Live practice

Principle and reasoning. Work from the governing physical relation and apply it to every stated part.

Periodic motion must reproduce the same displacement-time pattern after a fixed interval. Plot (a) never repeats and is nonperiodic. The full waveform in (b) repeats every \(2\,\mathrm{s}\). In (c) , the irregular shape from 1–\(7\,\mathrm{s}\) recurs from 7–\(13\,\mathrm{s}\), so its period is \(6\,\mathrm{s}\). Plot (d) is sinusoidal with period \(2\,\mathrm{s}\).

Principle. In the zero-pressure limit every gas approaches PV/T=nR. Stronger real-gas departure identifies the lower temperature.

The dotted horizontal behaviour represents an ideal gas. In the labelled plot T₂ has the deeper downward departure, so T₂ is lower and T₁>T₂.

Principle and reasoning. Work from the governing physical relation and apply it to every stated part.

(a) Liquid surface tension generally decreases as temperature rises because cohesive order weakens.

(b) Gas viscosity increases with temperature as momentum transfer grows, while liquid viscosity decreases as molecules overcome cohesive barriers more easily.

(c) In an elastic solid shear force is proportional to shear strain; in a Newtonian fluid it is proportional to the rate of shear strain.

(d) Faster flow in a constriction follows conservation of mass, A v=constant. Bernoulli's principle then relates this speed increase to a pressure decrease.

(e) Matching Reynolds number ρvL/η with a smaller model length requires a greater speed.

Principle. Stiffness is set by the initial slope Y=dσ/dε, whereas strength is judged by the largest stress sustained before failure.

Principle. Classical motion is allowed only where kinetic energy K=E-V(x) is nonnegative; the lowest possible total energy is the minimum of V.

Principle. Once released, only weight acts; horizontal source velocity or source acceleration does not add a force. While supported, the floor supplies the horizontal acceleration.

Principle. Each constant-speed leg is a straight segment; rest is horizontal. Determine the corner times from distance divided by speed.

Energy has dimensions ML²T⁻². Relate one new energy unit to the SI joule.

Use rectangular-coordinate definitions and keep corresponding coordinates aligned.

The final relation satisfies the original coordinate or distance condition, completing the verification.

Translate every condition into an equation and retain exact values throughout the conic calculation.

The last line satisfies the original conditions and supplies every requested part.