Forces, density and pressure：練習題

A solid cube with side length \(L\) and density \(\rho_{s}\) is completely submerged in a liquid of density \(\rho_{l}\). The top face of the cube is at a depth \(h\) below the surface of the liquid. Which expression represents the upthrust acting on the cube?

An object is suspended from a spring balance in air and the reading is 10.0 N. When the object is completely submerged in water (density \(1000\text{ kg m}^{-3}\)), the reading on the balance becomes 8.2 N. If the object is then submerged in an unknown oil, the balance reads 8.5 N. What is the density of the oil?

Two vertical forces of magnitude \(F\) act in opposite directions on a horizontal rod of length \(d\). One force acts at a distance of \(d/4\) from the left end, and the other acts at a distance of \(d/4\) from the right end. What is the torque of the couple acting on the rod?

A rectangular block of material has a base area of \(0.020\text{ m}^2\) and a height of \(0.15\text{ m}\). The block is made of two layers: the bottom layer is \(0.05\text{ m}\) thick with a density of \(1200\text{ kg m}^{-3}\), and the top layer is \(0.10\text{ m}\) thick with a density of \(800\text{ kg m}^{-3}\). The block is floating upright in a large tank of liquid with a density of \(1000\text{ kg m}^{-3}\). What is the depth of the bottom face of the block below the liquid surface?

A uniform beam of length 4.0 m and weight 200 N is supported horizontally by a pivot at one end and a vertical cable at a distance of 3.0 m from the pivot. A man weighing 800 N stands on the beam at a distance of 1.0 m from the pivot. What is the tension in the cable?
(Assume the beam is in equilibrium.)

A solid cube of side length \( 0.10 \, m \) is held stationary and fully submerged in a liquid of density \( 1200 \, kg \, m^{-3} \) by a vertical string attached to the bottom of the tank. If the density of the cube is \( 800 \, kg \, m^{-3} \), calculate the tension in the string.

Water flows vertically downwards from a tap. At the tap, the cross-sectional area of the stream is \( A_{0} \) and the speed is \( v_{0} \).
Using the principle of conservation of mass and equations of motion, show that the cross-sectional area \( A \) at a distance \( h \) below the tap is given by \( A = A_{0} \left( 1 + \frac{2gh}{v_{0}^{2}} \right)^{-1/2} \).

A uniform beam of length \( 4.0 \, m \) and weight \( 200 \, N \) is supported at its center. A mass of weight \( W \) is placed at one end, and a force of \( 150 \, N \) is applied downwards at a distance of \( 1.5 \, m \) from the center on the opposite side to keep the beam in equilibrium.
Calculate the value of \( W \).

A uniform wooden plank of length \( 4.0 \, m \) and mass \( 25 \, kg \) is supported by two vertical ropes. Rope A is at the left end and Rope B is positioned \( 1.0 \, m \) from the right end. A person of mass \( 65 \, kg \) stands \( 0.50 \, m \) from the left end.
(a) By taking moments about Rope A, calculate the tension in Rope B.
(b) Hence, find the tension in Rope A.
(c) State the two conditions required for the plank to be in static equilibrium.

A solid cylinder of height \( 0.15 \, m \) and cross-sectional area \( 2.0 \times 10^{-3} \, m^2 \) is submerged in a liquid of density \( 850 \, kg \, m^{-3} \). The top surface of the cylinder is at a depth of \( 0.40 \, m \) below the surface of the liquid.
(a) Calculate the hydrostatic pressure exerted by the liquid on the top surface of the cylinder.
(b) Calculate the upthrust acting on the cylinder.
(c) Explain, in terms of pressure, the origin of the upthrust acting on the cylinder.