URGENT SUMMER SCHOOL CALCULUS

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A force of magnitude 15 N is the resultant of two forces, one of which has a magnitude of 8 N and acts at an angle of 30° to the resultant. Find the magnitude and direction of the other force...how do I begin to solve this!?

  • URGENT SUMMER SCHOOL CALCULUS -

    Law of cosines:

    The side you are looking for is opposite the 30 degree angle.

    c^2=15^2+8^2-2*15*8cos30
    I get a magnitude of about 9
    For the angle, Iwould use law of sines. Draw a sketch first.

  • URGENT SUMMER SCHOOL CALCULUS -

    Well, I am taking the resultant to be along the x axis and the 8 N vector at 30 degrees above the x axis.
    Then our vector of length x will lie below the x axis an angle y
    Now there are two things that have to be true
    8 cos 30 + x cos y = 15
    8 sin 30 - x sin y = 0
    or
    6.93 + x cos y = 15
    4 - x sin y = 0
    or
    x cos y = 8.07
    x sin y = 4
    but we know sin y / cos y = tan y so divide
    tan y 4/8.07
    y = 26.4 degrees below x axis
    which is
    26.4 + 30 = 56.4 degrees from original vector
    x = 4/sin26.4 = 9 Newtons

  • URGENT SUMMER SCHOOL CALCULUS -

    You can also solve this problem purely algebraically without using geometry or drawing diagrams. If we denote the resultant force by R, the force of magnitude 8 N at an angle of 30° to the resultant by y and the unknown force by x, we have:

    x + y = R ------->

    x = R - y

    Square both sides:

    x^2 = R^2 + y^2 - 2 R dot y

    The inner product R dot y can be expressed as:

    R dot y = |R||y| cos(30°)

    This then gives |x| = about 9 N

    To find the angle of x to the resultant (let's call this theta), you can take the inner product of x with R and use that

    x dot R = |x| |R| cos(theta)

    We have:

    x = R - y

    taking the inner product of both sides with R gives:

    x dot R = R^2 - y dot R =

    R^2 -|y||R|cos(30°)

    If we divide both sides by |x||R| we get cos(theta) on the l.h.s.:

    cos(theta) = [|R| - |y|cos(30°)]/|x|

    this gives theta = 26.4°

  • URGENT SUMMER SCHOOL CALCULUS -

    My diagram doesn't work out, I have done it to scale, but I can't figure out how to draw it. I got the same answers as everyone above, I used the cosine and sine law to do mine (I got my answer after I posted :) ). But my diagram looks weird. Any thoughts?

  • URGENT SUMMER SCHOOL CALCULUS -

    15 cos

  • URGENT SUMMER SCHOOL CALCULUS -

    two forces of magnitude 10N and 30N act upon a body in direction making angle of 30 and 45 with x axis find the ressultant force?

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