Forces, Density and Pressure: Physics 9702 (Cambridge International AS & A Level)

A gate is hinged at one edge. A gardener pushes on the gate with a force of $40\text{ N}$, applied at right angles to the gate, at a point $1.2\text{ m}$ from the hinge.

What is the moment of this force about the hinge?

A uniform wooden beam AB has length $6.0\text{ m}$ and weight $100\text{ N}$. The beam rests on a single pivot at point P, which is $2.0\text{ m}$ from end A. A load of weight $40\text{ N}$ hangs from a hook at end A, and the beam is held horizontal by an additional vertical force $F$, applied upwards at end B.

(a) State the principle of moments. [1]

(b) Calculate the force $F$ needed to keep the beam horizontal and in equilibrium. [3]

(c) A separate rigid rod experiences a couple formed by two parallel forces of $6.0\text{ N}$, acting in opposite directions, whose lines of action are separated by a perpendicular distance of $0.50\text{ m}$. Calculate the torque of this couple. [2]

A uniform ladder AB has length $4.0\text{ m}$ and weight $200\text{ N}$. End A (the foot) rests on rough horizontal ground, and end B (the top) rests against a smooth vertical wall, so that the ladder makes an angle of $60^\circ$ with the ground. Because the wall is smooth, it can only push on the ladder perpendicular to itself (horizontally); because the ground is rough, it can exert both a normal reaction and a frictional force on the ladder. A person of weight $600\text{ N}$ stands on the ladder at a point $3.0\text{ m}$ from A, measured along the ladder.

(a) By taking moments about A, calculate the normal reaction force $N$ exerted by the wall on the ladder at B. [4]

(b) By resolving forces horizontally and vertically, calculate the frictional force and the normal reaction force exerted by the ground on the ladder at A. [3]

(c) Determine the magnitude and direction of the resultant force exerted by the ground on the ladder at A. [2]

A solid engineering component is made from a uniform metal alloy. The component has a mass of $3.6\text{ kg}$ and a volume of $4.0\times10^{-4}\text{ m}^3$.

(a) Calculate the density of the alloy. [2]

(b) The component is used as a piston with a cross-sectional area of $5.0\times10^{-3}\text{ m}^2$. Calculate the force the piston must exert to produce a pressure of $2.4\times10^{5}\text{ Pa}$ on the fluid beneath it. [2]

(c) The component is later fully submerged in a tank of oil of density $850\text{ kg m}^{-3}$, at a depth of $1.2\text{ m}$ below the oil's surface. Using $g = 9.81\text{ m s}^{-2}$, calculate the hydrostatic pressure due to the oil at this depth. [2]

A small solid sphere of volume $2.0\times10^{-4}\text{ m}^3$ is fully submerged in water of density $1000\text{ kg m}^{-3}$. Take $g = 9.81\text{ m s}^{-2}$.

What is the upthrust acting on the sphere?

A person of mass $55\text{ kg}$ stands momentarily on one heel while walking. The heel has a contact area with the floor of $1.2\text{ cm}^2$ ($1.2\times10^{-4}\text{ m}^2$). Using $g = 9.81\text{ m s}^{-2}$, what pressure does the heel exert on the floor?

A non-uniform metal rod PQ has length $2.4\text{ m}$ and weight $18\text{ N}$. Its centre of gravity is not at its midpoint, because the rod is thicker at one end. The rod is supported horizontally by two vertical spring balances, one under P and one under Q. When the rod is in equilibrium, the spring balance at P reads $11\text{ N}$ and the spring balance at Q reads $7\text{ N}$.

(a) State why the sum of the two spring balance readings must equal the weight of the rod. [1]

(b) By taking moments about P, calculate the distance of the rod's centre of gravity from P. [3]

(c) State the two conditions that must both be satisfied for a rigid body to be in equilibrium. [2]

A uniform horizontal beam AB has length $3.0\text{ m}$ and weight $120\text{ N}$. End A is attached to a wall by a smooth hinge, which can exert a force on the beam in any direction. A lamp of weight $60\text{ N}$ hangs from end B. A cable also runs from B to a point on the wall directly above A, making an angle of $40^\circ$ with the beam. The beam is horizontal and in equilibrium.

(a) By taking moments about A, calculate the tension $T$ in the cable. [3]

(b) By resolving forces horizontally and vertically, calculate the horizontal and vertical components of the force exerted by the hinge on the beam at A. [3]

(c) Calculate the magnitude of the resultant force exerted by the hinge on the beam at A. [2]

A driver turns a car's steering wheel of diameter $36\text{ cm}$ by applying two equal and opposite forces of $15\text{ N}$, one at each side of the rim, tangential to the wheel and in opposite directions, forming a couple.

What is the torque of this couple?

A rectangular wooden block floats upright in water, partially submerged. The block has a horizontal base area of $0.040\text{ m}^2$, a height of $0.25\text{ m}$, and a mass of $6.0\text{ kg}$. The density of water is $1000\text{ kg m}^{-3}$ and $g = 9.81\text{ m s}^{-2}$.

(a) State the condition, in terms of upthrust and weight, that must be satisfied for the block to float in vertical equilibrium. [1]

(b) Calculate the weight of the block. [1]

(c) Calculate the volume of water displaced by the block when it floats in equilibrium. [2]

(d) Calculate the depth to which the base of the block is submerged. [2]

(e) State one reason why the value of $g$ was not actually needed to answer part (c). [1]