Vector Addition
Content
University of Asia and the Pacific
physics laboratory activity 3
Vector Addition
A Theory
This experiment finds the resultant of adding two vectors by three methods: experimentally, by components, and graphically. NOTE: In all cases, the force caused by the mass hanging over the pulley is found by multiplying the mass by the acceleration due to gravity, g= 9.8 m/s2.
B Objective
The purpose of this experiment is to use the force table to experimentally determine the force which balances two other forces. This result is checked by adding the two forces by using their components and by graphically adding the forces. 1 Experimental Method Two forces are applied on the force table by hanging masses over pulleys positioned at certain angles. Then the angle and mass hung over a third pulley are adjusted until it balances the other r two forces. This third force is called the equilibrant ( FE ) since it is the force which establishes r equilibrium. The equilibrant is not the same as the resultant ( FR ). The resultant is the addition of the two forces. While the equilibrant is equal in magnitude to the resultant, it is in the opposite direction because it balances the resultant (see Figure 1). So the equilibrant is the negative of the resultant:
r r r r − FE = FR = FA + FB
r FR
>F
r
A
Figure 1 The Equilibrant Balances the Resultant
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University of Asia and the Pacific
y _ FB By _ FA x Ax Ry
y _ FR
x Rx
Figure 2 Components 2 Component Method Two forces are added together by adding the x- and y-components of the forces. First the two forces are broken into their x- and y-components using trigonometry:
r r ˆ ˆ ˆ ˆ FA = A x x + A y y and FB = B x x + B y y
r ˆ where Ax is the x-component of vector FA and x is the unit vector in the x-direction. See Figure r r 2. To determine the sum of FA and FB , the components are added to get the components of the r resultant FR :
r ˆ ˆ FR = (A x + B x ) x + (A y + B y ) y
To complete the analysis, the resultant force must be in the form of a magnitude and a direction (angle). So the components of the resultant (Rx and Ry) must be combined using the Pythagorean Theorem since the components are at right angles to each other:
r 2 2 FR = R x + R y
And using trigonometry gives the angle:
tan θ = Rx Ry
3 Graphical Method Two forces are added together by drawing them to scale using a ruler and protractor. The second r r r force ( FB ) is drawn with its tail to the head of the first force ( FA ). The resultant ( FR ) is drawn r r from the tail of FA to the head of FB . See Figure 3. Then the magnitude of the resultant can be
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University of Asia and the Pacific
measured directly from the diagram and convel1ed to the proper force using the chosen scale. The angle can also be measured using the protractor.
r FB r FA
Figure 3 Adding Vectors Head to Tail
θ
C Procedure
1 Setup a Assemble the force table as shown in the Assembly section. Use three pulleys (two for the forces that will be added and one for the force that balances the sum of the two forces). b Screw the center post up so that it will hold the ring in place when the masses are suspended from the two pulleys. c Hang the following masses on two of the pulleys and clamp the pulleys at the given angles: Force A = 50 g at 0° Force B = 100 g at 120° 2 Procedure (Experimental Method) a By trial and error, find the angle for the third pulley and the mass, which must be suspended from it that will balance the forces exerted on the strings by the other two masses. The third r force is called the equilibrant ( FE ) since it is the force which establishes equilibrium. The equilibrant is the negative of the resultant: r r r r − FE = FR = FA + FB b Record the mass and angle required for the third pulley to put the system into equilibrium in Table I in the Data Sheet. c To determine whether the system is in equilibrium, use the following criteria. 3 Method of Finding Equilibrium a The ring should be centered over the post when the system is in equilibrium. Screw the center post down so that it is flush with the top surface of the force table and no longer able to hold the ring in position. Pull the ring slightly to one side and let it go. Check to see that the ring returns to the center. If not, adjust the mass and/or angle of the pulley until the ring always returns to the center when pulled slightly to one side.
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University of Asia and the Pacific
b Draw and label the configuration on the attached force table template in the Data Sheet.
4 Analysis: To determine theoretically what mass should be suspended from the third pulley, and
r at what angle, calculate the magnitude and direction of the equilibrant ( FE ) by the component method and the graphical method. a Component Method: On a separate piece of paper, add the vector components of Force A and Force B to determine the magnitude of the equilibrant. Use trigonometry to find the direction (remember, the equilibrant is exactly opposite in direction to the resultant). Record the results in Table 1 in the Data Sheet. b Graphical Method: On the given graphing paper, construct a tail-to-head diagram of the vectors of Force A and Force B. Use a metric rule and protractor to measure the magnitude and direction of the resultant. Record the results in Table 1 in the Data Sheet. Remember to record the direction of the equilibrant, which is opposite in direction to the resultant.
Reference
PASCO Scientific. Instruction Manual and Experiment Guide. November 1991
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University of Asia and the Pacific
Data Sheet
Vector Addition
Name Section Date Group No.
Table 1 Result of the Three Methods of Vector Addition
r
Method
Experiment:
Equilibrant ( FE )
Magnitude Direction (θ)
Component:
Rx =_________________ Ry =______________
Graphical:
1. How do the theoretical values for the magnitude and direction of the equilibrant compare to the actual magnitude and direction? Compute for percent difference of each. Component-Experimental:
Graphical- Experimental:
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University of Asia and the Pacific
2. Use the attached graphing sheet for the graphical method. Make your drawings and labels clear and neat. Label with units in Newton. Sheet for Graphical method:
3. Draw and label the configuration of the experimental method on the attached force table template.
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physics laboratory activity 3
Vector Addition
A Theory
This experiment finds the resultant of adding two vectors by three methods: experimentally, by components, and graphically. NOTE: In all cases, the force caused by the mass hanging over the pulley is found by multiplying the mass by the acceleration due to gravity, g= 9.8 m/s2.
B Objective
The purpose of this experiment is to use the force table to experimentally determine the force which balances two other forces. This result is checked by adding the two forces by using their components and by graphically adding the forces. 1 Experimental Method Two forces are applied on the force table by hanging masses over pulleys positioned at certain angles. Then the angle and mass hung over a third pulley are adjusted until it balances the other r two forces. This third force is called the equilibrant ( FE ) since it is the force which establishes r equilibrium. The equilibrant is not the same as the resultant ( FR ). The resultant is the addition of the two forces. While the equilibrant is equal in magnitude to the resultant, it is in the opposite direction because it balances the resultant (see Figure 1). So the equilibrant is the negative of the resultant:
r r r r − FE = FR = FA + FB
r FR
>F
r
A
Figure 1 The Equilibrant Balances the Resultant
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physics laboratory activity 3: Vector Addition
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University of Asia and the Pacific
y _ FB By _ FA x Ax Ry
y _ FR
x Rx
Figure 2 Components 2 Component Method Two forces are added together by adding the x- and y-components of the forces. First the two forces are broken into their x- and y-components using trigonometry:
r r ˆ ˆ ˆ ˆ FA = A x x + A y y and FB = B x x + B y y
r ˆ where Ax is the x-component of vector FA and x is the unit vector in the x-direction. See Figure r r 2. To determine the sum of FA and FB , the components are added to get the components of the r resultant FR :
r ˆ ˆ FR = (A x + B x ) x + (A y + B y ) y
To complete the analysis, the resultant force must be in the form of a magnitude and a direction (angle). So the components of the resultant (Rx and Ry) must be combined using the Pythagorean Theorem since the components are at right angles to each other:
r 2 2 FR = R x + R y
And using trigonometry gives the angle:
tan θ = Rx Ry
3 Graphical Method Two forces are added together by drawing them to scale using a ruler and protractor. The second r r r force ( FB ) is drawn with its tail to the head of the first force ( FA ). The resultant ( FR ) is drawn r r from the tail of FA to the head of FB . See Figure 3. Then the magnitude of the resultant can be
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University of Asia and the Pacific
measured directly from the diagram and convel1ed to the proper force using the chosen scale. The angle can also be measured using the protractor.
r FB r FA
Figure 3 Adding Vectors Head to Tail
θ
C Procedure
1 Setup a Assemble the force table as shown in the Assembly section. Use three pulleys (two for the forces that will be added and one for the force that balances the sum of the two forces). b Screw the center post up so that it will hold the ring in place when the masses are suspended from the two pulleys. c Hang the following masses on two of the pulleys and clamp the pulleys at the given angles: Force A = 50 g at 0° Force B = 100 g at 120° 2 Procedure (Experimental Method) a By trial and error, find the angle for the third pulley and the mass, which must be suspended from it that will balance the forces exerted on the strings by the other two masses. The third r force is called the equilibrant ( FE ) since it is the force which establishes equilibrium. The equilibrant is the negative of the resultant: r r r r − FE = FR = FA + FB b Record the mass and angle required for the third pulley to put the system into equilibrium in Table I in the Data Sheet. c To determine whether the system is in equilibrium, use the following criteria. 3 Method of Finding Equilibrium a The ring should be centered over the post when the system is in equilibrium. Screw the center post down so that it is flush with the top surface of the force table and no longer able to hold the ring in position. Pull the ring slightly to one side and let it go. Check to see that the ring returns to the center. If not, adjust the mass and/or angle of the pulley until the ring always returns to the center when pulled slightly to one side.
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University of Asia and the Pacific
b Draw and label the configuration on the attached force table template in the Data Sheet.
4 Analysis: To determine theoretically what mass should be suspended from the third pulley, and
r at what angle, calculate the magnitude and direction of the equilibrant ( FE ) by the component method and the graphical method. a Component Method: On a separate piece of paper, add the vector components of Force A and Force B to determine the magnitude of the equilibrant. Use trigonometry to find the direction (remember, the equilibrant is exactly opposite in direction to the resultant). Record the results in Table 1 in the Data Sheet. b Graphical Method: On the given graphing paper, construct a tail-to-head diagram of the vectors of Force A and Force B. Use a metric rule and protractor to measure the magnitude and direction of the resultant. Record the results in Table 1 in the Data Sheet. Remember to record the direction of the equilibrant, which is opposite in direction to the resultant.
Reference
PASCO Scientific. Instruction Manual and Experiment Guide. November 1991
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University of Asia and the Pacific
Data Sheet
Vector Addition
Name Section Date Group No.
Table 1 Result of the Three Methods of Vector Addition
r
Method
Experiment:
Equilibrant ( FE )
Magnitude Direction (θ)
Component:
Rx =_________________ Ry =______________
Graphical:
1. How do the theoretical values for the magnitude and direction of the equilibrant compare to the actual magnitude and direction? Compute for percent difference of each. Component-Experimental:
Graphical- Experimental:
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University of Asia and the Pacific
2. Use the attached graphing sheet for the graphical method. Make your drawings and labels clear and neat. Label with units in Newton. Sheet for Graphical method:
3. Draw and label the configuration of the experimental method on the attached force table template.
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