Physics 8/18 NAME: TA: LAB PARTNERS: SECTION NUMBER: VECTORS (FORCE TABLE) LAB II Introduction In the Vectors I lab last week we used force tables to introduce the concept of vectors and how they are used in physics. In this second vector lab we will use the force tables again to explore some more vector concepts and reinforce your knowledge and understanding. To do well in physics it is very important that you understand how to use vectors so do not hesitate to ask your TA for assistance if you have any problems. You may want to read over the VECTORS I lab script too as a quick recap. Remember: 1) Vectors are used in physics to represent quantities that have a direction as well as a magnitude. Some examples of this are velocity, acceleration and force. 2) Because vector quantities have a specific direction associated with them, when we add them we must take this direction into account. 3) A system is in equilibrium when the vector sum of forces acting on it is equal to zero. For example: Two students are moving into a house together. They work together and push the sofa in the same direction each with a force of 20N. The sum of the force vectors is 20+20 = 40N. This is the net force applied. If the students get confused while wrestling the sofa through a doorway and end up pushing in opposite directions, each with a force of 20N. The sum of the force vectors in this case is 20 20 = 0N. That sofa is going nowhere! So you can see the direction matters. We use vectors to help describe systems where direction is important. Vectors are a neat mathematical tool to help solve many different problems. 1
Experimental Setup Materials needed Force table including central ring and retaining pin Three pulleys String Three mass hangers with additional masses Protractor Ruler Bubble level Transparencies and transparency pen Figure 1 - The force table Begin the same way that you did last week. Use the bubble level to be sure that the table is level. Use the leveling screws on the tripod to adjust the angle of the table if you need to. Place the retaining pin in the center of the force table. Attach two pulleys to opposite ends of the table, at angles of 0 and 180. Next, attach two pieces of string to the central ring, each being long enough that they can stretch over the pulley and hang several centimeters below the table. Attach the mass racks to the free ends of the string and verify that your table is level. Once you have done this check you are ready to start. 2
The experiment Last week, using the force table, you looked at balancing different force vectors in one and two dimensions. In this lab, we will look at some more advanced concepts: Part 1 Three vectors in equilibrium For this part of the lab, we will set up three force vectors in equilibrium. Define your x-y axes such that the positive x-axis points towards 0 on the force table. You may find it helpful to use a transparency to draw your axes on the table. Cover the top of the table with one of the transparent sheets provided and use the ruler to draw the x and y axes. Do not write directly on the force table! When hanging your masses, take care to always use either the inner or the outer number scale on the table (choose which you will use before you start). Each mass (m) hanging from the table gives a force of mg acting from the central point. (a) Hang 100g from a pulley so it acts along the positive x-direction (0 ). What is the magnitude of this force? (b) Hang 100g at 120 What is the magnitude of this force? (c) Define your x/y axes on the graph found on the next page. Sketch the two force vectors on the graph and label them a and b. Use your diagram to calculate the resultant vector for these two. Sketch it on the diagram and label it c. Now sketch on the graph a single vector, d, which you would add to your force table to balance a and b and reach equilibrium. 3
Try it on the force table. Have your TA check your results at this point before you go on. 4
Part 2 Resolving vectors into components In the previous section, we showed that two vectors can be combined to give a resultant vector. Similarly, any vector can be split up into two vectors. Often in physics we take a vector and split it into two perpendicular components, one on the x axis and one on the y axis. Any vector can be split up (or resolved ) into x and y components. (1) Place your x axis at the 0 line on the force table so the y axis is at 90º. Then hang 200g at 60. Hang another 200g at 240º. Your force table should now be balanced with two equal and opposite forces. Next we will calculate the x-y components of the force vector at 60 and show that we can replace this vector with these components, still balancing the vector at 240º. (a) First calculate the x-component of the force at 60º and write it down (b) Now calculate the y-component of the force at 60º and write it down (2) Remove the pulley at 60º and add two new pulleys, one for the x-component you just calculated and the other for the y-component. Verify that these two new forces can be used to replace the one you just took off i.e. they are equivalent. Note: A vector is equivalent to its components. (3) Remove the pulleys from the table and replace the two pulley set-up from (1) above. We will now show that the position of the x-y axis does not affect the result. (a) Rotate your transparency so the x-axis you drew previously now lines up with 45 on the force table. (b) Calculate the new x-component of the force at 60º and write it down. 5
(c) Calculate the y component of the force at 60º and write it down. (d) Remove the pulley at 60º and add two pulleys, one for the x-component you just calculated and the other for the y-component. Verify that these two new forces can be used to replace the one you just took off i.e. they are equivalent. So we can see that a vector is always equivalent to its x-y components wherever we put the x-y axes. Note though that the components will have different values for different axes. Part 3 - Comparing the graphical and mathematical methods for finding the resultant of two vectors. (1) Remove the pulleys from the experiment above. Mount a new pulley at 20º and hang 0.98N from the string. Mount another pulley at 140º and suspend 1.96N from it. Place your x-axis at 0 on the force table. (a) On the graph paper found of the next page, use a graphical method to find the resultant of these two vectors (find the magnitude and the direction of the resultant) and write it down below. (b) Now use the mathematical method to find the resultant (resolve the two vectors into their components, add the two sets of components then use trigonometry to find the resultant). 6
(c) Do you obtain different answers for the two methods used? Are there any errors involved in either of the methods? Which do you think is more accurate? 7
Questions (d) Use the force table to verify that your calculations are correct, have you TA check that you have calculated the resultant correctly Finally: Now answer these problems. 1) Mary needs to row her boat across a 100m wide river that is flowing to the east at a speed of 1.5m/s. Mary can row with a speed of 3m/s relative to the water. If Mary rows straight North how far downstream will she land? 2) The figure shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second rope with 5.0 units of force. (a) How hard must you pull on the third rope to keep the knot from moving? (b) In what direction must you pull on the third rope to keep the knot from moving? 8