Lab 3: Equilibrium of a Particle 1 Purpose To investigate force equilibrium for a particle at rest. To get practice in propagation of errors. 2 Theory Newton s 2nd law states that the vector sum of the forces on a particle is equal to the mass of the particle times its acceleration: F i = F 1 + F 2 + F 3 +... = ma. (1) i For a particle at rest (or moving with constant velocity), the acceleration is zero, i.e. a = 0. In this case, Eq. 1 states that the vector sum of the forces on a particle is zero F i = F 1 + F 2 + F 3 +... = 0. (2) i In this lab you explore experimentally the validity of Eq. 2. 3 Equipment Used Equipment List: The apparatus in this experiment consists of a force table, several pulleys, and some masses (see Fig. 1.). Set up: The force table should be set up so that the circular plate with angular markings is elevated above the table. Moveable pulleys should be put around its circumference. Markings around the circumference of the table give the angular position of the pulleys. A small metal ring with several strings tied to it should be placed at the center of the center of the table. A centering pin can be inserted in the middle of the table to hold the metal ring in place when the forces on it are not balanced. Each string attached to the ring can go over one of the pulleys and have a weight hanger attached to its other end. The weights provided can be added to the weight hangers. The angular position of the pulleys and the weights on the mass hangers are to be adjusted until the ring is centered on the table without touching the centering pin. 1
4 General Procedures Position 3 or 4 pulleys around the circumference of the table. Avoid making the position of the pulleys completely symmetric. There should be a string for each pulley. One end of each string is attached to the ring and the other end to a weight hanger. The centering pin should go through the ring. Add weights to the weight hangers until the ring does not touch the centering pin and is as close as possible to the exact center of the table. In recording the weight on each string, note that the weight hangers also have weight. Because the weights come in non-divisible units, it may be necessary to adjust the angular position of one or two pulleys to center the ring. The secret to success in this experiment is taking care to minimize error. Here are several things you should consider in trying to obtain the best results possible. Friction is the greatest source of error in this experiment. To minimize this lift the ring one or two centimeters and release it. Why is this effective? Parallax error. Devise strategies to minimize parallax error when trying to center the ring and when drawing lines to indicate the position of each string. Adjust pulley position. Because the weights come in not-so-small discrete increments, you may need to make fine adjustments in how well the ring is centered by moving the pulleys by small degrees to obtain the best balance possible. Estimate the angular position of each pulley as well as possible. What is the uncertainty in your ability to measure the angular position of each pulley? Use larger masses so that adding the smallest masses available gives the smallest fractional adjustment to the total mass on each pulley. Adjust string positions on the ring to minimize torques on the ring and thus make sure that string lines pass through a common point at the center of the ring. 5 Three Forces Set up the force table for 3 forces. Add weights to the weight hangers and adjust the positions of the pulleys until the ring is in equilibrium in the center of the table. 5.1 Graphical Analysis and Questions See Fig. 2. Fold a piece of paper in half. Remove the centering pin and place the folded paper under the strings on the force table. For each string carefully make several widelyspaced marks indicating the position of each string. Remove the paper and reconstruct the directions of the three forces by drawing lines through each pair of points. The lines should intersect at a common point (the center of the ring) and represent the directions of the three forces which we shall call A, B, and C. Now unfold the paper and use the other half of the paper to perform a vector addition of the three forces on the ring. A ruler and a triangle can be used to construct lines parallel to the 3 lines already on the paper. Place one leg of the triangle along one of the lines already 2
on the 1st half of the paper. Place the ruler along another leg of the triangle and hold the ruler firmly to the paper. Slide the triangle along the ruler until you reach a convenient place on the unused half of your paper. Draw a line parallel to the original line and make the length of this line proportional to the weight on the string corresponding to that line. This line segment represents one of the 3 vectors. Put an arrowhead on one end of the line segment so that it represents a vector pointing away from the center of the table. Repeat this process twice, starting the new line segments so that the tail of the vector added is at the head of the vector already on the 2nd half of the paper. If there were no errors, the 3 line segments should close to form a triangle. The error is the small vector necessary to close the figure. Determine the direction and magnitude of your error force. 5.2 Questions on the Graphical Analysis The instructions are to make several widely-spaced marks indicating the position of each string. Why is drawing several marks better than just drawing two? List the sources of error in your measurement and estimate the resulting uncertainties in the magnitude and direction of each of the forces. How does these compare with the size of your error force? 5.3 Analytical Analysis Consider two of the forces on the ring, A and B. The sum or resultant of these two forces must be equal and opposite to the 3rd force C. From the law of cosines we can calculate the magnitude of the resultant force R = A + B. Let θ be the interior angle between A and B legs of the triangle. This angle can be measured by reading the angles of the forces on the table. Then R = A 2 + B 2 2AB cos θ. (3) If the uncertainty in A, B, and θ are taken to be A, B, and θ, show that the uncertainty in R is R = 1 [ (A B cos θ) 2 ( A) 2 + (B A cos θ) 2 ( B) 2 + (AB sin θ) 2 ( θ) 2] 1/2, (4) R where A, B, and θ are assumed to be independent. 5.4 Questions on the Analytical Analysis How does the value of R compare with the magnitude of C for your experiment. (For a complete check the direction of R would have to be found anti-parallel to C.) Provide the reasons for error in the measurement of A, B, and θ and estimate the uncertainties A, B, and θ. Be careful to give appropriate estimates of A, B, and θ. For example, if you tell me that θ 1, I would say you re not a very good experimentalist! Using the expression above, estimate the uncertainty R in R. How does your estimate of R compare with the actual difference between the measured C and the one calculated using Eq. 3? 3
6 Four Forces Set up the force table for 4 forces and adjust the apparatus so that the ring is in equilibrium in the center of the force table. Position one of the strings along the 0 line and call this the x axis. 6.1 Analysis By Components See Fig. 3. Calculate the x components of the 4 forces and see how close their sum is to zero. Do the same for the y components of the 4 forces. Determine the error force (magnitude and direction). You can read the necessary angles off the force table. 7 Comment The rules for adding vectors, tails to heads or by components, come from mathematics. Here we see they make sense in a physical situation described by a vector equation. 7.1 Questions on Analysis By Components Propagate errors to determine the uncertainties in F x and F y. How do the uncertainties compare with the F x and F y you determined? Why are you not being asked to calculate the uncertainty on the magnitude of F? 4
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