- #1

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- Homework Statement
- To find the weight of a hemisphere

- Relevant Equations
- moments

Could I please ask for help with the following question:

A uniform lamina of weight W is in the shape of a triangle ABC with AB = AC = 2a and the angle BAC equal to 2ᾳ. The side AB is fixed along a diameter of a uniform solid hemisphere of radius a. The plane of the lamina being perpendicular to the flat surface of the hemisphere. The body rests in equilibrium with a point on the curved surface of the hemisphere in contact with a horizontal table and with BC vertical. Show that the weight of the hemisphere is (8/9)*W cos(ᾳ)

Here's a diagram of the object with the flat surface of the hemisphere horizontal, and I've set up cartesian axes as shown:

The orange dot represents the centre of gravity of the triangle and the blue dot that of the hemisphere.

Let the weight of the hemisphere be W_h

So:

A is the point

B is the point

C is the point

I can use the standard results for the centre of mass of a triangle and a hemisphere, so:

centre of mass of hemisphere is at

centre of mass of triangle is at

If I say that the centre of gravity of the whole body is at (x_bar, y_bar) then taking moments about x_bar I can say:

W * ((2a + 2a*cos(2ᾳ)) / 3 - x_bar) = -W_h *(x_bar - a),

This would give me W_h in terms of a, W and ᾳ but also x_bar.

I could find x_bar by equating moments of the triangle, hemisphere and composite body about the y-axis, this starts to get complicated.

Can anyone help me see the way forward?

Thanks,

Mitch.

A uniform lamina of weight W is in the shape of a triangle ABC with AB = AC = 2a and the angle BAC equal to 2ᾳ. The side AB is fixed along a diameter of a uniform solid hemisphere of radius a. The plane of the lamina being perpendicular to the flat surface of the hemisphere. The body rests in equilibrium with a point on the curved surface of the hemisphere in contact with a horizontal table and with BC vertical. Show that the weight of the hemisphere is (8/9)*W cos(ᾳ)

Here's a diagram of the object with the flat surface of the hemisphere horizontal, and I've set up cartesian axes as shown:

The orange dot represents the centre of gravity of the triangle and the blue dot that of the hemisphere.

Let the weight of the hemisphere be W_h

So:

A is the point

**(**0,0**)**B is the point

**(**2a,0**)**C is the point

**(**2a*cos(2ᾳ) , 2a*sin(2ᾳ)**)**I can use the standard results for the centre of mass of a triangle and a hemisphere, so:

centre of mass of hemisphere is at

**(**a,-3a/8**)**centre of mass of triangle is at

**(**(2a + 2a*cos(2ᾳ)) / 3, 2a*sin(2ᾳ) / 3**)**If I say that the centre of gravity of the whole body is at (x_bar, y_bar) then taking moments about x_bar I can say:

W * ((2a + 2a*cos(2ᾳ)) / 3 - x_bar) = -W_h *(x_bar - a),

This would give me W_h in terms of a, W and ᾳ but also x_bar.

I could find x_bar by equating moments of the triangle, hemisphere and composite body about the y-axis, this starts to get complicated.

Can anyone help me see the way forward?

Thanks,

Mitch.