of the disks and pulleys, as well as the mass of the hanging weight. 

Measured values of objects using caliper and electronic scale

Next, we turned on the Pasco rotational sensor and connected it to our computer and opened up logger pro. Because there is no defined sensor for this rotational apparatus, we had to create something that can work with the equipment. We chose Rotary motion and set the equation in the sensor settings to 200 counts per rotation to match the 200 marks on the disks. Collecting data from a rotating disk gives us the graphs of angular position, angular velocity, and angular acceleration vs. time. The latter graph was useless due to the poor timing resolution of the sensors.

We checked the hose clamp on the bottom to make sure it was closed so that the bottom disk would rotate independently of the top disk when the drop pin was in place. Then we turned on the air, just enough to keep the disks smooth and spun the disks freely to test the equipment. Finally, with the string wrapped around the torque pulley and the hanging mass at its highest point, we began measurements and released the mass. Using the slopes on graphs of angular velocity gave we were able to measure the angular acceleration as the mass moved up and down.

It was important that we take the average angular acceleration of each trial because when the hanging mass went up, there is a frictional torque and torque from the string slowing down the the disk. And when the mass goes down, there is a frictional torque slowing down the disk and a torque from the string speeding up the disk. So we had to add up the angular acceleration going up and going down and divide it by 2.

The objective was to look at the effects of various changes in the experiment on the angular acceleration of the system.
     - Trial 1, 2, and 3: Effects of changing the hanging mass.
     - Trial 1 and 4: Effects of changing the radius at which the hanging mass exerts a torque.
     - Trial 4, 5, and 6: Effects of changing the rotating mass.

Below, we have the graphs for angular angular acceleration for trials 4 and 5.

graph of angular velocity vs. time with angular acceleration given as slope for trial 4

graph of angular velocity vs. time with angular acceleration given as slope for trial 5

Results: As you can see, the motion for both trials were very similar, except that the angular acceleration for trial 5 was about 3 times larger than that of trial 4. This was interesting because trial 5 had a disk with a mass roughly 3 times smaller than that of the disk in trial 4. 

Here is the data for all six trials we performed.

data set for 6 trials

We decided for the sixth trial we would look at v (hanging mass) vs. ω (disk) and a (hanging mass) vs. α (disk). In order to measure the hanging mass's linear velocity, we placed a motion sensor directly below the hanging mass.

lab apparatus + motion sensor

Below we have our graphs for angular velocity of the disk the and linear velocity of the hanging mass. Finding the slope of the graphs gave us the angular acceleration of the disk and the linear acceleration of the hanging mass. 

graphs of angular velocity vs. time and velocity vs. time

Conclusion part 1: After performing the 6 experiments and reviewing our results, we were able to see how a change in the mass of the disk, the size of the torque pulley, and the amount of mass hanging effect the angular acceleration of the disk(s). Changing the hanging mass from 25 grams to 50 grams caused the average angular acceleration to change from 0.454 rad/s^2 to 0.926 rad/s^2. Doubling the hanging mass doubled the average angular acceleration. And changing the mass from 25 to 75 grams effectively tripled the average angular acceleration. Changing only the size of the torque pulley, from 2.836 cm diameter to a 5.380 cm diameter pulley, resulted in a change in average angular acceleration from 0.454 rad/s^2 to 0.872 rad/s^2. And changing the mass of the disks, from a 1360 gram steel disk to a 466 gram aluminum disk, caused the average angular acceleration to increase from 0.872 rad/s^2 to 2.489 rad/s^2.

Part 2: Use our data from part 1 to calculate an experimental moment of inertia and compare it to a theoretical moment of inertia.

Because there is some frictional torque in the system, the angular acceleration isn't the same when the mass ascends and descends. Below is our derivation of how we will calculate an experimental moment of inertia. The theoretical moment of inertia is simply the moment of inertia of a disk. 

Deriving experimental moment of inertia 

Results: Using those two equations, we were able to input our measurements and compare the experimental and theoretical moments of inertia below. 

experimental vs theoretical moments of inertia

Conclusion part 2: Our results showed that our experimental moment of inertia for each trial was significantly different from the moment of inertia we should have gotten theoretically. 

Final Conclusion/Uncertainty: Part one of our experiment required that we graph the angular velocity of 6 different trials with various changes in either hanging mass, pulley size, and mass of disks to see how it effected the average angular acceleration of the disks. We also set up a motion sensor to compare the linear velocity of the hanging mass with the angular velocity of the disk and the linear acceleration of the hanging mass with the angular acceleration of the disk. The results showed that increasing the mass hanging and the size of the pulley increased the angular acceleration, while increasing the size of the disk decreased the angular velocity. Part two required that we use our results from part one to calculate our experimental moments of inertia using our derived expression and compare it to the results using the theoretical moment of inertia. These results however were not very accurate, which could be due to a few factors. 

One reason our results for our experimental moments of inertia may be so off can be because we may have not cleaned the disks well enough to produce a nearly frictionless experiment. Also knots that we could not get out of the string could have slowed down the angular acceleration of the disks as it spun. Finally, there are various factors including the age of the device we were using, the integrity of our measurements and measuring tools, and energy lost to friction from the pulley.

27-Apr-2015: Ballistic Pendulum

Purpose: To measure the velocity of a metal ball shot into a ballistic pendulum using our knowledge of conservation of momentum and conservation of energy.

Procedure: We were given a device which had a spring loaded gun, a metal ball, and a ballistic pendulum. Firing the metal ball out of the gun and into the pendulum causes the pendulum to swing back in order to absorb the impact. The angle at which the pendulum recoils is recorded by a bar behind the pendulum. The idea was to measure certain variables in order to be able to solve for the velocity of the ball as it left the gun.

Ballistic Pendulum

So to begin the lab, we measured the mass of the ball (m), the pendulum (M), the length of the pendulum from its pivot (L), and the angle theta recorded by the bar after we shot the ball.

ballistic pendulum angle reading

m = 0.00763 kg +/- 0.0001
M = 0.0809 kg +/- 0.0001
L = 0.2000 m +/- 0.001
theta = 16.5 degrees +/- 0.00873 rad







Using our measurements and the conservation of momentum theory we then calculated for the velocity of the ball as it left the gun by using our equation. We understood that this was a elastic collision so we knew that momentum and energy was being conserved so by solving for the component of the distance the pendulum traveled and using our other known values, we could come up with the initial velocity of the ball. We also used the propagated uncertainty from our measuring devices by finding the partial derivative of each variable in relation to our equation to find a range of values for the velocity of the ball.

calculations for v +/- dv

Results: Our calculations gave us a velocity of 4.66 m/s +/- 0.2131, which sounds very reasonable. 

Conclusion/Uncertainty: By understanding the conservation of momentum in a collision from a ball and ballistic pendulum, we were able to solve for the initial velocity of the ball before hitting the pendulum. This was an elastic collision where momentum and energy were conserved. We did solve for uncertainty in our velocity but there still may be other errors that are unaccounted for. Things such as heat from friction from the ball and even the integrity of the pendulum can slightly effect the results. But overall the results do seem fairly reasonable. 

22-Apr-2015: Collisions in two dimensions

Purpose: To look at two-dimensional collisions using a marble ball and steel balls in order to determine if momentum and energy are conserved.

Procedure: We have a flat square surface where we observe the elastic collisions between two steel balls and another elastic collision between a steel ball and a marble ball. A camera is attached to a horizontal rod and hangs over the square surface by a vertical rod.

Lab apparatus of collisions in two dimensions

camera for video capturing the collisions

To set up the camera, we open up logger pro and go to the video capture tab. We click "camera settings", then "adjustments", then "image". From here we set the shutter to zero ,reduce the exposer, and increase Gain so that we get a reasonable image from the camera onto the screen.






For the experiment, we start by setting the stationary ball around the middle of the leveled glass table. We wanted to aim the rolling ball so that it hits the side of the stationary ball, so that we create a two dimensional collision where the balls roll off at some decent angle from one another. The aim is record the position over time of the two balls in order to test whether or not momentum and energy were conserved in the two-dimensional collision. The equation for the conservation of momentum and conservation of energy are as follows, respectively.

M1*U1 + M2*U2 = M1*V1 + M2*V2

1/2*M1*U1^2 + 1/2*M2*U2^2 = 1/2*M1*V1^2 +1/2*M1*V2^2 

The first collision involved two steel balls of the same mass. We measured both masses to be equal at 66.6 g. We roll the ball into the side of the stationary ball and record so we can plot the position at each corresponding time on the video capture.

video capture of steel vs steel balls with point series

After recording the collision, we create an an axis so that it matches up with our collision and set the origin at the point of collision between the balls. Then we create a point series for both balls for the duration of the experiment. Using these points we can create a graph of the x and y positions over a substantial time period.

graph of x and y positions of both balls over time

We then collect the data for the initial velocity of the rolling steel ball and the final velocities of both balls after the collision in order to calculate whether momentum and energy were conserved or not.

Our calculations showed that momentum and energy were both in fact conserved.

For the second collision we wanted to see whether momentum and energy are conserved between two balls of different masses. We use the marble ball as the stationary ball and roll the steel ball into its side and capture the collision.

video capture of steel ball colliding with the marble ball

Using those points, we were able to plot the graph of the x and y positions over a substantial time period.

graph of x and y positions of both balls over time

Finding the initial velocity of the rolling ball and the final velocity of the two balls helped us to test whether momentum was conserved in this collision. Our results verified that momentum and energy were both conserved.

Conclusion/Uncertainty: Using video capture to record the position of the balls over time gave us the velocity of the balls before and after the collisions. The velocities of the balls before and after the collisions and the masses of the balls were inputed into the conservation of momentum and energy equations to see whether momentum and energy were conserved in these 2 dimensional elastic collisions. Our values came out very close, verifying that momentum and energy were in fact conserved.

Uncertainties in the lab came in tracking the balls using the video capture because sometimes it was hard to be accurate with the plotting of the points. Human error in measuring masses also skew the results slightly.

15-Apr-2015: Impulse-Momentum Activity

Purpose: Using three impulse and momentum experiments, we were to convince ourselves that the impulse applied to an object is the area under a graph for a force acting over an interval of time and is equal to the change in momentum of the object after the collision.

Experiment 1: Observing Collision Forces That Change With Time

Procedure: To begin we set up our track so that a cart with a force sensor attached to it would collide into a springy bit of another cart clamped on the other end. The force cart has a rubber stopper replacing the hook on the sensor and springy cart is attached to a rod which is clamped to the table The carts should be set up such that the collision between the two carts happens with the rubber stopper and the springy bit. We did a couple test runs to make sure there was solid contact between the two ends. Then we placed a motion sensor on the force sensor cart side, with the positive direction reading in opposite direction of the cart's initial velocity.

The impulse-momentum theorem states that the impulse acting on the cart is equal to the amount of momentum change for the moving cart. So, by finding the change in momentum using the mass of the cart and the velocities before and after the collision, and the impulse by finding the area under the force vs. time graph, we can test whether impulse(J) = change in momentum (P).

Lab set up with carts placed at opposite ends.

Before starting the experiment we made sure that the force probe was calibrated horizontally and vertically. Then checked to make sure the motion sensor was reading the cart all the way through without any interference. We then began recording and graphing our data.

graph of velocity vs. time in green and force vs. time in blue

We compared our integral of the force vs. time graph to our change in momentum using the velocities from the velocity vs. time graph.

Results comparing impulse and change in momentum in an elastic collision

Results: Our results show that the Impulse = -.2268 which is nearly equal to the change in momentum =-.2452. Because the spring was not perfect, we can say that our results confirm we had a nearly elastic collision.

Experiment 2: A Larger Momentum Change

Procedure: We use the same set up as the first experiment, except we added 0.5 kg of mass to our cart in order to get a larger momentum change. We wanted to test whether the impulse and change in momentum still equaled to each other when using a more massive cart. We check to make sure the collision is solid again then begin recording our data.

graph of velocity vs. time in green and force vs. time in blue 

Using logger pro we found the integral of the force vs. time graph and compared to our change in momentum using the velocities before and after collision from green graph.

Results comparing impulse and change in momentum in elastic collision + mass

Results: Our results show that the impulse = -.827 which is very close to our change in momentum = -.793. Again because the spring is imperfect, we expected our result to not exactly reflect an elastic collision.

Experiment 3: Impulse-Momentum in an Inelastic Collision

Procedure: For this part, we wanted to examine the impulse-momentum theorem in a collision where the cart sticks to a wall and stops after the collision. We emulate this by replacing the springy bit on and the cart with clay attached to a vertical piece of wood clamped to the table, and replacing the rubber with a nail. We leave the extra 0.5 kg of mass on the cart and set up the track for the next collision. We begin recording data and wanted to see whether the impulse acting on the cart is equal to the change in momentum in inelastic collisions as well. 

graph of velocity vs. time in green and force vs. time in blue

Again we calculate the change in momentum and compare to the integral of the force vs. time graph for an inelastic collision.

results comparing impulse to change in momentum for an inelastic collision

Results: The impulse acted on the cart = -.408 which is very close to the change in momentum of the cart = -.371. Proves that J=change in P for inelastic collisions.

Conclusion/Uncertainty: After experimenting with two elastic collisions and one inelastic collision, we confirmed that the total net impulse acted on a cart, which is found by finding the integral of force over a interval of time, is in fact equal to the change in momentum of the cart for both elastic and inelastic collisions. The impulse-momentum theorem which states that J=change in P was accurately represented by all three of our results. However, our results were not perfectly equal due to a few possible uncertainties in our experiment. First, our track was not perfectly frictionless like we assumed in our calculations. Secondly, the spring bit for the first two experiments did not create a perfectly elastic collision. Also things like the accuracy our scale to measure the mass of the cart is +/- 1 g. But overall the experiments accurately represented the model that we were testing.