Rotational Dynamics

The torque of a force about a point O is τ = r×F, where r runs from O to the point where the force acts; its magnitude is the force times the perpendicular distance from O to the force's line of action. Use the supplied topic model Y_out = r*abs(sind(phi)), where Y_out is LEVER ARM r⊥ = r |sin φ| (m). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; abs(x) means |x|; sind/cosd/tand use degrees. phi is Angle φ from r to F, measured in °. Keep r (Distance r from O to P) = 0.3 m; f (Force magnitude F) = 80 N fixed. For this exercise use only -85.05 ° ≤ phi ≤ -9.45 ° on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.18852743 m, with a bounded reading uncertainty ±0.016553 m. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown phi by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00756 °, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The moment of inertia of a system of point masses about a chosen axis is I = Σmᵢrᵢ², each mass multiplied by the square of its perpendicular distance from that axis; it measures how strongly the system resists a change in its rotation about that axis. Use the supplied topic model Y_out = m1*s^2+m2*(s-0.9)^2, where Y_out is I ABOUT THE AXIS (kg m²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. s is Axis position s, measured in m. Keep m1 (Bead 1 mass m₁, at x = 0) = 2 kg; m2 (Bead 2 mass m₂, at x = 0.90 m) = 1 kg fixed. For this exercise use only 0.347625 m ≤ s ≤ 0.961875 m on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 1.0750562 kg m², with a bounded reading uncertainty ±0.079238 kg m². Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000061425 m, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The moment of inertia of a continuous body about an axis is I = ∫ r⊥² dm: split the body into mass elements dm, weight each by the square of its perpendicular distance r⊥ from the axis, and integrate over the whole body. Use the supplied topic model Y_out = 100*(1-h^4), where Y_out is I LEFT, % OF SOLID DISC (%). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. h is Hole radius R₁/R, dimensionless. Keep n (Number of rings n) = 4 fixed. For this exercise use only 0.085 ≤ h ≤ 0.715 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 95.149325 %, with a bounded reading uncertainty ±0.72682 %. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown h by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000063, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The parallel-axis theorem gives the moment of inertia about any axis as I_cm + Md², d being the perpendicular distance from the parallel axis through the centre of mass; the perpendicular-axis theorem gives a flat body's I_z as I_x + I_y for three perpendicular axes meeting at one point. Use the supplied topic model Y_out = 40*r^2/(1+r-p)^2, where Y_out is BALL OWN I ÷ ITS SHIFT TERM (%). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. p is Pivot distance p from left end, measured in m. Keep ms (Ball mass m_b) = 2 kg; r (Ball radius R) = 0.1 m fixed. For this exercise use only 0.10625 m ≤ p ≤ 0.89375 m on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 1.5177162 %, with a bounded reading uncertainty ±0.1669 %. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown p by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00007875 m, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.