COM of circular arc

Classical Mechanics Level 3

For the figure shown, a thin disc which is a part of circular disc of radius R R . The disc has an angle θ \theta . If the mass of the shown disc is m m , then locate its centre of mass.

Details and Assumptions:

  • Do not take numerical values in the diagram given.

  • Assume mass and area to be distributed equally among both planes.

  • Given θ \theta is for circular arc angle from the centre O O .

y com = 4 R sin ⁡ ( θ 2 ) 3 θ y_\text{com}=\frac{4R\sin \left(\frac{\theta }{2}\right)}{3\theta } y com = 4 R sin ⁡ ( θ 2 ) 4 θ y_\text{com}=\frac{4R\sin \left(\frac{\theta }{2}\right)}{4\theta } y com = 4 R sin ⁡ ( θ 5 ) 3 θ y_\text{com}=\frac{4R\sin \left(\frac{\theta }{5}\right)}{3\theta } y com = 4 R sin ⁡ ( θ 3 ) 2 θ y_\text{com}=\frac{4R\sin \left(\frac{\theta }{3}\right)}{2\theta }

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Sahil Silare by Sahil Silare , 20, India

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1 solution

First let us find the com of a portion of a ring.

Let us consider a small element of mass dm at angle φ with the horizontal and its length is Rdφ. Mass of the ring is M.

So , y c o m y_{com} = ∫ d m y ∫ d m \frac{\int dm y}{\int dm}

d m dm = M d φ θ \frac{Mdφ}{θ}

y c o m y_{com} = ∫ M d φ θ R s i n φ ∫ M d φ θ \frac{\int \frac{Mdφ}{θ}Rsinφ}{\int \frac{Mdφ}{θ}}

y c o m y_{com} = ∫ π − θ 2 π + θ 2 M d φ θ R s i n φ ∫ π − θ 2 π + θ 2 M d φ θ \frac{\int_\frac{π-θ}{2}^\frac{π+θ}{2} \frac{Mdφ}{θ}Rsinφ}{\int_\frac{π-θ}{2}^\frac{π+θ}{2} \frac{Mdφ}{θ}}

y c o m y_{com} = 2 R s i n θ 2 θ \frac{2Rsin\frac{θ}{2}}{θ}

Now let us find com of the disc.

Let us consider a small element of mass dm of thickness dx in the disc.Mass of the disc is M.

So , y c o m y_{com} = ∫ d m y ∫ d m \frac{\int dm y}{\int dm}

d m dm = 2 M x d x R 2 \frac{2Mxdx}{R^2}

y c o m y_{com} = ∫ 0 R 2 M x d x R 2 2 x s i n θ 2 θ ∫ 0 R 2 M x d x R 2 \frac{\int_0^R \frac{2Mxdx}{R^2}\frac{2xsin\frac{θ}{2}}{θ}}{\int_0^R \frac{2Mxdx}{R^2}}

y c o m y_{com} = 4 R s i n θ 2 3 θ \frac{4Rsin\frac{θ}{2}}{3θ}

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Instead of using " " a n d " " and " " as starting and ending brackets in LaTeX use " " a n d " " and " " to write it bigger. The characters are not clearly visible as they are too small.

Sahil Silare - 4 years, 4 months ago

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