EXPERIMENT 4 VECTORS AND EQUILIBRIUM

Transcription

1 EXPERIMENT 4 VECTORS ND EQUILIRIUM INTRODUCTION: The purpose of this experiment is to confirm the laws of vector addition, and to study the equilibrium of force vectors at a point. CKGROUND: scalar is a quantity that has magnitude only; examples are temperature, mass, and density. vector is a quantity that has both magnitude and direction; examples are velocity, acceleration, and force. vector may be represented by a straight line in the direction of the vector, with the length of the line proportional to its magnitude. Placing an arrowhead at the end of the line indicates the direction of the vector. Vectors may be added. The sum or resultant of two or more vectors is defined as the single vector that produces the same effect. Figure 1 shows the resultant of two forces and. 180 Equilibrant Resultant Figure 1: The resultant and equilibrant of two forces,. The resultant is defined as the force equal and opposite to the resultant as shown in Figure 1. If the resultant is added to the sum of and the sum of the forces equals zero, and the system of forces is in equilibrium. Vector addition may be accomplished graphically or analytically. Using the graphical method for more than two forces we have the polygon method of vector addition: the vectors to be added are placed so that the tail of the second is on the head of the first vector, maintaining their original directions. The tail of the third vector is placed on the head of the second vector, etc. when all the vectors are in place, the side which closes the polygon is the resultant of the vectors. This is shown in figure 2, for the addition of vectors,, C, and D. If the polygon closes by itself, the resultant is equal to zero and the vectors, if representing forces, are in equilibrium. R D Figure 2: Polygon method to add four vectors,, C, D. C R Resulant Figure.3: Polygon method for addition of two vectors, Fig. 4. The Parallelogram method to add two vectors,

2 Then addition of the two vectors is most conveniently carried out by the parallelogram method shown in figure 4. Vectors may also be added analytically, and this is preferred to the graphical method since one does not have to make precise drawings. The method is illustrated in figure 5 for the addition of two vectors and. The vectors are broken down into components: y = x i + y j y R where = x i + y j θ y θ θ x x = cos θ x = cos θ y = sin θ y = sin θ x i x x Figure 5. nalytic addition of two vectors, then R = R x i + R y j where R x = x + x, and R y = y + y The magnitude of R is then R = (R x ² + R y ²) 1/2 = [( x + x )² + ( y + y )²] 1/2 (1) while the angle that R make with the x-axis is given by θ = tan -1 (R y /R x ) = tan -1 [( y + y ) / ( x + x )] (2) This method may be extended easily to the sum of any number of vectors,, C, etc. by just replacing the appropriate quantities in equations (1) and (2) by sums of all the x and y components. THE EXPERIMENT: 1- Experimental pparatus: Vectors and the equilibrium of forces may be most easily studied in the lab by means of the force table shown in figure 6. The apparatus consists of: force table, weight hanger, slotted weights, ring attached to strings, and pulleys.

3 2- Experimental Procedure: Part Mount a pulley on the 30 mark and suspend a total of 200 g over it. y means of a vector diagram drawn to scale (choose your own scale) find the magnitude of the components along the 0 and 90 directions. Set up on the force table 0 and 90 forces you found from the diagram. These forces are equivalent to the original force. Test this statement by replacing the initial force at 30 by an equal force at 180 away from the initial direction, and check for equilibrium. Have your instructor check the equilibrium Part Mount a pulley on the 20 mark on the force table and suspend a total (including the mass holder) of 100g over it. Mount a second pulley on the 120 mark and suspend a total of 200 g over it. Draw a vector diagram to scale, using a scale of 20 g per centimetre, and determine graphically the direction and magnitude of the resultant using the parallelogram method. Check your results so far by setting up the resultant on the force table. Putting a pulley 180 from the calculated direction of the resultant, and suspending weights equal to the magnitude of the resultant does this. Have your instructor check the equilibrium Part C Mount the first two pulleys as in Part, with the same weights as before. Mount a third pulley on the 220 mark and suspend a total of 150 g over it. Draw a vector diagram to scale and determine graphically the direction and magnitude of the resultant, (Hint: This may be done by adding the third vector to the sum of the first two, which was obtained in Part.) Now set up the resultant on the force table and test it as before. NLYSIS:

4 1. Calculate analytically the magnitude and direction of the resultant in part and compare to the graphical determination.

5 PHY 1400 LORTORY REPORT EXPERIMENT 4 VECTORS ND EQUILIRIUM NME:.. DTE:.. SECTION:.. THIS PGE NEEDS TO E DONE T HOME EFORE COMING TO THE L. SESSION 1. EXPERIMENTL PURPOSE: State the purpose of the experiment.( 5 points ) 2. EXPERIMENTL PROCEDURES ND PPRTUS: ) riefly outline the apparatus used and the general procedures adopted. (5 points

6 3. DT and NLYSIS: Part (20 points) ttach the graphs and the analysis Part (25 points) ttach the graphs and the analysis Part C (25 points) ttach the graphs and the analysis NLYSIS: (20 points)