Live practice

Mineral: a naturally occurring element or compound in the earth's crust. Ore: a mineral containing enough of a metal for economical extraction. Gangue: unwanted earthy impurities such as sand, clay and rock associated with an ore.

The added base dissociates and supplies additional hydroxide ions. Provided the volume change is not dominant, the concentration of \(\ce{OH-}\) increases. The solution therefore becomes more strongly basic and its pH increases.

Principle. A product of a spatial sine and temporal cosine is a standing wave. Use sinA cosB=[sin(A+B)+sin(A−B)]/2.

Principle. The x-projection of uniform circular motion is SHM. Use ω=2π/T and set the phase from the initial point and sense.

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

Bernoulli's equation in its elementary form assumes steady, incompressible, non-viscous flow along a streamline. A rapid contains strong turbulence, eddies, air entrainment, and appreciable mechanical-energy dissipation. Those conditions violate the assumptions, so the simple equation cannot predict the full rapid flow. It may be used only as a rough local approximation in a smooth region where losses are separately allowed for.

Principle. At the bottom, tension supplies centripetal force in addition to balancing weight. Then apply the wire-extension relation.

Principle. Complete the hemisphere by adding its mirror half. The field inside a complete uniform spherical shell is zero, so the actual half-shell field is opposite the missing half-shell's field.

Principle. Release preserves the truck's instantaneous horizontal velocity; thereafter gravity changes only the vertical component.

Over an infinitesimal time interval, the path segment approaches the magnitude of displacement.

Treat the sheet as a thin cuboid. The least precise measured value has three significant figures; first convert thickness to metres.