The purpose of this experiment was to use the principles of conservation of energy and conservation of angular momentum to determine the final height a system reaches. The system used involves a meter stick pivoted near one end that is released from a horizontal position and is involved in an inelastic collision with clay at the vertical position (bottom of the swing). Determining a theoretical final height similar to an experimental also proves the principles of conservation of energy and angular momentum.
Apparatus:
Figure 1: Apparatus used in this experiment
The apparatus used in this experiment involves a meter stick pivoted near one of its ends, with tape on its other end. A clay piece, which was stood on three paper clips (used as legs), was also used and also had tape wrapped around it. The tape allows for the clay to attach to the meter stick upon collision. The meter stick was pivoted on a rotational sensor, which was used only for its minimal friction during rotation. A metal stand and a horizontal rod clamped to it were used to hold the rotational sensor and the meter stick.
A camera was also needed to record the collision between the end of the meter stick and the clay as well as the final height it reached. This experimental value for final height was used to prove that the method used to find the theoretical final height is correct. Lastly, a computer with LoggerPro software was also needed to obtain the recording of the experiment and to use video analysis in LoggerPro in order to determine the final height the system in the experiment reached.
Procedure:
First, some masses and dimensions of the system were measured. The mass of the meter stick was found to be 147 g, and the mass of the clay was found to be 28 g. The meter stick was found to be pivoted at 10 cm away from one end.
Once all of the equipment was set up, the actual experiment was performed and recorded, where the meter stick was released from the horizontal position and it collided with the clay when it reached the bottom of the swing. After the inelastic collision, both the clay and meter stick swung up to a final height. After recording, the video analysis software in LoggerPro was used to find the final height the system reached before falling back down, as shown below:
Figure 2: Video analysis of the recording of the experiment
The origin was placed at the point of collision between the meter stick and clay. The final point plotted right before the system starts falling back down to the vertical provides the final height of the system.
Figure 3: Data obtained from video analysis
Looking at the column of "y (m)" in the data obtained from the video analysis at the last data point (row 14), it can be seen that the final height the system reached was 0.3651 m. This experimental final height will be used to determine if the method used to find the theoretical final height is correct.
Next, the theoretical final height the system would reach, using the dimensions obtained, was found. The method used to obtain this height involves splitting the experiment into three phases, which includes:
- The use of conservation of energy from when the meter stick starts from rest at the horizontal position and reaches the bottom of the swing, but right before the collision with the clay. This method was used during this phase of the experiment was used to find the angular velocity of the meter stick at the bottom right before the collision.
- The use of conservation of angular momentum during the inelastic collision between the meter stick and the clay piece. This method was used for this phase in order to find the angular velocity of the clay and meter stick system after the collision.
- The use of conservation of energy from the point after collision to the final height the system reaches before falling back down to the vertical position. This method was used for this phase to find the theoretical final height the system (clay and meter stick) reaches.
Discussion:
The derivations used to final the theoretical final height are displayed below:
Figure 4: Derivation (Phase 1 and 2) of the theoretical final height the system reaches
First, the moment of inertia of the meter stick was derived since it is used throughout all three phases of the derivation for the final theoretical height the clay and meter stick system reaches. The parallel axis theorem was used to find the moment of inertia of the meter stick when the axis of rotation is 10 cm from one of the edges. This was derived by taking the moment of inertia of a thin rod rotating about its center, and then adding to it the mass of the meter stick multiplied by the distance the pivot moved from the center of mass squared. The moment of inertia about the pivot was found to be 0.03577 kg m^2.
Second, the conservation of energy principle in phase 1 was used. At the beginning of phase 1, there is only gravitational potential energy of the meter stick. At the end of phase 1, there is both rotational kinetic energy of the meter stick and gravitational potential energy of the center of mass of the meter stick. The angular velocity of the meter stick found right before the collision was 5.68 rad/s.
Third, the conservation of angular momentum principle was applied in phase 2 to find the angular velocity of the clay and meter stick system after the collision. Before the collision, there is only angular momentum of the meter stick. After the collision, there is angular momentum of the meter stick and clay piece. The inertia of the meter stick and clay system was found by treating the clay as a point mass, and then adding the moment of inertia of the meter stick to the moment of inertia of the clay. The moment of inertia of a point mass in mr^2, where r is the distance the mass is from the axis of rotation. In this case, it would be the length of the meter stick from the pivot to the end that collides with the clay, or 90 cm. The angular velocity of the meter stick and clay system was calculated to be 3.47 rad/s.
Figure 5: Derivation (Phase 3) of the theoretical final height of the system
Finally, the conservation of energy principle was used again in phase 3 of the experiment to find the final height the mass and end of the meter stick reaches. The point where gravitational potential energy is zero was placed at the pivot. At the beginning of phase 3, there is rotational kinetic energy of the system and gravitational potential energy of the meter stick. At the end of phase three, there is gravitational potential energy of the clay and the meter stick. The gravitational potential energy is negative in this phase since the center of mass of the meter stick and the clay are found under the point of zero gravitational potential energy. At the beginning, the height from the point of zero GPE of the meter stick is -0.4 m (the center of mass is in the middle of the meter stick, but it is pivoted 10 cm away from the edge). The height for the clay is -0.9 m. At the end, the height for the center of mass of the meter stick is -0.4cos(theta) since the meter stick moves an angle theta above the vertical, and the vertical distance of that movement is cosine of the angle times the hypotenuse, which is -0.4 in this case. The same reasoning is used for the height of the clay at the final height.
The relationship between theta and h was found using the following method:
Figure 6: Determining the relationship between h and theta
Substituting (h-L)/L for cos(theta) in the expression for phase 3, h becomes the only unknown variable and it can be solved for using algebra. It was found that the theoretical final height of the system was 0.386 m.
Conclusion:
Comparing the theoretical value (0.386 m) to the experimental value (0.3651 m), it can be seen that they are similar to each other. The percent error of the experimental value is -5.41% (it is negative since the experimental value is less than the theoretical, which is expected), which is small for such an experiment with many sources of uncertainty/error and is satisfactory. Therefore, it can be concluded that the conservation of energy and angular momentum principles are proven to be correct and the method used to find the theoretical final height of the meter stick and clay system is also correct.
The sources of uncertainty/error in this experiment are:
- Uncertainty in the plotting of data points during the video analysis
- Uncertainty in the measurement of the masses of the meter stick and clay as well as the length of the meter stick.
- Error in the assumption that the clay's center of mass is exactly at the end of the meter stick upon collision.
- Friction in the pivot of the meter stick.
- Sound and heat generated upon collision between the meter stick and clay.
- Friction between the paper clips acting as the stands of the clay and the ground upon collision.
- The meter stick used was not perfectly straight; there was some curving in it, affecting the center of mass of the meter stick and the position of the center of mass and the edge of the meter stick not being same in the horizontal position upon collision.
No comments:
Post a Comment