f N F_g êo 1 A slender rod that is the black line has a length L and a mass m. It is standing vertically on a rough, level surface with a friction coefficient mu. After it is slightly disturbed, the rod will start to rotate toward the ground. Find a relationship between the friction coefficient mu and the rod orientation angle theta where the rod starts to slip (mu = f(theta)). The red lines are the inertial frame N: {^₁, ^2, 3} and the blue lines are the rotating frame E: {ꃂ 굂 ê_3} and the purple lines are the forces acting on the rod. HINTS: This should be taken about the center of mass; therefore the moment of inertia is c = (1/12)*m*L^2. The torque equation is going to end up being - ½ * * m * sin(0) ñ³ + − ½ ½ * ƒ * cos(0) ñ³. Which means when L 2 2 い setting up the torque equation it should originally be: Lc = - ½₁₁X (N₁₂) + (fñ₁). This can be related to H₁ = Lc. This will allow you to find Ö. Then super particle theorem can be used. Sum of the forces (torque, gravity, normal) = M*Ïc, where R₂ = ½ê_L. Find Ⓡc. Then conservation of energy equation can be used: Ę i = Ę f. Where Ei = potential (mgh) and f = potential + kinetic energy of a rotating system which means it needs translational and rotational kinetic energy. This should have you find ė². These can all be simplified and plugged back into each other in order to find a relationship mu = f(theta).|
f N F_g êo 1 A slender rod that is the black line has a length L and a mass m. It is standing vertically on a rough, level surface with a friction coefficient mu. After it is slightly disturbed, the rod will start to rotate toward the ground. Find a relationship between the friction coefficient mu and the rod orientation angle theta where the rod starts to slip (mu = f(theta)). The red lines are the inertial frame N: {^₁, ^2, 3} and the blue lines are the rotating frame E: {ꃂ 굂 ê_3} and the purple lines are the forces acting on the rod. HINTS: This should be taken about the center of mass; therefore the moment of inertia is c = (1/12)*m*L^2. The torque equation is going to end up being - ½ * * m * sin(0) ñ³ + − ½ ½ * ƒ * cos(0) ñ³. Which means when L 2 2 い setting up the torque equation it should originally be: Lc = - ½₁₁X (N₁₂) + (fñ₁). This can be related to H₁ = Lc. This will allow you to find Ö. Then super particle theorem can be used. Sum of the forces (torque, gravity, normal) = M*Ïc, where R₂ = ½ê_L. Find Ⓡc. Then conservation of energy equation can be used: Ę i = Ę f. Where Ei = potential (mgh) and f = potential + kinetic energy of a rotating system which means it needs translational and rotational kinetic energy. This should have you find ė². These can all be simplified and plugged back into each other in order to find a relationship mu = f(theta).|
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Transcribed Image Text:f
N
F_g
êo
1
A slender rod that is the black line has a length L and a mass m. It is standing vertically on a
rough, level surface with a friction coefficient mu. After it is slightly disturbed, the rod will
start to rotate toward the ground. Find a relationship between the friction coefficient mu
and the rod orientation angle theta where the rod starts to slip (mu = f(theta)). The red lines
are the inertial frame N: {^₁, ^2, 3} and the blue lines are the rotating frame E: {ꃂ 굂 ê_3}
and the purple lines are the forces acting on the rod. HINTS: This should be taken about the
center of mass; therefore the moment of inertia is c = (1/12)*m*L^2. The torque equation
is going to end up being - ½ * * m * sin(0) ñ³ + − ½ ½ * ƒ * cos(0) ñ³. Which means when
L
2
2
い
setting up the torque equation it should originally be: Lc = - ½₁₁X (N₁₂) + (fñ₁). This can
be related to H₁ = Lc. This will allow you to find Ö. Then super particle theorem can be
used. Sum of the forces (torque, gravity, normal) = M*Ïc, where R₂ = ½ê_L. Find Ⓡc. Then
conservation of energy equation can be used: Ę i = Ę f. Where Ei = potential (mgh) and f
= potential + kinetic energy of a rotating system which means it needs translational and
rotational kinetic energy. This should have you find ė². These can all be simplified and
plugged back into each other in order to find a relationship mu = f(theta).|
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