PHYS 1111L LAB 2. The Force Table

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1 In this laboratory we will investigate the vector nature of forces. Specifically, we need to answer this question: What happens when two or more forces are exerted on the same object? For instance, in the picture below, suppose each player kicks the ball at the same time with different strengths. Which way will the ball move? (Please ignore the fact that the player on the right is fouling the player on the left!) From Newton s Laws of Motion, we know that each kick will impart an acceleration to the ball. But the effective acceleration of the ball will be in the direction of the net force on it, and this net force will not be, in general, the arithmetic sum of the individual forces. Instead, experiments show and such questions can only be answered by experiment that the net force will be the vector sum of the individual forces. In our experiments we will demonstrate that the direction of a force is as important to determining its effect as its magnitude, and we will develop a rigorous method for adding forces.

2 Principles Weight and mass The forces in these experiments will be the weights of various masses. The weight W of any mass m is given by: W = mg (g = 9.80 m/s 2 ) where g is the acceleration of gravity. The above expression follows from Newton s Law of Gravitation and Newton s Second Law of Motion and is valid near the surface of the Earth. Scalars and Vectors Quantities like mass, length and time can be specified by a single number (and unit). When we add such quantities, we can use simple arithmetic. Such quantities are called scalars. 2 kg + 3 kg = 5 kg Quantities like displacements, velocities, accelerations and forces have directions associated with them. The magnitudes of these quantities are scalars, but the directions of these quantities also 5 kg matter. These quantities are best specified by arrows. The length of the arrow represents the magnitude of the quantity, and the Weight = direction of the arrow represents the direction of the quantity. Such quantities are called vectors. When more than one force acts on an object, we have to use a new type of addition to find the magnitude and direction of the net force: We have to add arrows. 50 N Downward The two forces on the mass 5 kg can be added like vectors to find the net force. 5 kg + =

3 Adding Vectors The operation of adding vectors is defined by the following procedure. Suppose we want to add forces A and B, represented by the arrows below: Force A Force B First, we must decide on a scale with which to represent the vectors. For instance, we might chose to let each centimeter in our drawing represent one unit of force. To find the sum of these two forces, we then: Draw the first arrow to scale in the direction in which it points; Draw the second arrow to scale in the direction in which it points, starting at the tip of the first arrow; The net force is the arrow that runs from the tail of the first to the tip of the last. A B A + B The vector A+B in the above is called the resultant of A and B. Its magnitude can be found by measuring its length with a ruler and converting this length to force units according to the scale at which A and B were drawn. Its direction relative to A or B or to some reference direction can be measured with a protractor. Any number of vectors can be added in this way. The above illustrates the graphical method for adding vectors. Analytical Method for Adding Vectors Although the above procedure defines vector addition, we can use trigonometry and the Pythagorean Theorem to add vectors analytically. That is, we can add vectors without actually drawing them on paper. To do this, we first need to define the components of a vector.

4 The diagram at right shows the vector A as the vector sum of A x and A y. A x and A y are the projections of A on the x- and y-axes, respectively. (A y has been drawn to the right of the y- axis to make the vector sum clear, but what defines A y is its length and the fact that it is parallel y-axis.) A θ A y A x and A y are the components of A in the x- and y- directions. They show the length of A in these directions. From trigonometry, we see that A x A x = Acosθ and A y = Asinθ where A is the magnitude of A and θ is the angle with Magnitude the x-axis. and Direction of a Force The second thing we need is to find the magnitude and direction of a force from its components. We can use the Pythagorean Theorem to find the magnitude of the vector, since by definition the components and the vector itself form a right triangle, 2 2 A = ( A x + A y ) The direction of A can be found from the tangent of the angle that A makes with the x- axis: Ay tan θ = A x The analytical method for adding vectors consists of breaking each vector into its components, adding these components to find the components of the sum, then finding the magnitude and direction of the sum as above. The procedure will be developed in the experiments that follow. Equilibrium If two or more forces sum to zero length, then the net force will be zero and the object is said to be in equilibrium. For instance, in the diagram below, the red force arrows sum to zero length when added by vector addition.

5 The blue ring must be in equilibrium, since there is no net force on it. Also, each of the three forces must be equal and opposite to the sum of the other two. A force vector that puts a system in equilibrium is called an equilibrant. Experiments Set up & Preliminary Notes 1. Set up the force table with four pulleys around its circumference and with the center pin inserted into the center of the platform. Adjust the feet of the table so that it is level with the ground. 2. Included in your set up will be a set of metal rings tied together with strings. Find the average mass of one ring + string by dividing the total mass by the number of rings. If the rings in your setup are not all the same type, then weigh each ring individually. Record these values. 3. The ring with several strings tied to it is the center ring. Place this over the center pin on the force table. Drape the other rings over the pulleys. Rings not used in your experiment should rest on top of the force table. Measure all masses to the nearest gram. Measure all angles to the nearest tenth of a degree. Never untie the strings. Avoid creating knots in the strings: carry the ring set by holding the center ring and allowing the other rings to dangle. Leave the rings on the force table when you are done. When hanging a weight over a pulley, make sure that the string is perpendicular to the edge of the table. Experiment 1: Equilibrium Objective: Find Equilibrium for Two Weights 1. Place a pulley at the 53 mark on the force table and drape a string and ring over it. Hang a mass hanger from the ring and place 200-grams on the hanger. Observe the effect on the center ring: in which direction is it pulled? 2. Calculate and record the weight in Newtons of the total mass hanging from this pulley. Also record the direction of this force. Label this force F 1.

6 3. Place another pulley at the 0 mark and hang an identical amount of mass from it. Label this force E 1. Slowly slide the pulley with the second weight around the rim of the table until you find equilibrium. Observe what happens to the center ring as you slide the pulley. At equilibrium, the pin will be well centered within the metal ring. Calculate and record the magnitude and direction of E 1. Note: In the experiments that follow, the center ring will be your equilibrium indicator. At equilibrium, the center ring should be well-centered with no tendency to move to one side or another. Since some frictional forces may act on the system, the exact position for weight at equilibrium may be ambiguous. A good test for equilibrium is to pull the center ring to the side and release it. At equilibrium, the ring will exhibit no tendency to be pulled to one side or another. 4. In your conclusions, address these questions: At equilibrium, what is the arithmetic sum of the forces acting on the center ring? At equilibrium, what is the vector sum of the forces acting on the center ring? Experiment 2: Components of a Force Objectives: Find the components of a force experimentally, graphically, and analytically. Compare the effects of the components of a force to the effect of the force itself. 1. Remove F 1 from the force table and set its ring on top of the force table, so that the only force acting on the ring is E Hang masses at 0.0 degrees and at 90.0 degrees so that the center ring returns to equilibrium. Do this by trial and error. Narrow down the needed set of masses to within 5 grams. These special weights are called the components of F 1, relative to a coordinate system that has axes in the 0-degree and 90-degree directions on the force table. Convince yourself that no other set of masses will balance E 1 when hung at these two directions. 3. Calculate the magnitudes of these two forces and record them as experimental values for F 1x and F 1y.

7 Note that by definition, the directions of these forces are 0- and 90-degrees respectively. To help you organize your work, here is a possible format for your data and calculations: Components of F 1 F 1x F 1y Magnitude F 1 F 1 Direction Experimental Values Graphical Values Analytical Values (= mg) (given) 4. Draw a vector diagram of this system of forces. Use dashed lines to draw x- and y-coordinate axes on graph paper, with the origin at the center of the page. The positive x-axis represents the 0- degree direction on the force table and the positive y-axis represents the 90-degree direction. Set up a scale for your diagram: let each Newton of force be represented by five centimeters in the diagram. Label the diagram with this scale factor: 1 cm = 0.2 Newtons, and give the diagram a title. Draw F 1 and E 1 to scale on the diagram in the appropriate directions. Use bold lines to draw the forces and draw small arrows at the ends of each to indicate their directions. Draw F 1x along the x-axis, using a bold line with an arrow at the end. Draw F 1y by drawing a bold line from the end of F 1x in the 90-degree direction. 5. Calculate the magnitude and direction of the vector sum of F 1x and F 1y (experimental values). Compare these calculated values with the actual values for the magnitude and direction of F 1.

8 6. Determine the values of F 1x and F 1y directly from your force diagram: use a ruler placed at the tip of F 1 to find its projection on the coordinate axes. Call these values the graphical values for F 1x and F 1y. 7. Finally, use analytical means to determine F 1x and F 1y directly from the magnitude and direction of F Draw a conclusion about the relationship between F 1 and its components. Experiment 3: Components of a Force in Different Coordinates Objective: Are the components of a force unique? In this experiment, you will analyze the same force as above (F 1 ) using different coordinate axes. Let the new x-axis run along the 30-degree line and the new y-axis run along the 120-degree line. In effect, the old coordinate system has been rotated counterclockwise by 30-degrees. 1. By trial and error, find masses that balance E 1 when they are hung at 30 degrees and at 120 degrees. Calculate the weights of these masses and record their magnitudes. Call these new components F 1x* and F 1y*. 2. On the same force diagram as above, draw the new x- and y-axes. In this new coordinate system, what has changed about the original force F 1? What is its magnitude in this new system? What is its direction relative to the new axes? Does it need to be redrawn? 3. As above, add F 1x* and F 1y* (experimental values) graphically by drawing them on your diagram. Calculate the magnitude and direction of their vector sum and compare their sum to the original F Also determine F 1x* and F 1y* graphically and analytically, as above. 5. Draw a conclusion about the relationship between a force vector and the coordinate system in which it is analyzed. Consider these questions: What would happen to the values of F 1x* and F 1y* if we had chosen an x-axis along the 53-degree line on the force table? What if we had chosen the y-axis to be along the 53-degree line? Could we choose axes such that either or both F 1x* and F 1y* are negative numbers?

9 Experiment 4: Addition of Two Forces Objective: Find the net force when two forces act. Now we want to hang two weights from the center ring that do not sum to zero. This time, we will use a force diagram to predict their vector sum the net force. We will then hang the equilibrant for this predicted net force and test our results. We will also determine the net force by the analytical method. 1. Hang 80 grams on a hanger at 45.0 degrees; and hang 130 grams on a hanger at degrees. Call these weights F 2 and F 3 respectively. Calculate and record the magnitudes of these weights. Set up a table, similar to what you used above, to record your data and calculations. You will need to record the magnitude and direction for each force, and the magnitude and direction of the x- and y-components of each force. All these values will be determined by the graphical method. Here s a possible format: Graphical Addition of Two Forces Mag. Dir. Mag. Dir. Mag. Dir. F 2 F 3 R F 2x F 3x R x F 2y F 3y R y 2. Draw a vector diagram of the system on graph paper as you did above, using the 0- and 90-degree directions for your coordinate axes. Draw F 2 and F 3 to scale, with their tails at the origin. (The origin represents the center ring.) 3. Add these two forces graphically by redrawing F 3 at the tip of F 2 (or F 2 at the tip of F 3 ). The vector sum of these two is the vector that runs from the origin to the tip of the second vector. Draw this sum and label it R (for resultant). 4. Using your diagram and a ruler and protractor, determine the magnitude and direction of R. (The direction should be measured from the x-axis.) Record these values. 5. Determine the mass that corresponds to R and hang this from the force table in the direction opposite to the direction of R. As before, we will call this balancing

10 force E, for equilibrant. If your system does not balance, or nearly so, go back and find your mistake. 6. Now find R in another way. On your diagram, indicate the x- and y- components of F 2 and F 3 and R. As before, use a ruler to find the projection of the tip of each vector on the coordinate axes, and mark where each vector projects to each axis. Record the values of these components, including their sign ( + or - ). The sign indicates the direction the component points along its axis. 7. Add the two x-components to each other, and add the two y-components. Show in your calculations that: F 2x + F 3x = R x and F 2y + F 3y = R y 8. Draw these two component sums as vectors on your diagram and show in your diagram that: R x + R y = R 9. Finally, find the magnitude and direction of R using analytical means only: a. Find the x- and y- components of each given force using its given magnitude and direction; b. Find the x- and y-components of R by summing the components of the forces to be added; c. Find the magnitude of R using the Pythagorean Theorem; d. Find the direction of R using trigonometry. (Note on the inverse tangent function in calculators: For vectors that lie in the 2 nd or 3 rd quadrants, you must adjust your calculator s answer for the inverse tangent by adding 180 degrees. This arises because your calculator cannot distinguish between tan -1 (-y/x) and tan -1 (y/-x).) Set up a new table to record these analytical results. Compare these analytical results for the magnitude and direction of R with those found by the graphical method. 10. Draw conclusions from your work in this experiment. How do your results from the graphical diagram compare to your analytical results? What can you say about the components of the force vectors and their sum?

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