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course phy 201
10/21 8:30 pm
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?
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F = 25 newtons; theta = roughly 54 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?
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fX: 48.25 newtons; fY: -13.1 newtons
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You might have had your calculator in degree mode here.
50 degrees isn't that far from 45 degrees. A sketch will show you that both the x and y components are positive, that the y component is somewhat greater than the x component, but that it's not all that much greater.
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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?
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length of vector: 8.5 cm; length of x: 7.5 cm; length of y: 3.25 cm (all rounded to the nearest ¼ of a centimeter)
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What therefore are the magnitude and angle of that vector?
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magnitude: 8.5 cm; angle: roughly 23.42 degrees
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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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if we know how much weight with the acceleration of gravity pulls the rubber band to that length, then we know how much force it should take to bring the rubber band to that magnitude.
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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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force exerted by base vector: 0.462 newtons; force exerted by x-vector: 0.065 newtons; force exerted by y-vector: 0.457 newtons;
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At 23 degrees, the x component will be greater than the y component.
Check the mode of your calculator.
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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?
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R_x: -0.2;
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What is the total R_y of the forces in the y direction?
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R_y: 0.3;
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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?
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angle: 123.662 degrees; magnitude: 0.36 newtons
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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?
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80%
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The vectors R_1 = P_0-P_1 and R_2 = P_0-P_2 are <-6, 2> and <2, 9>.
These vectors are nearly perpendicular, The projection would be small compared to the vector.
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What do you estimate to be the angle between P_1 and P_2?
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85 degrees
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This is a reasonable estimate of the angle.
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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?
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105%
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The projection can't be longer than the vector being projected.
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What do you estimate to be the angle between P_1 and P_2?
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70 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?
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115%
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At 70 degrees, the projection would be less than half as long as the vector being projected.
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&#Your work looks good. See my notes. Let me know if you have any questions. &#