Tuesday, June 9, 2015
Lab #20: Conservation of Energy and Angular Momentum
Purpose: Using the conservation of energy and momentum we will predict the height a system will swing up to after experienceing an inelastic collision.
Procedure: We set up C-clamps, bars, other clamps and a rotation sensor for our meterstick to rotate on. We also wrapped tape around the swinging end of the meter stick and tape around a small mass of clay that will stick to meter stick. The clay mass will be positioned directly where the meter stick will be at the bottom of its swing. We also set a small block of wood behind where the collision will take place as a marker.
This experiment will be broken up into three seperate parts for calculations:
Part 1 is the conservation of energy when we transfer our potential energy from our horizontal position to the kinetic energy the ruler will experience at the bottom of its swing.
Part 2 is the conservation of momentum of the swinging ruler having an inelastic collision with the mass of clay.
Part 3 is conservation of energy again but this time starting with the kinetic energy of the swinging system transfering into gravitational potential energy at its mas swing height.
One very important thing that we had to take account of during these calculations was that the center of mass of the ruler was not a perfect .5 meters away from the pivot. To adjust for this we used the parallel axis theorem for the shift whenever we calculated inertia of the meterstick. Likewise the mass of clay would not be a perfect 1m away from the pivot, but all we had to adjust for this was the distance we used because it is a point-mass.
Finally after our calculations we came up with a prediction for the max angle and height of the swing. Now we set up our camera and logger-pro to capture and analyze the experimental run.
We compared our theoretical value to what we collected in logger pro and calculated our percent error.
Our percent error was fairly small and possibly due to things like not releasing the meterstick from a perfectly horizontal position, the clay not beint a perfect point mass and the motion capture also distorting the data just a bit.
Overall it is fairly easy to show that both energy and momentum were conserved it this inelastic collision and in an entertaining way where we get to swing a meter-stick into a small mass.
Procedure: We set up C-clamps, bars, other clamps and a rotation sensor for our meterstick to rotate on. We also wrapped tape around the swinging end of the meter stick and tape around a small mass of clay that will stick to meter stick. The clay mass will be positioned directly where the meter stick will be at the bottom of its swing. We also set a small block of wood behind where the collision will take place as a marker.
This experiment will be broken up into three seperate parts for calculations:
Part 1 is the conservation of energy when we transfer our potential energy from our horizontal position to the kinetic energy the ruler will experience at the bottom of its swing.
Part 2 is the conservation of momentum of the swinging ruler having an inelastic collision with the mass of clay.
Part 3 is conservation of energy again but this time starting with the kinetic energy of the swinging system transfering into gravitational potential energy at its mas swing height.
One very important thing that we had to take account of during these calculations was that the center of mass of the ruler was not a perfect .5 meters away from the pivot. To adjust for this we used the parallel axis theorem for the shift whenever we calculated inertia of the meterstick. Likewise the mass of clay would not be a perfect 1m away from the pivot, but all we had to adjust for this was the distance we used because it is a point-mass.
Finally after our calculations we came up with a prediction for the max angle and height of the swing. Now we set up our camera and logger-pro to capture and analyze the experimental run.
We compared our theoretical value to what we collected in logger pro and calculated our percent error.
Our percent error was fairly small and possibly due to things like not releasing the meterstick from a perfectly horizontal position, the clay not beint a perfect point mass and the motion capture also distorting the data just a bit.
Overall it is fairly easy to show that both energy and momentum were conserved it this inelastic collision and in an entertaining way where we get to swing a meter-stick into a small mass.
Wednesday, May 27, 2015
Lab #18 Inertia of a Rotating Triangle
Purpose: To find the moment of inertia of an irregular shape about a central axis. We will do this both theoretically by setting up integrals and experimentially by using a rotating air table.
Procedure: Because this triangle has an irregular shape we do not have an inertia formula that we can just plug values into. We will have to create our own by setting up an integral where we selected a vertical slice of the shape as our dm and integrated from the tall edge to the point. However, by doing this we found the inertia as if we were rotating about the outter edge. Thankfully we can now use the parallel axis theorem to find the moment of inertia and solve for the inertia about its central axis!
Now for the eperimental calculations we set up an air table with rotating disks and a hanging mass like in our previous angular acceleration/ inertia lab. The difference is that now we already know what the moment of inertia for the disk should be, so when we set up a system with the additional tiangle mounted on top we can subtract the inetria provided by the disk to find the inertia of the triangle.
We set up logger pro to collect and graph data on the angular velocity of the rotating system. We had logger pro analyze linear fits for the velocity vs time graphs and we averaged the values to find out angular acceleration.
Now that we have collected all the values we need to solve for the system and triangle inertia, we set up tangential and angular equations and manipulated them to solve fot the total inertia of the system. As stated before we calculated the inertia of the triangle by subtracting the inertia of the disk from the inertia for the entire system.
Once finding Our theoretical value for the triangle's inertia we compared it to the theoretical value and came up with very small percent errors. These errors could be due to the air pressure not being set perfectly or minor frictions throughout the system.
Lab #17: Moment of Inertia and Frictional Torque
Purpose: To calculate the moment of inertia of a large disk and two cylynder system that spins around its central axis. We also calculated the angular decceleration due to frictional torque. Lastly we used these and other concepts to predict how long it would take for a cart to travel a certain distance when wound around the small radius of the system.
Procedure: To find the moment of inertia for the disk-cylinder system we broke down the object into pieces we could handle seperately to find each one's moment of inertia. We used symetry to split the shape into one large disk and two long cylinder of identical volume on each side of the disk. We then measured and labeled the diameter, radius and thickness of the disk and cylinders.
One issue we faced was not being able to take the object apart to measure the mass of each piece. To work around this we used the total mass(which was etched into the disk) of the object and set up equilevant equations comparing the total mass and total volume to the individual mass and volume of each shape. We then solved for the masses of the disk and cylinders seperately.
To find the moment of inertia for the system as a whole, we found the inertia for each of its parts and combined them. Because each indivisual shape was either a wide or elongated cylinder we used I=1/2*m*r^2 to calculate each individual moment of inertia.
We know this apparatus will have a frictional torque that will work against the later pull of the cart. To calculate the angular decceleration of the system we ran several runs where the wheel spun at different speeds and slowed down due to friction reaching final velocity of zero while we measured how many rotations it completed to stop. We then averaged our trials and found a good value for the decceleration of the system.
Know that we have the values for the forces in this experiment we set up likear kinematic equations for the motion of the cart and rotational kinematics for the disk and through clever manipulation we solved for our predicted time it would take the cart to travel down the track.
Finally after all our calculations we can run the actual experiment and measure how long it takes the cart to make it down the ramp. We ran two seperate trials to collect and compare our teoretical and actual values.
Our percent error for each trial was very small and most likely caused by minor experimental errors such as friction between the cart and track or our tangential string not being perfectly parallel to the track. However, this experiment did produce very good results while still being conceptually rich.
Procedure: To find the moment of inertia for the disk-cylinder system we broke down the object into pieces we could handle seperately to find each one's moment of inertia. We used symetry to split the shape into one large disk and two long cylinder of identical volume on each side of the disk. We then measured and labeled the diameter, radius and thickness of the disk and cylinders.
One issue we faced was not being able to take the object apart to measure the mass of each piece. To work around this we used the total mass(which was etched into the disk) of the object and set up equilevant equations comparing the total mass and total volume to the individual mass and volume of each shape. We then solved for the masses of the disk and cylinders seperately.
To find the moment of inertia for the system as a whole, we found the inertia for each of its parts and combined them. Because each indivisual shape was either a wide or elongated cylinder we used I=1/2*m*r^2 to calculate each individual moment of inertia.
We know this apparatus will have a frictional torque that will work against the later pull of the cart. To calculate the angular decceleration of the system we ran several runs where the wheel spun at different speeds and slowed down due to friction reaching final velocity of zero while we measured how many rotations it completed to stop. We then averaged our trials and found a good value for the decceleration of the system.
Know that we have the values for the forces in this experiment we set up likear kinematic equations for the motion of the cart and rotational kinematics for the disk and through clever manipulation we solved for our predicted time it would take the cart to travel down the track.
Finally after all our calculations we can run the actual experiment and measure how long it takes the cart to make it down the ramp. We ran two seperate trials to collect and compare our teoretical and actual values.
Our percent error for each trial was very small and most likely caused by minor experimental errors such as friction between the cart and track or our tangential string not being perfectly parallel to the track. However, this experiment did produce very good results while still being conceptually rich.
Lab #16: Angular Acceleration
Purpose: To explore how angular acceleration changes when we use different hangin masses, rotating disks and combinations with different masses, and torque pulleys with different radii.
Procedure: For this experiment we used an apparatus that provides a constant air pressure that will keep the disks and hanging mass pulley "floating/hovering" so that we have virtually no friction.
When collecting data for each seperate trial logger pro showed that angular velocity steadily increased for the portion where the hanging mass is accelerating downward and then alternates to sloping downward as the mass reaches its max distance and disks inertia then begins to wind the string pulling the mass back upward to its initial position.
We then ran each trial with varying hanging mass, disk mass, torque pulley radii and recorded our data in the provided chart.
We then set up a motion sensor underneath the hanging mass so that we could collect data on its velocity and compare both angular and tangental acceleration.
We used the relationship between tangential acceleration and angular acceleration of (a = αr). Our data reflected that both accelerations are direclty related by a factor of the radius with only slight percentage errors due to small unaccunted frictions within the experiment.
Finally we calculated and compared our actual inertia values to our theoretical
values. Calculating each by either using our data and measurements and solving for what we needed by seting up, altering and substituting in out three experiment related equations: ma= mg-T, T*r=I*α, and a = α*r. Our theoretical and actual values did differ by a small bit because of small errors with friction and in our measurements.
Procedure: For this experiment we used an apparatus that provides a constant air pressure that will keep the disks and hanging mass pulley "floating/hovering" so that we have virtually no friction.
When collecting data for each seperate trial logger pro showed that angular velocity steadily increased for the portion where the hanging mass is accelerating downward and then alternates to sloping downward as the mass reaches its max distance and disks inertia then begins to wind the string pulling the mass back upward to its initial position.
We then ran each trial with varying hanging mass, disk mass, torque pulley radii and recorded our data in the provided chart.
We then set up a motion sensor underneath the hanging mass so that we could collect data on its velocity and compare both angular and tangental acceleration.
We used the relationship between tangential acceleration and angular acceleration of (a = αr). Our data reflected that both accelerations are direclty related by a factor of the radius with only slight percentage errors due to small unaccunted frictions within the experiment.
Finally we calculated and compared our actual inertia values to our theoretical
values. Calculating each by either using our data and measurements and solving for what we needed by seting up, altering and substituting in out three experiment related equations: ma= mg-T, T*r=I*α, and a = α*r. Our theoretical and actual values did differ by a small bit because of small errors with friction and in our measurements.
Lab #15 Conservation of Energy and Momentum
Purpose: To use our knowledge of conservation of momentum and energy to work a ballistic pendulum problem backwards to solve for the initial speed of a projectile.
Apparatus: The apparatus consist of a spring propelled cannon that launches a metal ball into a hanging block with a slot so that they can then swing together as one object and push a angle reader up as it travels to a max height.
Procedure: We fired the projectile and observed at what the angle reader marked.
Using the angle and the length of the srting we calculated how high the ballistic pendulum traveled. Using the conservation of energy we work backwards to find the initial velocity of the combination ball and block. That initial swing velocity now becomes out system velocity final when setting up our conservation of momentum equation. Lastly we solved for what the initial velocity of the projectile would have to be!
Apparatus: The apparatus consist of a spring propelled cannon that launches a metal ball into a hanging block with a slot so that they can then swing together as one object and push a angle reader up as it travels to a max height.
Procedure: We fired the projectile and observed at what the angle reader marked.
Using the angle and the length of the srting we calculated how high the ballistic pendulum traveled. Using the conservation of energy we work backwards to find the initial velocity of the combination ball and block. That initial swing velocity now becomes out system velocity final when setting up our conservation of momentum equation. Lastly we solved for what the initial velocity of the projectile would have to be!
Tuesday, May 26, 2015
Lab #14 2D Collisions
Purpose: To observe and collect data on two dimensional collisions to explore the conservation of momentum and energy. For this experiment we will conduct two separate runs. One with two metal balls of equal mass and the second with one metal ball and one glass marble.
Procedure: We set up a camera directly above a glass table(so that there would be very little friction) to capture video of the collisions. These videos can then be analyzed in Logger Pro frame by frame.
For our first run we used a metal ball and a marble. We had one ball stationary and rolled the other one at an angle so that they would collide and bounce away in different directions each with its own new velocity and direction.
Using logger pro we went through the video frame by frame plotting points for each ball's position. We then created calculated columns for the x and y components of each ball and plotted each component on a distance vs time graph. We also set up a linear fit for each component before and after the collision.
Using each linear fit to find each components velocity before and after the collision we now calculated the momentum each ball and set up conservation of momentum equations for both the x and y components to see if momentum really was conserved.
For our second run we used two metal balls and conducted the same steps; capturing video, plotting points, analyzing linear fits to get velocities, and finally setting up and calculating conservation of momentum equations.
Our data safely confirmed the conservation of momentum during these two collisions, but what about energy? was it also conserved? To verify this we had to use our plotted points to set up new calculated columns for the kinetic energy(1/2*m*v^2) for the components of each ball and plotted each one on a graph of energy vs time.
We observed that for the energy to be conserved before and after the collision the potential energies for the x and y components acted similarly while the kinetic energy acted in an inversely proportional way.
Though our experiment had some small degrees of error this is most likely due to fundamental errors with the experiment. Such as the glass table not being completely frictionless and the curved camera lens distorting our video slightly.
Overall our experiment was a success in showing how both momentum and energy are conserved in our 2D collisions.
Procedure: We set up a camera directly above a glass table(so that there would be very little friction) to capture video of the collisions. These videos can then be analyzed in Logger Pro frame by frame.
For our first run we used a metal ball and a marble. We had one ball stationary and rolled the other one at an angle so that they would collide and bounce away in different directions each with its own new velocity and direction.
Using logger pro we went through the video frame by frame plotting points for each ball's position. We then created calculated columns for the x and y components of each ball and plotted each component on a distance vs time graph. We also set up a linear fit for each component before and after the collision.
Using each linear fit to find each components velocity before and after the collision we now calculated the momentum each ball and set up conservation of momentum equations for both the x and y components to see if momentum really was conserved.
For our second run we used two metal balls and conducted the same steps; capturing video, plotting points, analyzing linear fits to get velocities, and finally setting up and calculating conservation of momentum equations.
Our data safely confirmed the conservation of momentum during these two collisions, but what about energy? was it also conserved? To verify this we had to use our plotted points to set up new calculated columns for the kinetic energy(1/2*m*v^2) for the components of each ball and plotted each one on a graph of energy vs time.
We observed that for the energy to be conserved before and after the collision the potential energies for the x and y components acted similarly while the kinetic energy acted in an inversely proportional way.
Though our experiment had some small degrees of error this is most likely due to fundamental errors with the experiment. Such as the glass table not being completely frictionless and the curved camera lens distorting our video slightly.
Overall our experiment was a success in showing how both momentum and energy are conserved in our 2D collisions.
Monday, May 25, 2015
Lab #13 Impulse - Momentum
Purpose: To explore both impulse and momentum by using a system in which can examine both elastic and inelastic collisions.
Definitions/Equations:
Momentum = Mass * Velocity ( p=mv )
Impulse = Force applied * Time (J=F*t)
Impulse without any external forces conserves momentum thus allowing us to use
Impulse = the change in momentum = intergal of (F*dt)
Procedure:
We set up a metal cart on a track that when pushed alongst the track it would collide into a plastic cart positioned so that the spring and bumper in it bounces the metal cart back. We also set up a motion sensor on the other side of the track to record position and time information for us and also set up calculated columns for both velocity and Force on the cart. We then ran this expirement twice, once with only the cart and the second time with added mass
Once we took data we created both Velocity vs Time and Force vs Time graphs, focusing on the velocity before and after the collision. We then integrated the area within the part of the force graph where the collosion occurs.
Inelastic collision: For the second part of the experiment we had a very similar set up except instead of colliding with the plastic cart and bumber it will have a nail that will srick into a block of clay mounted to an upright wood with a base that wil be underneath the track itself.
We finally calculated the theoretical values each expirement had, by multiplying each mass by the change in velocity( Vafter - Vinitial ) it experienced due to the collision.
Conclusions: Comparing our Theoretical and Actual values for the impulse in each experiment we can safely agree with the theorem that impulse = the change in momentum an object experiences. Our data might be slightly off due to experimental flaws such as the cart experiencing a frictional force and slowing down on its own, but overall we now better understand how impulse, momentum, velocity and mass are connected in collisions.
Definitions/Equations:
Momentum = Mass * Velocity ( p=mv )
Impulse = Force applied * Time (J=F*t)
Impulse without any external forces conserves momentum thus allowing us to use
Impulse = the change in momentum = intergal of (F*dt)
Procedure:
We set up a metal cart on a track that when pushed alongst the track it would collide into a plastic cart positioned so that the spring and bumper in it bounces the metal cart back. We also set up a motion sensor on the other side of the track to record position and time information for us and also set up calculated columns for both velocity and Force on the cart. We then ran this expirement twice, once with only the cart and the second time with added mass
Once we took data we created both Velocity vs Time and Force vs Time graphs, focusing on the velocity before and after the collision. We then integrated the area within the part of the force graph where the collosion occurs.
Inelastic collision: For the second part of the experiment we had a very similar set up except instead of colliding with the plastic cart and bumber it will have a nail that will srick into a block of clay mounted to an upright wood with a base that wil be underneath the track itself.
We finally calculated the theoretical values each expirement had, by multiplying each mass by the change in velocity( Vafter - Vinitial ) it experienced due to the collision.
Conclusions: Comparing our Theoretical and Actual values for the impulse in each experiment we can safely agree with the theorem that impulse = the change in momentum an object experiences. Our data might be slightly off due to experimental flaws such as the cart experiencing a frictional force and slowing down on its own, but overall we now better understand how impulse, momentum, velocity and mass are connected in collisions.
Friday, April 24, 2015
Lab #12: Magnetic Potential Energy
Purpose: To verify that conservarion of energy still applies for a system involving magnetic potential energy.
Set Up: Place an Air track on table and make sure to level it using something like notecards. Make sure there are magnets on the end supports of the air track and also on the glider. The vacuum will provide a force through the holes in the track and keep the glider afloat so we can oppose any drag due to the glider touching the track.
Procedure:
1) Because we have no set equation for magnetic potential energy we must build our own for it. We will use the relationship of potential energy being equal to the negative integral of force over distance.
2) If we raise one end of the air track the glider will slide and find an equilibrium point at which the magnetic repulsion is equal to the parallel gravitational component.
3) We stacked different heights of books underneath one end of the track and measured the distance between the glider and the opposing magnet and also recorded each new angle due to the raising of one end.
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4) To plot this data in logger pro we first had to convert the angle measures into radians so that we could calculate each force using trig functions in logger pro.
5) Once we graphed each force vs the separation distance we used a power fit(Ar^B) to calculate a coefficient and exponent to be able to later integrate.
6) Now that logger pro has given us our A and B we can find the negative inegral of the force over distance to calculate an equation for magnetic potential energy.
Set Up: Place an Air track on table and make sure to level it using something like notecards. Make sure there are magnets on the end supports of the air track and also on the glider. The vacuum will provide a force through the holes in the track and keep the glider afloat so we can oppose any drag due to the glider touching the track.
Procedure:
1) Because we have no set equation for magnetic potential energy we must build our own for it. We will use the relationship of potential energy being equal to the negative integral of force over distance.
2) If we raise one end of the air track the glider will slide and find an equilibrium point at which the magnetic repulsion is equal to the parallel gravitational component.
3) We stacked different heights of books underneath one end of the track and measured the distance between the glider and the opposing magnet and also recorded each new angle due to the raising of one end.
4) To plot this data in logger pro we first had to convert the angle measures into radians so that we could calculate each force using trig functions in logger pro.
5) Once we graphed each force vs the separation distance we used a power fit(Ar^B) to calculate a coefficient and exponent to be able to later integrate.
6) Now that logger pro has given us our A and B we can find the negative inegral of the force over distance to calculate an equation for magnetic potential energy.
Thursday, April 23, 2015
Lab #11: Conservation of Energy - Mass-Spring System
Purpose: To explore the conservation and transfer of energy of a vertically-oscillating mass-spring system.
Procedure:
1) We need to identify all the different forms energy will take in this experiment. We know that th hanging mass will have both a kinetic energy and a potential energy. However, the mass itself is not the only thing interacting in this eperiment. The spring itself will also have its own KE and GPE, but because it is a spring it has a third energy form of elastic potential energy. Below are the equations we used for each:
2) We set up a column style apparatus with a force sensor attached to the top of our clamped bars, a spring with a hanging mass attached, and at the bottom we have a motion sensor reading upward. We then calibrated and oriented all the equipment as needed.
3) The only variable we will have to do extra work to find is the spring constant. To do this we slowly stretch the spring downward and plot the force vs distance stretched in logger pro. Once this is complete we simple have logger pro calculate a linear fit to the graph whereas our slope is equal to our spring constant.
4) Now we used our equations to set up calculated columns within logger pro and simply release the system from a sthertched position and have logger pro collect data for each seperate energy.
5) Observing each different energy oscilate we can see how the energy is trasfered from one type to another as they fluxuate in opposite paterns.
6) Our column and graph for the total energy stored in the system was reassuring that the energy is conserved.
Procedure:
1) We need to identify all the different forms energy will take in this experiment. We know that th hanging mass will have both a kinetic energy and a potential energy. However, the mass itself is not the only thing interacting in this eperiment. The spring itself will also have its own KE and GPE, but because it is a spring it has a third energy form of elastic potential energy. Below are the equations we used for each:
2) We set up a column style apparatus with a force sensor attached to the top of our clamped bars, a spring with a hanging mass attached, and at the bottom we have a motion sensor reading upward. We then calibrated and oriented all the equipment as needed.
3) The only variable we will have to do extra work to find is the spring constant. To do this we slowly stretch the spring downward and plot the force vs distance stretched in logger pro. Once this is complete we simple have logger pro calculate a linear fit to the graph whereas our slope is equal to our spring constant.
4) Now we used our equations to set up calculated columns within logger pro and simply release the system from a sthertched position and have logger pro collect data for each seperate energy.
5) Observing each different energy oscilate we can see how the energy is trasfered from one type to another as they fluxuate in opposite paterns.
6) Our column and graph for the total energy stored in the system was reassuring that the energy is conserved.
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