Physical Pendulum Lab

The objective of this lab was to find the period of a physical pendulum in Simple Harmonic Motion by taking measurements and finding the moment of inertia.

The first part of this lab was to predict using calculations the period of the ring rotating about a point halfway between its inner and outer radii.  To do this, I used Newton's Second Law equation for torque.  I found the total torque to be a product of the component of the weight, -mgsin(theta), times the distance from the pivot which was, r+(R-r)/2.  Next I had to find the moment of inertia for a ring with its pivot moved a distance, r+(R-r)/2.  After substituting in the values, I solved for alpha to get the equation, alpha=-2g*sin(theta)/3(R+r).  It is known that for small theta, sin(theta)=theta.  With this we can switch out sin(theta) for theta.  The equation now reads, alpha=-2g*theta/3(R+r).  This equation resembles the equation for radial acceleration, a=-(omega^2)*r.  With this comparison, we can say that omega=(2g/3(R+r))^1/2.  Work seen below.

Now that we have omega, the period can be solved for with the relation T=2pi/omega.  The equation now reads T=2pi(3(R+r)/2g)^1/2.  The radii were found using a pair of calipers; which turned out to be R=0.0696m and r=0.0576m.  Plugging these values into the equation for the period gives T=0.876s.  Work is seen below.

Now, to verify my prediction I placed the ring on a pivot and using a photo gate, found the period of its oscillation about its equilibrium position.  Not only was a photo gate used, but a laptop and Logger Pro were used to take data.  I placed a small piece of tape on the bottom of the ring and positioned the photo gate to pick up the tape passing through it.  Below is my setup as well as the experimental period.

As you can see from the photo above, my calculated period is considerably higher than the experimental period.  Calculating the percent error gives a value of 21.5% which is unusually high.  Below are my calculations.

It could be that I measured the radii of my ring incorrectly making the period come out larger from my calculations.

For Part 2,  we were asked to cut out a triangle and semicircle and find the calculated and experimental periods for either of them based on their orientation.  I chose to use the triangle for my lab.  After cutting out a triangle from cardboard, I took measurements using a pair of calipers and found that the height of my triangle was 15.3cm and the base was 15.0cm with a mass of 7g.

After taking measurements, I set about calculating a period for the triangle's motion as a pendulum hanging from its point.  The calculated period came out to be 0.618s.  Calculations for the period seen below along with calculation of the center of gravity for the triangle.

As can be seen from the pictures my calculated period for the triangle hanging about it's tip was quite off.  A percent error calculation shows that I was 13.3% under my experimental value.  Seen below.

Angular Acceleration

The objective of this lab was to find angular acceleration using a torque we knew the value of.  The apparatus used air in between the flywheels and pulley to create nearly frictionless revolutions.  Below is a picture of the apparatus.

A string was wrapped around the torque pulley above the flywheels and hung over a pulley which suspended a 24.6g weight.  There were three flywheels we used in our experiments; two steel and one aluminum.  For each trial there was a steel flywheel on the bottom and either a steel or aluminum flywheel on the top.  The steel flywheel on the bottom had a mass of 1.348kg and the top steel flywheel had a mass of 1.357kg.  The aluminum flywheel had a mass of 0.466kg.  All the flywheels had diameters of 12.65cm.  Other options for components were a large torque pulley with mass 36g and diameter 5cm and a small torque pulley with mass 9.9g and diameter 2.5cm.  On the top was a pin that stopped the flow of air from going out the top.

Also used in this lab was a Lab Pro and Laptop to count the lines on the top flywheel as it spins.  After following the instructions for the setup of Logger Pro in the lab handout, the hose clamp on the bottom of the apparatus was left open to allow the top flywheel to spin freely.

The air was then turned on to a moderate level and a test run was performed to make sure things went as we wanted them to.  The first trial had the two steel flywheels and a small torque pulley with the 24.6g weight hanging.  Here is a picture of the experiment.

The alpha down is 0.1470rad/s^2 and the alpha up is 0.1648rad/s^2 with their average being 0.1559rad/s^2.

The second experiment was doubling the mass with the same components.  Below is a picture of the experiment.

Alpha down is 0.2985rad/s^2 and alpha up is 0.3315rad/s^2 with their average of 0.315rad/s^2.

The third experiment saw triple the mass with the same components.  An image of the trial is below.

The alpha down is 0.4467rad/s^2 with alpha up 0.4852rad/s^2 and an average of 0.466rad/s^2.

The fourth trial changed the torque pulley but kept the steel flywheels with 24.6g hanging only.  Here is a picture of the trial.

The alpha down here is 0.2843rad/s^2 and alpha up is 0.3253rad/s^2 with their average of 0.3048rad/s^2.

The fifth trial only had the top flywheel changed out to the aluminum flywheel with 24.6g hanging and a large torque pulley.  Below is the trial picture.

Here alpha down is 0.8033rad/s^2 and alpha up is 0.9191rad/s^2 and their average is 0.8612rad/s^2.

The last trial had both the steel flywheels spinning with the large torque pulley and 24.6g of mass hanging.  Heres a picture of the trial.

The alpha down is 0.1438rad/s^2 and alpha up is 0.1615rad/s^2 and their average is 0.1527rad/s^2.

It seems that changing the mass to twice and three times its values increased the values of alpha by its respective increase.  Trial two's alpha with twice the mass is twice the alpha of trial one.  The same can be said of trial 3.  When the torque pulley was increased to the larger pulley, more torque was applied to the system which resulted in a larger value of alpha due to the relation of torque=moment of inertia times alpha.  With an increase in the torque, the alpha must also increase since moment of inertia stays the same.  When a lighter flywheel was substituted for the top flywheel, the moment of inertia was reduced so the alpha had to increase because the torque was the same value with only 24.6g hanging.  The opposite happened when both steel flywheels were spinning and caused the moment of inertia to increase dramatically with the torque staying the same the alpha had to be reduced.

Using the derivations from the lab manual, the moment of inertia for can be found with respect to the hanging mass, the radius of the torque pulley and the value of alpha.

Impulse-Momentum

The objective of this lab is to see how impulse changes the motion of a cart.

The first experiment has a cart with a force sensor on it traveling towards a cart with its spring extended.  The cart makes contact with the spring, the spring applies an impulse to the cart with the motion detector on it and sends it back the way it came.  Logger Pro was used to measure the magnitude of the push and the time the push was applied for.  An impulse is defined as the sum of the forces applied to an object times the amount of time applied for.  In the case of this cart, the only force on the cart was the spring that pushed it away as all other forces either canceled out or were negligibly small.

Ballistic Pendulum

The objective of this lab was to observe an inelastic collision and determine how high the two masses that collided would travel.  Below is the ballistic pendulum that was used in the demonstration.  In the picture, you can see a "cannon", a wire suspended block, a ball below the block, and a wire which will come into contact with the block and ball and will stay at the position of highest swing reached.  The wire is used to measure an angle from the vertical the block and ball travel.

Here is a closer view of the block and ball for visual aid.

The ball was placed into the cannon and fired into the block which rose a height from its original position and made an angle with the vertical at its peak swing.

Weighing the ball determined its mass to be 0.00763kg and the block came out to 0.0809kg.  Next, the length the block hung down was measured and was determined to be 0.21 meters.  After firing the ball, the block/ball combo created a 16 degree angle with the vertical.

We are asked to determine how fast the ball was traveling before it hit the block.  The problem was divided into two parts; before the ball hit and after.

Here we see the ball after it was fired and when it gets stuck in the block.  This is a momentum problem with momentum being conserved as the outside forces cancel out.  Initial momentum has the ball with it's initial velocity.  Final momentum has the ball and block moving as one with their new speed.  Solving for the final speed gives me an expression.

In the second part of the problem, the ball and block are moving together as one and have kinetic energy that will be transformed into potential energy as it swings an angle with the vertical and reaches a height above it's initial position.  Since the angle the combo created is known, the height it reached can be solved for.  Setting the initial and final kinetic energy and potential energy equal to each other gives an equation that reads that the initial kinetic energy is equal to the final potential energy.  Substituting in the expression for the velocity of the combo allows for the initial velocity to be solved for after inputting values.

Magnetic Potential Energy Lab

The objective of this lab was to confirm that conservation of energy still applies in situations with magnets and to find an equation for magnetic potential energy.

In order to check conservation of energy still applies in situations with magnets, we created a system that would directly test magnetic potential energy without other forces such as friction complicating things.  We used an air track with a cart on it that was attached to a machine that would blow air through the holes in the track to lift the cart.  Most importantly, we placed a magnet on both the cart and the end of the track so the magnets would repel each other.  We then placed books and other objects beneath the track to raise it.

The reason we were lifting the track was to get the cart to be supported by the magnetic force parallel with the track.  We would then use the component of gravity parallel to the track to equate it to the magnetic force.  We need to find the angle that the track is making with the table which we will call our horizon.  For this, we used our phones which were accurate to a tenth of a degree.  After the cart settled in it's new elevated position, we used a ruler to measure the distance between each face of the magnets.  Also important was the mass of the cart.  Below is a table with all our findings for this part of the lab.

We then put all our data into Logger Pro.  In Logger Pro, we graphed the component of the force of gravity that was parallel to the track, calculated from our recordings, against the distance we measured between the magnets.  Below is a picture of the graph.

From here, we assume that the line is the graph of some power law and do a power fit on the line.  From here, we find our constants A and B and plug them into the equation that states that the negative of a force integrated over a distance is equal to the potential energy.  We now have an equation for magnetic potential energy for this system.  Below is our computation.

Next, we wanted to see if energy was conserved in the system.  We attached a motion detector behind the magnet attached to the now level track.  We then turned on the air and gave the cart a modest push so that the cart would be bounced back by the magnets and our number would turn out nice.  From the graphs below, you can see in the velocity vs time graph that velocity starts at a number then becomes it's negative after the magnets push each other away.  From the KE, magnetic potential energy and total energy vs time graph above it, we can see our magnetic potential energy depicted by the red line and our kinetic energy as the light blue line become mirror opposites of each other after the magnets push against each other.  Our total energy, dark blue, unfortunately spikes slightly and we see magnetic potential energy as being larger than kinetic energy.

In the end, my group and I were satisfied with the first part of the lab but wished our magnetic potential energy was more the mirror image of the kinetic energy.  This problem could have spawned from the first part of our lab with our constants relying on measurements taken with a ruler.  It could have been that I pushed the cart too hard and we got the magnets closer than we ever did in the first part of the lab and saw it respond with greater force than anticipated.

Collisions in Two Dimensions

The purpose of this lab was to simulate collision and determine if both momentum and energy are conserved through the ordeal.

The setup for this lab was both easy and difficult.  What I mean by this is that a lab technician had already set up the camera, rod stand, firewire cable and glass that we were to conduct the collision on which was the easy part.  The difficult came when the camera didn't want to communicate with the laptop we were using so our professor was kind enough to lend us his.  Below is a picture of the setup.

After following the instructions to setup our camera, we chose three marbles, two of equal mass and a smaller marble, and weighed them for their masses(seen below). The results were, m1=0.021kg, m2=0.019kg and small=0.005kg.  For our first collision, we used the two marbles of equal mass.  One we had stationary under the camera and the other we rolled into the stationary at an angle so they would go off in different directions.  Next, we collided the stationary large marble with the smaller marble with the same intent of them going in different directions.  Since there were groups waiting to use the station, we conducted both our collisions before we analyzed our video.

Seen below is the analysis of our video capture with the segment highlighted being before the collision.  The red dots represent the x component of motion for our moving marble and the blue represent the y component.  The y-axis is position and the x-axis is time.  The velocity components of our moving marble, displayed as the slope of the linearization,  are shown in the boxes respectively.  These are the initial velocities of the collision.  As a reminder the mass of the moving marble, m1, was 0.021kg.

Next below is a picture of the analysis with the highlighted portion being after the collision for m1.  As you can see by the slopes, the ball has slowed down and changed direction after the collision.

Displayed below is a picture of the same collision but this time, I have analyzed the motion of m2.  Before the collision, m2 was stationary and had no velocity.  Now though it has gained some velocity displayed in the boxes as the slopes thanks to the collision.  The mass of m2=0.019kg just to remind.

Next, we use our masses and recorded velocities to see whether or not both momentum and energy are conserved after the collision.

Above we can see that momentum is almost conserved in the elastic collision between the two similar marbles.  The velocities were obtained from Logger Pro after tracking their motion before and after the collision.

Seen above are the calculations for kinetic energy of the two similar mass marbles.  Most of the energy is in the Y direction as we chose to have M1 progress along this axis.

In the next experiment, we used marble M1 and S, a smaller marble.

In the picture above, we can see that M1 is the only marble moving with it's velocities displayed as slopes in the boxes above.

Now displayed above is the velocities of M1 after the collision with the smaller marble.

Seen above are the velocities of S after M1 runs into it.

Above are the calculations for momentum.  For this collision, we decided to use M1 again to collide with stationary S.  The velocities were obtained from Logger Pro after tracking M1 as it went through the action of colliding with S.

These are the calculations for kinetic energy.  For both initial components of kinetic energy, only M1 is moving with velocities obtained from Logger Pro.  After the collision, both M1 and S are moving but M1 is moving a little slower while S starts moving after the collision.

In conclusion, since these are considered elastic collisions both momentum and kinetic energy should be conserved.  However, kinetic energy can be seen to lose about half its value after the collision.  It could have been that M1 was projected too fast and didn't roll but skid along the glass surface creating a small amount of friction that took away from the already small values of kinetic energy.  It is also possible that the marbles were not tracked very well when the collisions were examined.

The major sources of error from this lab came from the tracking of the marbles.  Although great effort was put into obtaining data, the values that came from the data were less than desirable.  Another small source of error came from the weighing of the marbles.  The scale used to weigh the marbles was accurate to one gram.

Centripetal Force with a Motor

The objective of this lab was to find a relationship between the angle created by swinging a rubber stopper with an angular speed.

For this lab, our professor had setup a rotating apparatus and conducted the experiment while the groups recorded data.  To record the height of the swinging stopper, our professor attached a piece of paper to a ring stand and slowly raised the paper until the top of the paper was barely grazed by the bottom of the stopper.  It is also important to mention that there was a meter stick at the top of the rotating apparatus which the rubber stopper was attached to the end.  This created a distance from the apparatus that would form an angle with the vertical when swung.  We then proceeded to measure the distance of the string to the apparatus, height of the apparatus, and length of the string.

Our group collected six trials with our periods consisting of ten rotations.  Our group members individually timed the ten rotations with our cell phones and our periods generally agreed to a hundredth of a second.

From there, we drew a free body diagram of the rubber stopper and divided the forces into their x and y components.  From the x and y components we were able to derive a relationship between the angle the string makes with the vertical and its angular speed (shown below).

Using our measured heights, we were able to find the angles of each trial (shown below).

Next we calculated our angular speed using our trial periods (shown below).

Using our equation relating angular speed and the angle, we found values for calculated angular speed (seen below).

Here is a table with all our data for convenience.

Here is the error in our calculated rotational speed compared to our experimental rotational speed.

Where our error most likely comes from in this lab are our measurements taken with the meter stick and our period times taken with our phones.