Experiment 3: Force Vectors Object: To find the total force on a mass in mechanical equilibrium. The apparatus: You should make labeled pictures of the items described below. Show how the apparatus works in one of your pictures. • A force table. This is a round, horizontal, circular table, with angles marked from 0° to 360°. It has a metal bolt in the center to keep a small ring (see below) from escaping. • A small circular plastic or metal ring. • Three vertical pulleys, which can be attached to the force table with thumbscrews. • Three mass supports, which are are connected to the ring by strings. The strings fit over the pulleys. • Various masses to place on the mass supports. The weight produced by a mass m is mg. For convenience, you can use a unit of gr (grams) for the masses. One gr weighs one ggr. In units of Newtons, the weight of one gram is 1 ggr = (g)(1 gr) = (9.81 m/s²)(10−³ kg) = 9.81 × 10−³ N. You do not need to convert your weights to Newtons! Scule angles. (in units degrees) WILL 112. F₂ Force Table Fi Bolt ring String 7 •Thumbscrew 1 pulley ← Support /holder for weights. 82 Ces View of Force Table from Above -pulley 0₁=00 What you need to do with the apparatus: Place one pulley at 0°. The other two will be placed at two angles of your choosing. Call the angles 0₁ = 0, 0₂ and 03. These angles should obey 0₁ <0₂ < 03. Next choose three UNEQUAL masses to place on the mass supports of each of the pulleys. No two of these masses may be (even approximately) equal. The experiment will work better if all the masses are greater than 100 gr, and differ from each other by at least 30 gr. NOTE: it may be obvious, but you must include the mass of the supports themselves! Adjust the angles (and the masses, if you wish, but keep them unequal, as stated above) until the ring is centered in space around the bolt. The ring should just float, in contact only with the three strings. The weights at 01, 02 and 03 are F₁, F2 and F3, respectively. Weights are magnitudes of forces, not masses. Data: Write your data: 0₁ = 0º, F₁ =, 0₂ =, F2 =, 03 =, F3 =. Calculations: Each force is given by its magnitude and angle (direction). Forces are vectors, which may be written as pairs of numbers. A vector must always have a vector sign (arrow) above. A number must never have a vector sign above. With your numbers, first find the following force vectors: and F₁ = (F₁ cos 0₁, F₁ sin 0₁), F2 = (F2 cos 02, F2 sin 0₂), F3 = (F3 cos 03, F3 sin 03). These vectors have magnitudes and directions F₁, 01, F2, 02, F3, 03, respectively. Now add the three force vectors, that is, find the total force F. This is F = F₁+F₂ +F3 = (Fx, Fy). Ideally, the total force F, should be zero. In practice, the magnitude F, of F will be very small compared to the magnitudes F₁, F2 and F3. Now the magnitude of the force is given by the Pythagorean formula You should find that the three ratios F = F2+F2. F F F 2 2 2 F₁ F2 F3 are small, meaning that the sum F, you found above is close to zero. Each of the ratios should be less than 5% = 0.05. Are they? If not, you have done something incorrectly. Questions: 1. Suppose 0₁ = 0, 0₂: 120° and 03 = 240°. If the three forces have the same magnitude, what is the total force? You must justify your answer, to receive credit (as always). 8