Measuring g

Lab 06 – Measuring g Purpose Students will attempt to measure g, the acceleration due to gravity. Theory Historically, measuring g has been tricky due to the short time intervals involved with easily useable distances. Larger heights introduce an error from wind friction. The modern brute-force solution uses digital systems that take out the guesswork. Unfortunately, they replace direct experience with circuit magic, and one might as well just look up the answer online. Galileo Galilei attempted to find the value by rolling marbles down tilted boards, but the friction and rotation of the marbles can be quite problematic. The approach here takes advantage of circular motion. A mass is swung in a vertical circle by an attached string just slowly enough so the string goes slack at the top of the arc. When that is the case, only two forces are present at that point: weight and the centrifugal pseudoforce: mg = mv²/r. Mass divides out and a measure of g emerges. parabola, and the resulting speed will be used as the average. After freefalling one radius from its peak, the mass’s vertical speed will be vy = √(2gr). The horizontal component was already found to be vx = √(gr). If the magnitude is now found using the Pythagorean Theorem, the result will be v = √(3gr). Installing this result into the original formula gives this: 𝑔 = 4𝜋 % 𝑟/3𝑡 % where r = 1. Procedure 1. Find a piece of string at least 1.5 m long. Dental floss will work, but thread might break. 2. Tie a compact weight to the end of it. The roll of tape out of a scotch tape dispenser will work very well. A large nut or washer also serves. A ball of foil is too light, and a pen is too oblong. A wine cork will work. A water bottle is too big and too light. Improvise! 3. Mark a spot on the string one meter from the center of mass of the weight. This spot must be just outside where the string will be held. If the speed of the mass is always the same, with time t to complete one circle, speed squared should come out v² = 4π²r²/t² since distance traveled is the circumference 2πr. Unfortunately, the mass speeds up on the way down and slows on the way up. Presumably, the average speed will happen about halfway down the circle, where gravity has increased its top-of-arc value with a vertical component due to freefall. 4. Practice swinging the weight in a vertical circle at the slowest possible speed to barely keep it going. It’s safer to do this outside, just in case. Many subtle assumptions are being made here, but the falling ¼-circle of the path will be treated like a freefall 2. What’s the percent error? (Assume the “true” value of g is 9.8 m/s².) 5. Time ten revolutions at that speed, and divide the result by 10 to find t. Analysis Please answer each of the following in Canvas using complete sentences: 1. What is the computed value for g?

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