Question

Find the centre of mass of a hemispherical shell of radius R

Answer 1

Answer:

We are having a hollow hemisphere of mass M and radius R. The centre of mass of the hollow hemisphere will lie on the y-axis, which is the line passing through the centre of the base of the hollow hemisphere.

We are considering an elemental strip of width Rdθ and has a mass dM. The radius of the elemental ring is r = Rsinθ.

The elemental mass dM = (M/2πR2) (2πRsinθ.Rdθ)

dM = Msinθdθ

The y-coordinate of the centre of mass yc =(1/M)∫ydM

Putting the values of y and dM, we get

yc =(1/M)∫(Rcosθ)(Msinθdθ)

y_{c}=\frac{1}{M}\int_{0}^{\pi /2}(Rcos\theta )(Msin\theta d\theta )y

c

=

M

1

∫

0

π/2

(Rcosθ)(Msinθdθ)

y_{c}=\frac{1}{M}\int_{0}^{\pi /2}(Rcos\theta )(Msin\theta d\theta ) \times \frac{2}{2}y

c

=

M

1

∫

0

π/2

(Rcosθ)(Msinθdθ)×

2

2

y_{c}=\frac{R}{2}\int_{0}^{\pi /2}(sin2\theta d\theta )y

c

=

2

R

∫

0

π/2

(sin2θdθ)

= R/2(½ + ½) = R/2

Therefore, the centre of mass of a hollow hemisphere will be at R/2 along the y-axis. Here R is the radius of the hollow hemisphere.

Answer 2

Answer:

We are having a hollow hemisphere of mass M and radius R. The centre of mass of the hollow hemisphere will lie on the y-axis, which is the line passing through the centre of the base of the hollow hemisphere.

We are considering an elemental strip of width Rdθ and has a mass dM. The radius of the elemental ring is r = Rsinθ.

The elemental mass dM = (M/2πR2) (2πRsinθ.Rdθ)

dM = Msinθdθ

The y-coordinate of the centre of mass yc =(1/M)∫ydM

Putting the values of y and dM, we get

yc =(1/M)∫(Rcosθ)(Msinθdθ)

y_{c}=\frac{1}{M}\int_{0}^{\pi /2}(Rcos\theta )(Msin\theta d\theta )y

c

=

M

1

∫

0

π/2

(Rcosθ)(Msinθdθ)

y_{c}=\frac{1}{M}\int_{0}^{\pi /2}(Rcos\theta )(Msin\theta d\theta ) \times \frac{2}{2}y

c

=

M

1

∫

0

π/2

(Rcosθ)(Msinθdθ)×

2

2

y_{c}=\frac{R}{2}\int_{0}^{\pi /2}(sin2\theta d\theta )y

c

=

2

R

∫

0

π/2

(sin2θdθ)

= R/2(½ + ½) = R/2

Therefore, the centre of mass of a hollow hemisphere will be at R/2 along the y-axis. Here R is the radius of the hollow hemisphere.

Explanation:

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