Results 1 to 5 of 5

Math Help - Proof Questions.

  1. #1
    Member
    Joined
    Apr 2006
    Posts
    201
    Awards
    1

    Post Proof Questions.

    Need help on two proof questions I hope you guys can help!

    There is alot on the internet but it's hard the way they explain it.

    1) Prove that the centre of mass of a uniform wire in the form of a semi-circle of radius r is at a distance 2r/pie (sorry don't know how to insert symbol) from the centre.

    2) Prove that the centre of mass of a uniform semi-circular laminar of radius r is at a distance 4r/3pie from the centre.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Apr 2006
    Posts
    201
    Awards
    1

    Post Re:

    I guess everyone hates doing proof questions.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by dadon
    1) Prove that the centre of mass of a uniform wire in the form of a semi-circle of radius r is at a distance 2r/pie (sorry don't know how to insert symbol) from the centre.

    The centre of mass is the average position of the mass of a body.

    In the case of the semi-circle of m\ \mbox{kg/m}, the total
    mass is m.r. \pi\ \mbox{kg}.

    Introduce \theta as in the diagram, then an element of the wire
    has length r.d\theta and the centre of mass is:

    <br />
R=\frac{1}{m.r. \pi}\int_{wire} \rm{r}  \  \ dm=\frac{1}{m.r. \pi}\int_0^{\pi} r {\sin(\theta) \choose \cos(\theta)} m.r d\theta=\frac{r}{\pi} \int_0^{\pi} {\sin(\theta) \choose \cos(\theta)} d\theta =\frac{r}{\pi} {2 \choose 1} d\theta<br />
.

    I'm sure you can complete the problem from here.

    RonL
    Attached Thumbnails Attached Thumbnails Proof Questions.-wire.jpg  
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,829
    Thanks
    123
    Quote Originally Posted by dadon
    ...
    1) Prove that the centre of mass of a uniform wire in the form of a semi-circle of radius r is at a distance 2r/pie (sorry don't know how to insert symbol) from the centre.
    2) Prove that the centre of mass of a uniform semi-circular laminar of radius r is at a distance 4r/3pie from the centre.
    Thanks
    Hello,

    to 1) Put the semicircle into a coordinate system, so that the diameter lies on the x-Axis and the centre of the semicircle is the origin. Then the semicircle is described by:
    y=\sqrt{r^2-x^2}

    The centroid of a homogenous plane area, which has a graph of a function and the x-axis as boundaries, can be calculated in this problem here by:

    x_{c}=\frac{\int^r_{-r}{xydx}}{\int^r_{-r}{y dx}} and

    y_{c}=\frac{\int^r_{-r}{y^2dx}}{2 \cdot \int^r_{-r}{y dx}}

    x_{c}=\frac{\int^r_{-r}{x\cdot \sqrt{r^2-x^2}dx}}{\int^r_{-r}{\sqrt{r^2-x^2} dx}}

    Using the substitution method on the integral of the numerator, you'll get the value 0 (zero). So the x-coordinate of the centroid is 0, which could be expected.

    y_{c}=\frac{\int^r_{-r}{(r^2-x^2)dx}}{2 \cdot \int^r_{-r}{\sqrt{r^2-x^2} dx}}

    you'll get:

    y_{c}=\frac{\left[r^2 \cdot x -\frac{1}{3} x^3  \right]^r_{-r}} {\left[ x \cdot \sqrt{r^2-x^2} +r^2 \cdot \arcsin \left( \frac{x}{r} \right) \right]^r_{-r} }

    y_{c}=\frac{2r^3}{\frac{\pi}{2} r^2 - \left(- \frac{\pi}{2} r^2 \right)}=\frac{2r}{\pi}

    I'm awfully sorry, but I'm a little bit in hurry. I hope this was of some help.

    Greetings

    EB
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Apr 2006
    Posts
    201
    Awards
    1

    Post re:

    Thank you! You were a great help!

    I will try both problems this way and ask for help if I still need it. Iíll post my solutions as well to get your feedback. Thanks again

    Take care,

    dadon
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: December 28th 2011, 12:53 AM
  2. [SOLVED] Proof and questions about a subgroup theorem
    Posted in the Advanced Algebra Forum
    Replies: 11
    Last Post: April 13th 2011, 11:51 AM
  3. Strategy to Solve Proof Questions
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: November 8th 2009, 08:56 AM
  4. cdf proof questions
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: October 11th 2008, 08:06 PM
  5. 2 Proof Questions
    Posted in the Number Theory Forum
    Replies: 19
    Last Post: January 3rd 2007, 07:46 PM

Search Tags


/mathhelpforum @mathhelpforum