f ñ₂ Ө N F_g êL fi₁ 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: {₁₁, n₂, 3} and the blue lines are the rotating frame E: {ê₁, ê, ê³} and the purple lines are the forces acting on the rod. = Start with super particle theorem equation which is: Σ Fexternal MR where the sum the forces is Normal (N) = N n-hat_2, friction force is (f) = μ*N n-hat_1, and gravity force (Eg) = - mg n-hat_2. R. - (Öcose - ²sinė)ñ₁-(Ösinė + 0³cos), and M is just mass (m). N, Ö, and 8^2 = are all unknown. N can be found using Newton's second law F = ma, where the position is y which is y = cose, so the Normal equation becomes: N = mg mL 2 (0 sin(0) 02 cos(0)) Use Euler's equation H₂ = L₂ where H = ml²öñì¸ and L₂ = N(µcosė – sinė)k₂ Solve for Ö 12 1 Use conservation of energy → mgy₁ = mgy; + ½ my² + ½±0² the initial is just L/2 Solve for 02 Plug back into find relationship between mu = function(theta)

f ñ₂ Ө N F_g êL fi₁ 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: {₁₁, n₂, 3} and the blue lines are the rotating frame E: {ê₁, ê, ê³} and the purple lines are the forces acting on the rod. = Start with super particle theorem equation which is: Σ Fexternal MR where the sum the forces is Normal (N) = N n-hat_2, friction force is (f) = μ*N n-hat_1, and gravity force (Eg) = - mg n-hat_2. R. - (Öcose - ²sinė)ñ₁-(Ösinė + 0³cos), and M is just mass (m). N, Ö, and 8^2 = are all unknown. N can be found using Newton's second law F = ma, where the position is y which is y = cose, so the Normal equation becomes: N = mg mL 2 (0 sin(0) 02 cos(0)) Use Euler's equation H₂ = L₂ where H = ml²öñì¸ and L₂ = N(µcosė – sinė)k₂ Solve for Ö 12 1 Use conservation of energy → mgy₁ = mgy; + ½ my² + ½±0² the initial is just L/2 Solve for 02 Plug back into find relationship between mu = function(theta)