The Study of Concurrent Forces with the Force Table Apparatus: Force table with 4 pulleys, centering ring and string, 50 g weight hangers, slotted weights, protractors, and rulers. Discussion: The force table is designed to help you study the properties of forces at known angles. Only when forces are along the same line do they add by ordinary algebra. If two or more forces on the same body form angles with each other, it is necessary to use geometry to find the amount and direction of their combined effect. Prior to Lab: Complete the example problem attached at the end of the lab instructions. Overview of the experiment: This experiment has three parts. For the first part of the experiment you will take two forces at right angles to each other and find their resultant. Next you will find the components of a force. Finally, in the third part of the experiment you will combine several forces using their components. Procedure: Part 1. Consider two forces at right angles to each other, one equal to the weight of 300 g and the other to the weight of 500 g (2.94 and 4.90 Newtons, respectively). Let the 4.90 N weight be at 0 o, and the 2.94 N weight be at 90 o. Set these up on the force table and find by experiment the magnitude and direction of the force on a third string which will be equal and opposite to the effect of the two forces working together. This force is called the equilibrant. Write the magnitude (in Newtons) and direction of the equilibrant in your lab notebook. The equilibrant is equal and opposite to their resultant. Write down the magnitude and directions of the resultant, expressing the magnitude in Newtons. Next compute the resultant of these same two forces by use of the Pythagorean theorem, which applies in the case where the forces are at right angles to each other. Draw a force parallelogram (a rectangle in this case) to scale letting 1 Newton equal a length of 1 cm and, using one triangle, find the angle of the resultant, using: 2.94N sinθ = R How does this (theoretical) calculated value compare with the resultant determined using the force table? Calculate the per cent difference in magnitude and angle. Part 2. Consider a force of 9.80 Newtons (the weight of 1 kg.) acting at an angle of 35. Is it possible for two forces acting at 0 o and at 90 o to produce the same effect when acting together as this force when acting alone? If so, then this 9.80 n. force can be represented by a diagonal of a rectangle with the two component forces represented by 1/11/2009 1
the sides from the same vertex. Compute the component along the zero degree line; i.e., the x-component, as follows: Using the lower triangle in the figure, F x o cos35 9.80N = o F = 9.80Ncos35 = x Compute the y-component, using: F y F = 9.80 N F y o sin 35 9.80N = o F = 9.80Nsin 35 = y 35 o F x If these two forces acting together are actually equivalent to the original force, than their opposites acting together should be opposite and equal to the original force. Set up the 9.80 N force at 35 o on the force table and also the forces equal in magnitude and opposite in direction to the two computed components. Do they counteract each other to produce equilibrium? Part 3. Finally, the resultant of three given forces is to be found by the method of components. The three forces you will use will be given by your instructor. 1. Set up the forces given you on the force table. 2. Using the fourth string, determine by trial and error the direction that the force should act to balance the vector sum of the first three forces. 3. Clamp a fourth pulley at the location of the equilibrant force found in step 2. 4. Using a weight hanger and slotted weights determine the magnitude of the equilibrant required to balance the three forces. 5. Show your lab instructor your result. 6. Using the method of components (see example at beginning of instructions) calculate the vector sum of the three forces. What should the angle of the equilibrant be? 7. Calculate the per cent difference between the direction (angle) of the calculated equilibrant and the equilibrant measured using the force table. 8. Repeat 7 for the magnitudes of the calculated and measured equilibrants. Questions: A. What do you conclude about the nature of vector sums? Can you use simple algebra to determine a vector sum? B. What may have contributed to your experimental error? 1/11/2009 2
Prior to Lab Problem: Complete the following problem: The component method of adding vectors is given here for three sample forces as follows: A = 2.45 N @ 40 o B = 3.92 N @ 165 o C = 3.43 N @ 330 o Overview of the component method of vector addition. First make a neat drawing, not necessarily to exact scale, but reasonably accurate as to sizes and angles, placing the three forces on a diagram with a pair of x and y axes. Find the angle of each force with the x-axis. This angle is called the reference angle, and is the one used to calculate sines and cosines. Next compute the x- and y-components of the three forces, placing like components in columns. Place plus or minus signs on the various quantities according to whether an x-component is to the right or left of the origin, or whether a y-component is up or down relative to the origin. Add the columns with regard to sign (subtracting the minus quantities), and place the correct sign on each sum. The resulting quantities are the x- and y-components of the resultant. Since they are forces at right angles to each other, they are combined by the Pythagorean Theorem. The angle of the resultant is found by the fact that the ratio of the y-component to the x-component is the tangent of the angle of the resultant with the x-axis. Make a new drawing showing the two components of the resultant along the x- and y-axis, and showing their resultant. This drawing will make clear whether the reference angle obtained by use of the Arctan (tan -1 ) is the actual angle with the x-axis. (If the resultant is in the second, or third quadrant the reference angle must be added to 180 o to obtain the actual angle of the resultant.) The procedure is illustrated below: Draw a sketch of the vectors and a xy coordinate system here: 1/11/2009 3
Example Problem: Fill in the following table: A x = 2.45 N cos 40 o = A y = 2.45 N sin 40 o = B x = 3.92 N cos = B y = 3.92 N sin = C x = cos = C y = sin = R x = R y = Draw a sketch of the components of the resultant in an xy coordinate system here (include the magnitude and direction of the resultant also): The magnitude of the resultant is R= R + R = 2 2 x y The angle the resultant makes with the x axis (reference angle) is R 1 y θ = tan = Rx According to your sketch above does the angle place the resultant in the correct quadrant? If not, what is the correct angle for the resultant vector? Write the resultant in both polar and unit vector notation. R = = 1/11/2009 4
Set 1 Set 2 Force θ (deg) (N) A 1.96 45 A 2.94 140 B 2.94 260 B 3.92 220 C 3.92 300 C 1.96 300 Set 3 Set 4 Force θ (deg) (N) A 2.45 60 A 1.96 40 B 2.45 150 B 3.92 150 C 2.94 200 C 2.94 270 Set 5 Set 6 Force (N) θ (deg) A 2.94 80 A 3.92 60 B 3.92 120 B 2.94 180 C 1.96 190 C 1.96 320 1/11/2009 5
Set 7 Set 8 A 3.92 70 A 2.94 170 B 1.96 135 B 3.92 210 C 3.92 290 C 1.96 280 Set 9 Set10 A 3.92 30 A 1.96 50 B 2.94 60 B 3.92 150 C 1.96 120 C 2.94 270 1/11/2009 6