Question

Find the centre of mass of a metal circular sheet of diameter 6 cm which has a 2 cm square cut out of it . The two sides of the square lie along diameters of the circle.

Answer 1

R

4π-2

 

R

2π

 

R

π-2

 

R

2π-2

 

Answer :

A

Solution :

A1(ℂ1)=A2(ℂ2)

side of square will be  

R

√

2

 

∴ℂ2=

A1

A2

 

(ℂ1=

(

R

√

2

 

)2

πR2-(R/

√

2

)2(

R

2

 

)=

R

4π-2

Answer 2

Given that,

Diameter of sheet = 6 cm

Side of square = 2 cm

Consider a circular sheet with radius r and mass M.

A square sheet with diagonal r and mass m is cut off from it.

Center of mass of the circle at center is zero.

Center of mass of the sheet at $\dfrac{r}{2}$

We need to calculate the area of square

Using formula of area

$A=a^2$

Put the value into the formula

$A=(\dfrac{r}{\sqrt{2}})^2$

Metal sheet have density is

$m=M\times\dfrac{\dfrac{r^2}{2}}{\pi r^2}$

$m=\dfrac{M}{2\pi}$

We need to calculate the center of mass of remaining sheet

Using formula of center of mass

$x=\dfrac{M\times0-\dfrac{M}{2\pi}\times\dfrac{r}{2}}{M-\dfrac{M}{2\pi}}$

$x=\dfrac{-\dfrac{Mr}{4\pi}}{M(\dfrac{2\pi-1}{2\pi})}$

$x=\dfrac{-r}{2(2\pi-1)}$

Put the value of r in to the formula

$x=\dfrac{-3}{2(2\pi-1)}$

Negative sign shows the direction of the center of mass of remaining part is  towards right from center of the circle.

$x=0.283\ cm$

Hence, The center of mass of remaining sheet is 0.283 cm.