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course Phy 201
The rules for force vectors are as follows:If a force F is directed at angle theta, as measured from the positive x axis, then the components F_x and F_y of that force in the x and y direction are respectively
• F_x = F cosine(theta)
and
• F_y = F sine(theta).
If the components of a force F are F_x and F_y, then the magnitude of the force is
• F = sqrt( F_x^2 + F_y^2 )
and the angle of the force with the positive x direction is
• theta = arcTangent ( F_y / F_x ), plus 180 degrees if F_x is negative.
These four rules are all that is required to analyze force vectors in two dimensions.
`q001. If F_x = 15 Newtons and F_y = 20 Newtons, what is F and what is theta?
15^2+20^2 = c^2
F = 25
Arctan (20/15) = 53.1 degrees
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`q002. If a force of magnitude F = 50 Newtons is directed at 50 degrees as measured counterclockwise from the positive x direction, what are its components F_x and F_y?
Mgx = cos(270-50) = -.766
Mgy = sin(270-50) = -.642
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You have the right angle, and the right sines and cosines.
But you have to multiply those sines and cosines by the magnitude of the force.
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Identical rules apply to any vector quantity:
If a vector R is directed at angle theta, as measured from the positive x axis, then the components R_x and R_y of that vector in the x and y direction are respectively
• R_x = R cosine(theta)
and
• R_y = R sine(theta).
If the components of a vector R are R_x and R_y, then the magnitude of the vector is
• R = sqrt( R_x^2 + R_y^2 )
and the angle of the vector with the positive x direction is
• theta = arcTangent ( R_y / R_x ), plus 180 degrees if R_x is negative.
These four rules are all that is required to analyze force vectors in two dimensions.
`q003. At least one of the rubber bands you used in the experiment with the push pins had nonzero 'rise' and 'run'. Select one such rubber band.
The length vector for that rubber band runs from the pinhole closest to the origin to the pinhole furthest from the origin. What are the x and y components of that vector?
I missed this experiment and Morgan did not have her data with her the day that we worked on this assignment together. I am planning to get data from her for this question,
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What therefore are the magnitude and angle of that vector?
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How can you use the magnitude of the vector and the calibration graph for that rubber band to find the tension corresponding to your two pinholes?
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The force exerted by that rubber band on the paperclip in the middle is equal to its tension, and acts in the direction of that rubber band. What therefore are the x and y components of that force?
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`q004. In an experiment with three rubber bands, the observed forces F_1, F_2 and F_3 are found to have x and y components as follows:
F_1_x = 2.3 Newtons, F_1_y = 3.2 Newtons.
F_2_x = -3.0 Newtons, F_2_y = 1.8 Newtons.
F_3_x = 0.5 Newtons, F_3_y = -4.7 Newtons.
What is the total R_x of the forces in the x direction?
Theta = arctan(3.2/2.3) = 54.29
Theta = arctan(1.8/-3) = -30.96
Theta = arctan(-4.7/.5) = -83.93
Cos(270-54.29) = -.81
Cos(270-30.96) = .51
Cos(270-83.93)= .99
Total F in x direction = .69
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You've got the right angles and correct cosines, but some of those forces are bigger than others, so their cosines sort of count more.
Precisely, to get the total force in the x direction you have to find the x components of all the forces and add them. Each force would have to be multiplied by the cosine of its angle.
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What is the total R_y of the forces in the y direction?
Sin(270-54.29) = -.58
Sin(270- - 30.96) = -.86
Sin(270- - 83.93) = -.11
Total F in y direction = -1.55
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To get the total force in the y direction you have to find the y components of all the forces and add them. Each force would have to be multiplied by the sine of its angle.
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What is the magnitude of the net force R exerted by the three rubber bands, and at what angle is this net force directed?
Arctan (-1.55/.69)
Angle about 66 degrees
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This result will be modified when you take the magnitudes of the forces into consideration, but do note that the arcTan of -1.55 / .69 is negative, indicating an angle that goes counterclockwise from the positive x axis.
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`q005. For each set of three points, sketch the vector R_1 from P_0 to P_1 and the vector R_2 from P_0 to P_2. Then sketch the projection of R_2 on R_1, and answer the specified questions.
P_0 = (-4, 1), P_1 = (-10, 3), P_2 = (-2, 10).
What do you estimate the length of the projection to be, as a percent of the length of R_2?
About 75 %
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These vectors would be nearly perpendicular. The projection would be much shorter than the length of R_2.
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What do you estimate to be the angle between P_1 and P_2?
About 41 degrees
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R_1 would be pretty close to horizontal, and R_2 pretty close to vertical. The angle would be quite a bit greater than 41 degrees.
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P_0 = (2, 3), P_1 = (5, 12), P_2 = (10, 4).
What do you estimate the length of the projection to be, as a percent of the length of R_2?
About 85%
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What do you estimate to be the angle between P_1 and P_2?
About 58 degrees
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What do you estimate to be the length of the projection line, as a percent of the length of R_1?
About 95%
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You're on the right track but you'll want to rework some of these problems according to my notes.
&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).
Be sure to include the entire document, including my notes. &#
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