2015-May-20 Lab 19: Conservation of Energy/Conservation of Angular Momentum

PURPOSE:  To use conservation of energy and conservation of angular momentum to predict how high a meter stick will rise, pivoted about its end, after hitting and sticking to a clay blob.

PROCEDURE:  

We set up a system in which a meter stick rotated about (close to) one end (2.5 cm away from the end of the meter stick).  The meter stick was held up to its horizontal position, released from rest, and swung down to hit a clay blob so that the clay blob would stick to the end of the stick and the stick-clay system would continue moving to a maximum height.

Diagram of stick-blob system.

In the time that the stick is released from its horizontal position until just before it hits the clay blob, energy is conserved.  Then, from right before the meter stick hits the clay, angular momentum is conserved through the inelastic clay-stick system.  Finally, energy is again conserved in the stick-clay system as it continues to rotate together to its maximum height, when kinetic energy becomes zero (angular velocity is zero).

We used our measurements to calculate angular velocity just before the stick hits the clay blob, when it is in its vertical position.  Then we used this ωfinal as ωinitial in our conservation of angular momentum equation (equation 2) to solve for ωfinal just after the meter stick hit the clay.  Again, this ωfinal became ωinitial in our last conservation of energy equation, for initial angular velocity of the system as it rotated to its maximum height.

For gravitational potential energy, we set zero at the pivot, so when the meter stick was in its horizontal position its center of mass was at zero, and when the meter stick was in its vertical position its center of mass was 0.475 meters below GPE = 0, so -0.475 meters.  

For our first conservation of energy equation:

On the left hand side, gravitational potential energy is at 0 and the stick is at rest so it has no kinetic energy.  On the right hand side when the meter stick is in its vertical position, its center of mass is at height -0.475 meters.  The moment of inertia of a rod about its center of mass is 1/12*mass*length^2, and using the parallel axis theorem we moved the pivot 0.475 meters from the center of mass, so we must add mass*distance^2 (shown above).  Solving for angular acceleration, we find that it is 5.477 rad/s just before impact with the clay blob.

Our next equation is conservation of angular momentum:

Initial angular momentum equals final angular momentum, or Iωinitial = Iωfinal.  Moment of inertia of the stick is the same as in the previous equation.  Our initial angular velocity comes from our previous equation.  After the collision, the moment of inertia of the stick is the same but we must add the inertia of the clay, which is the mass of clay * distance of clay from the pivot, squared (equation for inertia of a particle).  Solving for our new final angular velocity of both the stick and clay system, we get 2.733 rad/s just after collision.

For part 3, we use conservation of energy again.  Final kinetic energy is zero because we want to find the maximum height that the meter stick rotates, which will be when angular velocity is zero.

Our inertia on the left hand side of the equation is the same as the inertia of the clay-stick system from the previous equation.  However, we have new potential energies.  We can get the gravitational potential energy of the system by finding the GPE of each individual part and adding them together.  On the left hand side of the equation, the meter stick is vertical, so its height is -0.475 meters (0.475 meters below the pivot, where GPE=0).  The clay is at the very end of the meter stick, so its height is -0.975 meters.  On the right hand side of the equation when the stick-clay system is at its highest, it will be at an angle θ.  The vertical height of the stick's center of mass at angle θ will be -0.475cosθ, or the vertical component of the stick's center of mass at this angle, and the vertical height of the clay at angle θ will be -0.975cosθ, or the vertical component of the distance the clay is from the pivot at this angle. 

Plugging all this into our equation for conservation of energy, we find that when the stick-clay system reaches its highest point it is at an angle of 45.36 degrees from the vertical.  Taking the cosine of this angle gives us a vertical distance from our pivot towards the table, which is 0.7026 meters.  If we subtract this from 0.975 meters, we can see that the end of the meter stick reached a height of 0.272 meters above the ground, or 27.2 cm.

Using video capture to plot the origin and maximum y distance the stick-clay system reached, we found it to be 0.290 meters above the ground, which was within 6.2% of our calculated height.

CONCLUSION:

Conservation of energy and conservation of angular momentum allowed us to calculate the angular velocity before the stick and clay collided and just after they collided, to find the gravitational potential energy when angular velocity was at zero at the stick/clay's highest point.  We also used moments of inertia, the parallel axis theorem, and sum of individual components of GPEs to calculate our prediction.  In the end, we had a 6.2% error with our experimental result, but this could have come from a number of factors.  The height at which we released the meter stick may not have been perfectly horizontal, and the video freeze frame may not have stopped exactly at the highest point on our grid.  The angle that the camera was set up may not have been positioned directly in front of the rotating system, which could cause distances to be skewed because of the angle it was at.  Another important factor could be that the clay blob's center of mass may not have been exactly at the end of the meter stick.