University Physics III

Simple harmonic motion is oscillation in which the acceleration is proportional to the displacement from equilibrium and directed opposite to it, a = −ω²x, with ω² one positive constant of the system. Use the supplied topic model Y_out = 2*pi/w, where Y_out is PERIOD T₀ (LINEAR) (s). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. w is Angular frequency ω₀, measured in rad s⁻¹. Keep n (Cubic term at 0.20 m ÷ linear term) = 0.15; a (Amplitude A) = 0.12 m fixed. For this exercise use only 3.6375 rad s⁻¹ ≤ w ≤ 8.3625 rad s⁻¹ 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 = 0.96371568 s, with a bounded reading uncertainty ±0.031854 s. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown w by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0004725 rad s⁻¹, 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.

A travelling wave is a disturbance whose shape moves through a medium at a fixed speed, while each element of the medium only oscillates about its own equilibrium position and is left where it started. Use the supplied topic model Y_out = v/f, where Y_out is WAVELENGTH λ = v/f (m). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. f is Frequency f (set by the source), measured in Hz. Keep v (Wave speed v (set by the medium)) = 12 m s⁻¹; d (Probe position x along the cord) = 0.6 m fixed. For this exercise use only 7.85 Hz ≤ f ≤ 14.15 Hz 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 = 1.026255 m, with a bounded reading uncertainty ±0.02673 m. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown f by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00063 Hz, 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 superposition principle says that where two or more waves overlap in a linear medium, the displacement of the medium at each point and instant is the algebraic sum of the displacements each wave would produce there on its own. Use the supplied topic model Y_out = 15*exp(-((110-2.75*t)/26)^2), where Y_out is PULSE 1 AT MEETING POINT (mm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. t is Time t, measured in ms. Keep b (Second pulse height A₂) = -10 mm; eps (Nonlinear mixing ε) = 0 fixed. For this exercise use only 6.375 ms ≤ t ≤ 37.875 ms 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 = 1.4697082 mm, with a bounded reading uncertainty ±0.22045 mm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00315 ms, 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.

Fermat's principle states that between two points light follows a route whose optical path length ∫ n ds, and so its travel time, is stationary against small deformations of the route: unchanged to first order, not necessarily least. Use the supplied topic model Y_out = 33.356*(n1*sqrt(16+xc^2)+n2*sqrt(16+(6-xc)^2)), where Y_out is TRAVEL TIME T(x) (ps). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. xc is Crossing point x, measured in cm. Keep n1 (Index n₁ above) = 1; n2 (Index n₂ below) = 1.5 fixed. For this exercise use only 0.6375 cm ≤ xc ≤ 3.7875 cm 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 = 422.39157 ps, with a bounded reading uncertainty ±1.4901 ps. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown xc by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315 cm, 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.