Gravity and the Laws of Motion: Galileo Ramp Lab^{1}
Introduction: Until the time of Galileo, everyone accepted the Aristotelian concepts of physics. Aristotle taught that everything tries to seek its “proper place”. In Aristotle's view, lighter objects fall more slowly than heavier ones, simply because lighter things contain more of the lighter elements, air and fire which tend to rise, and heavier things tend to fall faster because they contain more earth and water. Similarly, Aristotle said that the natural state of an object is to be at rest, so that an object will stop moving as soon as the force acting on it is removed. Aristotle had no concept of acceleration. Galileo did a series of experiments to show that all bodies accelerate toward the earth and that they all accelerate in the same way. He also showed that in the absence of friction, bodies would keep moving forever. In effect, he achieved a partial understanding of gravity, and he discovered the concepts of acceleration and inertia. Galileo (and Newton after him) showed that the acceleration of gravity is a constant for all objects at the surface of the Earth, which means that the velocity of a falling object increases continuously. In Newtonian terms, Galileo found that the following relations hold: V_{f}= a*t when V_{i} = 0 d = ˝ at^{2 }when V_{i} = 0 Where V_{f}_{ } is the final velocity of the object after falling for time, t, the acceleration is a and d is the distance the object has fallen. The average velocity (of any object) is just V_{a}= d / t In this experiment, you will be able to calculate the average velocity, V_{a}. The final velocity, V_{f}, will be twice as large (2V_{a}) Available equipment: 1. Balls of different materials and sizes 2. Scale for weighing the balls 3. Ramps (“Inclined planes”) 4. Stop watch 5. Ruler 6. NO PROTRACTORS!! Procedure: Part 1: Galileo's Experiments (Velocity and Acceleration) Galileo found that he could not measure motions of falling bodies directly, because they accelerated too quickly. Instead, he did experiments by rolling balls down inclined planes, so that the acceleration was much slower. This is one approach that you will follow in this exercise. However, there are two problems with this technique. The first is friction. A rolling ball is not subject to very much friction, but the friction is not zero. However, there is a more serious issue that Galileo probably did not recognize. As a ball starts to roll down the ramp, some of the gravitational potential energy goes into making the ball spin - and of course the ball will continue to roll when it gets to the end of the ramp. So Galileo could not have gotten the right value for the acceleration of gravity. But he had the right idea. Set up a ramp at an angle shallow enough so that you can measure how long it takes to roll down, and not so shallow that friction becomes the dominate force. Record the angle of the ramp and the length of the ramp in your lab notes. With the help of a partner, measure how long it takes an object to go down the ramp. Start with the ball one quarter of the way up from the bottom of the ramp, then one half, then three quarters and finally start with the ball at the top of the ramp. In each position, make several measurements of the time it takes the ball to roll down to the end. Use the above formulas to calculate the final velocity and the acceleration of the ball. Repeat this procedure using a total of three different angles for one ball. Perform another trial using a ball of a different mass at a previously tested angle. Data: Ramp: 3 different angles (with constant mass) + 2 different masses (with constant angle) = 5 scenarios total Be sure to follow the directions above, with respect to dividing your Ramp and Air Tracks into proportional lengths!!! Data Table: Create a data table for all five scenarios that includes values for mass, angle, average velocity, final velocity, average acceleration, and calculated g for each scenario, average g for the ramp, and percent error of your average g. Graphing Your Results:
Analysis: Calculate the actual acceleration of gravity, g, with the data you collected in the lab and the relation g = a / sin(i) where a is your measured acceleration in m/s^{2}, and i is the angle of inclination of the air track. Present all of your data and results for V_{a} , V_{f }, a, and g in a properly labeled data table. For each calculated value of g, determine the % error from the accepted value (9.8 m/s^{2}) Questions:
^{1}Adapted and partially copied verbatim from http://www.bu.edu/astronomy/manuals/AS101/gravity.pdf |