Forces, density and pressure：練習題

Which statement correctly describes the centre of gravity of a rigid body?

Two parallel forces, each of magnitude \( 15 \text{ N} \), act in opposite directions as a couple. The perpendicular distance between the lines of action of the forces is \( 40 \text{ cm} \). What is the torque produced by this couple?

A block of mass \( 5.0 \text{ kg} \) is held in equilibrium on a frictionless plane inclined at an angle of \( 30^\circ \) to the horizontal by a force acting parallel to the plane. What is the magnitude of the normal contact force exerted by the plane on the block?

An object has a mass of \( 500 \text{ g} \) and a volume of \( 250 \text{ cm}^3 \). What is the density of the object in SI units?

A uniform horizontal beam of length \( 2.0 \text{ m} \) and weight \( 100 \text{ N} \) is supported by a pivot at its centre. A downward force of \( 50 \text{ N} \) is applied at a point \( 0.50 \text{ m} \) from the left-hand end. What vertical force must be applied at the right-hand end to keep the beam in equilibrium?

Calculate the hydrostatic pressure exerted by a column of mercury of height 760 mm. The density of mercury is \( 1.36 \times 10^4 \text{ kg m}^{-3} \) and the acceleration of free fall is \( 9.81 \text{ m s}^{-2} \).

A block of wood is floating in water of density \( 1000 \text{ kg m}^{-3} \). The volume of the block is \( 2.0 \times 10^{-3} \text{ m}^3 \) and 60% of its volume is submerged. Calculate the upthrust acting on the block.

A couple is applied to a circular disc of radius 0.25 m by two tangential forces, each of magnitude 40 N, acting in opposite directions at the edges. Determine the torque of the couple and state the conditions required for the disc to be in equilibrium.

(a) State Archimedes' principle.
(b) A spherical buoy of volume \(0.24\text{ m}^3\) is completely submerged in seawater of density \(1025\text{ kg m}^{-3}\). The buoy is held in place by a cable attached to the seabed. Calculate the upthrust acting on the buoy.
(c) If the weight of the buoy is \(850\text{ N}\), calculate the tension in the cable.

A U-tube with a uniform cross-sectional area contains mercury of density \(1.36 \times 10^4\text{ kg m}^{-3}\). Water of density \(1.00 \times 10^3\text{ kg m}^{-3}\) is poured into the left limb until the water column is \(20.0\text{ cm}\) high. An unknown oil is then poured into the right limb until the mercury levels in both limbs are again equal.
(a) Show that the pressure difference \(\Delta p\) between two points at different depths in a liquid is given by \(\Delta p = \rho g \Delta h\).
(b) Calculate the pressure at the water-mercury interface due to the water column.
(c) If the oil column in the right limb is \(25.0\text{ cm}\) high, calculate the density of the oil.