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Question

Hello I need help with finding the solutions for these parts with explanations. Please and thank you. 

Two vertical walls are separated by a distance of x. The wall on the left is perfectly smooth, while the wall on the right is rough. A uniform board is propped between them, as shown. Your objective: find the length of the longest board that can be thus propped between the two walls without falling. (i) Draw a free-body diagram of the board. Label clearly the pivot point you will use for calculating torques. Please draw forces directly on the diagram provided. Hints: Call the normal forces from the left and right walls L and R, respectively. One of the walls will also provide a friction force. Call the angle that the board makes with the horizontal 0. (ii) Write the equilibrium equations for the board, including force components (in x and y) and torque. You will also need to include a constitutive relation for the static friction. (iii) Solve the equilibrium equations for the tangent of the "critical angle," or the smallest angle that allows the board to be supported between these two walls. If you have chosen your axis of rotation wisely, this will only involve some fairly mple bra. The answer should be ind ndent of any force or mass. (iv) Use the fact that y =x tan 0, along with the Pythagorean Theorem and the result of your analysis in (iii), to obtain the longest supportable board length. (v) Evaluate your answer in (iv) assuming that the walls are 6.84 m apart, the coefficient of static friction is equal to 0.818, and that the board's mass is 42.5 kg.
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Transcribed Image Text:Two vertical walls are separated by a distance of x. The wall on the left is perfectly smooth, while the wall on the right is rough. A uniform board is propped between them, as shown. Your objective: find the length of the longest board that can be thus propped between the two walls without falling. (i) Draw a free-body diagram of the board. Label clearly the pivot point you will use for calculating torques. Please draw forces directly on the diagram provided. Hints: Call the normal forces from the left and right walls L and R, respectively. One of the walls will also provide a friction force. Call the angle that the board makes with the horizontal 0. (ii) Write the equilibrium equations for the board, including force components (in x and y) and torque. You will also need to include a constitutive relation for the static friction. (iii) Solve the equilibrium equations for the tangent of the "critical angle," or the smallest angle that allows the board to be supported between these two walls. If you have chosen your axis of rotation wisely, this will only involve some fairly mple bra. The answer should be ind ndent of any force or mass. (iv) Use the fact that y =x tan 0, along with the Pythagorean Theorem and the result of your analysis in (iii), to obtain the longest supportable board length. (v) Evaluate your answer in (iv) assuming that the walls are 6.84 m apart, the coefficient of static friction is equal to 0.818, and that the board's mass is 42.5 kg.
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