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!
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