deadline till 10.00 PM canadian time

deadline till 10.00 PM canadian time

1114 Forces and Equilibrium – 1 Saved: May 12, 2020 Forces and Equilibrium (Vectors) Purpose Verify the equilibrium condition of concurrent forces . Introduction and Theory When an object is in static equilibrium, Newton’s First Law states that the sum of all the external forces (or the resultant force) acting on the object must add up to zero . This is the equilibrium condition of concurrent forces, which are forces acting through a single point of the object . If the forces are not concurrent, there will be another equilibrium co ndition, which is not the topic of today . Today, we will add three concurrent forces in static equilibrium and show their sum equals to zero: Σ⃗= 1⃗⃗⃗⃗+ 2⃗⃗⃗⃗+ 3⃗⃗⃗⃗= 0 There are two ways to add vectors, numerically by their components, or graphically by the “ he ad -to- tail method ”. In Problem 1, we verify that the resultant force has zero components in both horizontal and vertical directions ( the third dimension is not involved) . In Problem 2, we verify by graphing that the resultant force is a vector with zero length, or experimentally, the length is no bigger than the uncertainty . Problem 1 Add the force vectors by components Apparatus A cardboard box, 3 identical rubber bands, 3 push pins, a small metal ring with an opening (like a key ring) , a 30 -cm ruler. Figure 1 1114 Forces and Equilibrium – 2 Saved: May 12, 2020 Data In this lab, we use the extension of rubber bands as t he measurement of the magnitude of forces. This is not ideal, because the rubber bands are not perfect springs, so their extensions are not exactly proportional to the stretching force. B ut it is a good approximation , especially a t moderate stretch es. To find the extension, we first need to measure the unstretched length of the rubber bands . This could be tricky if the rubber bands are not straight without any stretch , but are like circle s. If that is the case for you , you should use the average of the diameter of the circle and the straight length just before stretching ( show n in Figure 2 ) to be the unstretched length . For the rubber ban ds in Figure 2, it is 0= 1 2(3.6 cm + 5.2 cm )= 4.4 cm Figure 2 Get three rubber bands with the same length . M easure the unstretched length and record that in your report. We will use one unstretched length for all rubber bands. If any rubber band is different by more than 0.1 cm, you should change the rubber bands so that they are all similar . Lay a piece of bla nk paper on the cardboard box. Put the three rubber bands onto the key ring, and fix the other end to the cardboard box using push pins, allowing each rubber band stretch a couple centimeters . Do not stretch too hard as it creates a danger of flying push p ins. For the third pin, adjust its position so that two rubber bands ar e perpendicular with each other, as shown in Figure 1. The key ring should be free to move and not under any friction . We need to record both the direction and the magnitude of each fo rce because force is a vector . Mark two dots between each rubber band : they give the directions of the three forces . Then write in pen the length of eac h rubber band for the magnitude of each force. The length is from the end of the rubber band to the cent re of the push pin. After you record three directions and three lengths, take a picture to show your setup and data for Problem 1. When you do this experiment, you will know that there are many uncertainties in your data. The directions can be off a few degrees; the extensions have uncertainties. Overall it is not a very precise experiment, and you can get an overall uncertainty of 0.5 cm or so. We will use that as the uncertainty for both problems. 1114 Forces and Equilibrium – 3 Saved: May 12, 2020 Calculate the resultant force Remove the white paper from the cardboard box. Connect the dots and the pin holes to draw three straight lines on the paper . The y should meet on a single point O, called the point of concurrency , or at least not more than 0.1 cm apart, otherwise you should retake the data . For the magnitudes, subtract the unstretched length (4.4 cm on my example ) from each measured length, and write these magnitudes on the paper, by each force. W ith the direction and the magnitude known, draw the three force vectors, as shown by the red arrows in Figure 3. These arrows represent the three force s acting on the key ring, although the lengths reflect the relative size of the forces only and we do not know the forces in newton . Figure 3 Now we will find the horizontal and vertical components of all thre e forces . 1⃗⃗⃗⃗ and 2⃗⃗⃗⃗ have only one component each . To find the components of 3⃗⃗⃗⃗, draw lines perpendicular from the head of 3⃗⃗⃗⃗, to either axis, as shown in Fig ure 3 . The projections are the components of 3⃗⃗⃗⃗, F3x and F3y. Calculate the horizontal and vertical components of the resultant force Rx and Ry. They are equal to the sum of the horizontal and vertical components of the three forces , Σ and Σ. The results are in unit s of cm instead of N , but they are proportional to th e magnitude of the force s. Conclusions State the horizontal and vertical component of the resultant force in a statement like: “The horizontal component and the vertical component of the resultant force are − 0.2 cm and 0. 1 cm in length . Because they are both smaller th an uncertainty of our force measurements, the resultant force agrees with 0, and we verified the equilibrium condition of forces .” 1114 Forces and Equilibrium – 4 Saved: May 12, 2020 Problem 2 Add the force vectors graphic ally by the “head to tail method” The apparatus, setup and the proce dures are all the same as in Problem 1, only this time, no two forces form a right angle. See Figure 4 . Figure 4 Calculate the resultant force graphically Again, you must find the direction and magnitude of each force and draw three vectors from point O, the point of concurrency. To add three forces 1⃗⃗⃗⃗, 2⃗⃗⃗⃗ and 3⃗⃗⃗⃗, we first move 2⃗⃗⃗⃗ so that its tail touches the head of 1⃗⃗⃗⃗, then move 3⃗⃗⃗⃗ so that its tail touches the “new head ” of 2⃗⃗⃗⃗. Finally we mark a letter “S” at the “new head ” of 3⃗⃗⃗⃗ and vector ⃗⃗⃗⃗⃗⃗ is the resultant force. Remember that when moving a vector, you must keep both its magnitude and direction unchanged . You can check whether the moved vector and the original vector form a parallelogram. If both opposite sides are equal in length, then the quadrilateral is a parallelogram. Conclusions State whether the resultant force agrees with 0, by comparing its magnitude to the uncertainty , and whether you have verified the equilibrium condition of forces. Complete report Please se e the online sample report for what need s to b e hand ed in for this lab. It contains 5 pages: (1) report ; (2) photo of Problem 1; (3) data and calculations of Problem 1; (4) photo of Problem 2; (5) data and calculations of Problem 2. Please note that this sample report is for formats only, you must not try to match the forces. You should not put any two forces close to a straight line as the sample report does.