Magnitude of the Acceleration Due To Gravity on Earth

Gravity Goals • To measure the magnitude of the acceleration due to gravity on Earth • To observe Newton’s laws of motion and utilize them to make predictions Equipment • Computer • Lab Pro Motion Detector • Cart • Incline Track • Rulers • Weights • Pulley • String Methods • Measure the acceleration of a cart down an inclined plane • Measure the acceleration of a cart on a level track due to a hanging weight Page 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 time it was thought that matter was made of four elements: air, fire, earth and water. Accordingly, he thought lighter objects fall more slowly than heavier ones simply because lighter things contain more of the lighter elements, air and fire, 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 found that he could not measure motions of falling bodies directly, because they accelerated too quickly. Instead, Galileo did experiments by rolling balls down inclined planes, so that the acceleration was much slower. 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 for an object that travels a distance ∆x in a time ∆t, the following relations hold: vs “ ∆x ∆t vf “ 2s v “ a∆t (1) (2) vf (3) ∆t In the above equations, vs is the average velocity of the object, vf is the final velocity of the object when it reaches the distance ∆x, and a is the acceleration of the object. (The symbol ∆ is used to represent “change in”, so ∆t means “change in t”, where t is time.) So far we have not addressed gravity at all. It is the force of gravity that actually causes the ball to roll. On Earth, the downward acceleration caused by gravity is a“ g “ 9.8 m{s2 The acceleration of a free-falling body due to gravity is g “ 9.8 m{s2 and it is the same for all objects, independent of the objects’ mass. This was quite profound in Galileo’s time. Newton proved with his mathematical formulation of gravity what Galileo had shown experimentally, that if you drop a heavy object and a lighter object off the top of a building at the same time, they will hit the sidewalk Page 2 at the same time (neglecting the effects of air resistance). Gravitational force and potential energy are dependent on mass, but gravitational acceleration is not. The last thing we have to consider in our experiment is that a ball rolling down an inclined plane is not in free-fall. The inclined plane exerts some force on the ball. The larger the inclination of the plane, the smaller the force exerted by the plane, and the closer the ball will be to free-fall. The acceleration of the ball in the direction it rolls down the ramp (a) is not the full acceleration of gravity. The acceleration is some fraction of g, and that fraction must have something to do with the inclination of the ramp. Let’s define the angle θ as the angle of inclination of the ramp. When θ “ 00 , the acceleration of the ball is zero (the ramp is flat), when θ “ 900 the acceleration of the ball is g, because it is in free-fall. What trigonometric function (sine, cosine) behaves like this? Sin(θ) is zero when θ “ 00 , and is equal to one when θ “ 900 . So we might simply guess that the way to relate a to g would be: a “ g sinpθq (4) This is exactly correct, and can be shown with a little bit of trigonometry and a discussion of vectors, which we will do in lab. You now know how to calculate a. If you can calculate sinpθq of your ramp, you can get an experimental value for g. Page 3 1 Acceleration of Gravity Procedure You will repeat Galileo’s experiment, but instead of rolling balls you will use a cart on an inclined plane or track. There are two problems with the technique that Galileo used. The first is friction. A rolling ball is not subject to much friction, but the friction is not zero. Additionally, it takes energy to make things spin. As the ball rolls down an inclined plane, some of the gravitational potential energy goes into making the ball spin, and that energy isn’t helping the ball accelerate down the ramp. The main difference between the balls used by Galileo and the cart that you will use is that the carts have very little in the way of rotating mass, less mass that needs to spin (they have wheels but they are quite small and made of light plastic). With less mass that needs to spin up, less energy will be sapped from the acceleration down the track. You will use a computer to measure the position of the cart versus time. To do this make sure the motion sensor is plugged into the LabPro device and that the LabPro’s USB cable is plugged into the computer. Open Logger Pro using the icon on the desktop (looks like a vernier caliper and paper). Logger Pro should automatically recognize the sensors, if it doesn’t: • Under the Experiment tab mouse over to Set Up Sensors and under the pop up tab select LabPro: 1 • A window will open with a picture of the LabPro • Select the Dig/Sonic 1 box in the upper right hand corner and choose the Motion Detector • You can close the LabPro window now You can take measurements of the carts position versus time by clicking the Collect button (a little green button with and arrow). It is located above the graph. When you hear the motion detector start to tick it is taking measurements. You should zero the motion detector by going to the Experiment tab, and within the drop down menu you will see Zero. Make a Zero position by setting the cart 5 cm away from the detector and click Zero (if the cart is too close to the detector when you zero, it won’t get a good zero point). If you move the detector or the start position of the cart you will have to re-zero it. It takes the computer about 2 seconds to begin taking data after you click Collect. Use this time to start the cart’s motion. You should see the data represented by a line on the graph. You can open another graph window by clicking Graph under the Insert tab. By double clicking on the graph you will Page 4 bring up the Graph Options window. Under Axes Options you can choose to display the Acceleration, Velocity, or Position. To find the average slope of a line on the graph: First you need to decide from what part of the graph you want the slope. In our example it is a graph of Position vs. Time. We want the average velocity while the cart was moving. We need to click and drag the mouse over the portion of the line that represents the moving cart thus highlighting it. Be careful to only highlight the part of the line that you need. The bump at the end of the graph is where the cart was kept from hitting the sensor by someone’s hand and shouldn’t be included in our slope calculations. Now click the linear fit button. The computer will fit the line and give you the formula of the line in the f pxq “ mx ` b format, where x is the independent variable (t in our example), m is the slope, and b is the y intercept. Print a graph from one of your runs. First you will have to connect to the wireless network. Click the Cisco VPN Client on the desktop and doubleclick bu-oncampus. Enter in your username and Kerberos password to connect. To print, under File go to Page Setup and select the icon which represents landscape printing, then under File go to Print Graph. Page 5