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    Work, vectors, and the inner product

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    4. A 1.5 kg object moving along the x axis has a velocity of +4.0 m/s at x = 0. If the only force acting on this object is shown in the figure, what is the kinetic energy of the object at x = +3.0m?

    5. If the vectors A and B have magnitudes of 10 and 11, respectively, and the scalar product of these two vectors is -100, what is the magnitude of the sum of these two vectors?

    6. Two vectors A and B are given by A = 4i + 8j and B = 6i - 2j. The scalar product of A and third vector C is -16. The scalar product of B and C is +18. The z component of C is 0. What is the magnitude of C?

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    SOLUTION This solution is FREE courtesy of BrainMass!

    Problem 4

    Let's first compute how much work the force performs on the object as it moves from x = 0 meters to x = 3 meters. This is given by the integral of F dx. From x = 0 meters to 2 meters F is positive and the contribution to the integral is the area enclosed by the x-axis, the y-axis and the graph of the force which is 8 J.

    From x = 2 meters to 3 meters the contribution to the integral is minus the area anclosed by the graph, the segment on the x-axis from x = 2 meters to x = 3 meters and the line parallel to the y axis given by x = 3 meters. This contribution is thus minus 2 J.

    So, the total gain in kinetic energy equals 8 J - 2 J = 6 J. The kinetic energy at x = 0 meters was

    1/2 * 1.5 kg (4.0 m/s)^2 = 12 J

    So, it follows that the kinetic energy at x = 3 meters is 12 J + 6 J = 18 J which is answer a)

    Problem 5

    If A is a vector then by A^2 one means A dot A = |A|^2, the square of the length of the vector A.

    In general, one can expand (A + B)^2 as follows:

    (A + B)^2 = A^2 + B^2 + 2 A dot B

    So, we can compute the magnitude of A + B if we know the magnitudes of A and B and the scalar product of A and B.
    In the problem it is given that A^2 = 100 and B^2 = 121 and A dot B = -100, so we find that:

    (A + B)^2 = 21, so the magnitude of A + B is sqrt(21) which is approximately 4.6, answer b).

    Problem 6

    You can solve this problem by writing C = p i-hat + q j-hat. Note that it is given that the z-component of C is zero, so you know that C is of this form. Then, from the given scalar products, you can solve for p and q and then compute the magnitude of C from that. However, this problem can also be solved without solving for C first. This works as follows. From A and B you construct a new orthonormal basis using the Gram-Schmidt process, see here:


    Let's take our first basis vector, e1, to be in the direction of A. Then we have to normalize e1 such that it has a magnitude of 1, so we put:

    e1 = A/|A| = A/sqrt(80)

    Then, using B, we construct a vector orthogonal to A by subtracting from B the component of B in the direction of A. Let's call this vector u2:

    u2 = B - (B dot e1) e1 = B - (B dot A)/A^2 A = B - 8/80 A = B - A/10

    Here I used that the scalar product of A and B is 8 (A = 4 i + 8 j, 6 i - 2 j, so A dot B = 4*6 - 8*2 = 24 - 16 = 8).

    The next step is to normalize u2. We put:

    e2 = u2/|u2| = (B - A/10)/| B - A/10|

    We can simplify this:

    (B - A/10)^2 = B^2 + A^2/100 - A dot B/5 = 40 + 0.8 - 1.6 = 39.2

    So, we have:

    e2 = (B - A/10)/sqrt(39.2)

    Now, we can expand C in terms of the new orthogonal unit vectors e1 and e2 using the general formula:

    C = (C dot e1) e1 + (C dot e2) e2.

    We have:

    C dot e1 = C dot A/sqrt(80) = -16/sqrt(80)

    And, using the linearity of the scalar product:

    C dot e2 = [C dot B - 1/10 C dot A]/sqrt(39.2) = [18 + 16/10]/sqrt(39.2) = 19.6/sqrt(39.2)

    The square of the magnitude of C is given by the sum of the squares of these inner products (just like when you express a vector in terms of the usual unit vectors i and j). To see this, just write

    C^2 = [(C dot e1) e1 + (C dot e2) e2]^2, and expand using that e1^2 = e2^2 = 1 and e1 dot e2 = 0, yielding:

    C^2 =(C dot e1)^2 + (C dot e2)^2 = 16^2/80 + 19.6^2/39.2 = 13. This means that:

    |C| = sqrt(13) = approximately 3.6 = answer c)

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