A ball of mass M rests between the grooves A and B of the incline and against a vertical wall at C. If all three surfaces of contact are smooth, determine the reactions of the surfaces on the ball. Hint: Use the x, y, z axes, with origin at the center of the ball, and the z axis inclined as shown.

Question

A ball of mass [latex]M[/latex] rests between the grooves [latex]A[/latex] and [latex]B[/latex] of the incline and against a vertical wall at [latex]C[/latex]. If all three surfaces of contact are smooth, determine the reactions of the surfaces on the ball. [latex]Hint:[/latex] Use the [latex]x, y, z[/latex] axes, with origin at the center of the ball, and the [latex]z[/latex] axis inclined as shown.

Given:

[latex]M[/latex] = [latex]2[/latex] [latex]kg[/latex]

[latex]{ heta }_{ 1 }[/latex] = [latex]10[/latex] [latex]deg[/latex]

[latex]{ heta }_{ 2 }[/latex] = [latex]45[/latex] [latex]deg[/latex]

Accepted Answer

[latex]\sum { { F }_{ x } } =0;\quad \quad { F }_{ C }\cos { \left( { heta }_{ 1 } ight) } -Mg\sin { \left( { heta }_{ 1 } ight) } =0[/latex]

[latex]\quad \quad\quad \quad \quad \quad \quad { F }_{ C }=Mg. an { \left( { heta }_{ 1 } ight) }[/latex]

[latex]\quad \quad \quad \quad \quad \quad \quad { F }_{ C }=0.32kg.m[/latex]

[latex]\sum { { F }_{ y } } =0;\quad \quad { N }_{ A }\cos { \left( { heta }_{ 2 } ight) } -{ N }_{ B }\cos { \left( { heta }_{ 2 } ight) } =0[/latex]

[latex]\quad \quad \quad \quad \quad \quad \quad { N }_{ A }={ N }_{ B }[/latex]

[latex]\sum { { F }_{ z } } =0;\quad \quad 2{ N }_{ A }\sin { \left( { heta }_{ 2 } ight) } -Mg\cos { \left( { heta }_{ 1 } ight) } -{ F }_{ C }\sin { \left( { heta }_{ 1 } ight) } =0[/latex]

[latex]\quad \quad \quad \quad \quad \quad \quad { N }_{ A }= \frac { 1 }{ 2 } .\frac { Mg.\cos { \left( { heta }_{ 1 } ight) } +{ F }_{ C }.\sin { \left( { heta }_{ 1 } ight) } }{ \sin { \left( { heta }_{ 2 } ight) } } [/latex]

[latex]\quad \quad \quad \quad \quad \quad \quad { N }_{ A }=1.3kg.m[/latex]

[latex]\quad \quad \quad \quad \quad \quad \quad N = { N }_{ A }={ N }_{ B }[/latex]

Question : A ball of mass M rests between the grooves A and B of the in...

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