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Dynamics, forces and pressure

4. Forces, Density and Pressure

5 min read 13 parts Lesson 24 of 24 Unit quiz

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4. Forces, Density and Pressure — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use centre of gravity, moments, couples and equilibrium conditions, and calculate density, pressure, hydrostatic pressure and upthrust.Centre of gravity — the point at which the weight o…

  • Centre of gravity versus centre of mass
  • Torque of a couple
  • Common forces
  • To draw the triangle
  • Unit conversions
  • Calculation

Section 1 of 13Centre of gravity

  • Centre of gravity
    • the point at which the weight of an object may be considered to act.
    • For a uniform symmetrical object it is at the point of symmetry, e.g. the centre of a sphere.
    • It can lie outside the body and can move when the shape or orientation changes; leaning forward lowers a person's centre of gravity.
  • Centre of gravity versus centre of mass
    • they are the same in a uniform field,
    • but the centre of mass does not depend on the field whereas the centre of gravity does.
      • (the Moon's centre of gravity is pulled slightly towards Earth)

Section 2 of 13Stability

More stableLess stable
Wide baseNarrow base
Low centre of gravityHigh centre of gravity
Centre of gravity above the baseCentre of gravity outside the base, so it topples
  • An object is stable while its centre of gravity lies vertically above its base.
  • The most stable objects have wide bases and low centres of gravity.

Section 3 of 13Moment of a force

  • Moment
    • the turning effect of a force about a pivot.
  • Equation
    • moment = Fd, in N m, where d is the perpendicular distance from the pivot to the line of action of the force.
    • If the force is at an angle, use the perpendicular component of the force or distance.
    • A door handle far from the hinge gives a larger moment, so less force is needed.
    • Carry extra significant figures until the final answer to avoid rounding errors.

Section 4 of 13Couples and torque

1Couple2 points
  • a pair of forces equal in size, opposite in direction and acting along different lines, which produces rotation only.
  • The resultant force is zero, so the object does not accelerate linearly.
2Torque of a couple2 points
  • τ = Fd, where F is one of the forces and d is the perpendicular distance between them.
  • The forces of a couple cannot share a line of action.

Section 5 of 13Principle of moments

  • Statement
    • for a system in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about the same point.
  • Method
    • apply the principle in these steps.
    • Choose a pivot and mark every force and its perpendicular distance.
    • Treat the weight of a uniform beam as acting at its centre.
    • Equate clockwise and anticlockwise moments and solve for the unknown.

Section 6 of 13Conditions for equilibrium

  • Equilibrium
    • there is no resultant force and no resultant torque, so the object stays at rest or moves with constant velocity and does not rotate.
    • A pivot can be chosen at the point of one force to remove its moment (d = 0).
    • Coplanar forces in equilibrium form a closed vector triangle (tip to tail) when drawn to scale.
  • Common forces
    • weight, normal reaction, tension and friction.
  • To draw the triangle
    • make a free-body diagram, then place the arrows tip to tail keeping lengths and directions; the path must close.

Section 7 of 13Density

  • Density
    • mass per unit volume: ρ = m/V, in kg m−3.
    • Gases are less dense than liquids, which are less dense than solids, because there are fewer particles per unit volume.
  • Volumes
    • cuboid lwh; cylinder πr2h; sphere 4/3 πr3.
  • Unit conversions
    • larger to smaller unit: multiply; smaller to larger: divide; cube the factor for volumes (1 cm3 = 10−6 m3, 1 mm3 = 10−9 m3).

Section 8 of 13Pressure

  • Pressure
    • force per unit area: p = F/A, in pascals (1 Pa = 1 N m−2).
      What the symbols mean: p = pressure (Pa = N m−2) · F = force acting at right angles to the surface (N) · A = area (m2)
    • It is a scalar; a force spread over a small area gives a large pressure (a high-heeled shoe) and over a large area gives a small pressure.
    • Use the cross-sectional area in contact with the surface.

Section 9 of 13Hydrostatic pressure: Δp = ρgΔh

  • Consider a column of fluid of area A and height h above a point; its weight is W = mg.
  • Using ρ = m/V with V = Ah gives m = ρAh, so W = ρAhg.
  • Pressure = W/A = ρhg, so the change in pressure with depth is Δp = ρgΔh.
  • The pressure acts in all directions at a point in a fluid at rest and increases with depth, density and g.
  • Total pressure = hydrostatic pressure + atmospheric pressure (101 325 Pa).

Section 10 of 13U-tube manometer

  • A U-tube manometer measures pressure differences using the heights of liquid in the two arms.
  • With equal pressure on both sides the levels are equal.
  • If the pressure on one side rises, that level falls and the other rises; the height difference gives the pressure difference (Δp = ρgΔh).
    What the symbols mean: p = pressure due to the liquid (Pa) · ρ = density of the liquid (kg m−3) · g = 9.81 N kg−1 · h = depth (m)
  • With two different liquids, the pressure at a point equals atmospheric pressure plus the weight of liquid above it, which lets densities be compared.

Section 11 of 13Upthrust

  • Upthrust
    • the upward force on an object immersed in a fluid, caused by the difference in hydrostatic pressure between the bottom and top of the object.
  • Calculation
    • F = (pbottom − ptop)A, where A is the cross-sectional area of the object.
    • Upthrust is much greater in liquids than in gases because liquids are denser.
    • It is why objects seem lighter in water; the object rises if upthrust exceeds its weight, and floats if its density is less than the fluid's.

Section 12 of 13Archimedes' principle

  • Statement
    • the upthrust on an object in a fluid equals the weight of the fluid displaced.
  • Equation
    • F = ρgV, where V is the volume of fluid displaced and ρ is the fluid density.
      • (the immersed volume)
  • Floating
    • the object floats when the upthrust equals its own weight.
  • Tip
    • you need the immersed volume only to find the mass (m = ρV) and then the weight (mg) of the displaced fluid.

Section 13 of 13More from the specification

  • Newton
    • Unit of force, kg m s-2.
  • Gravitational field
    • Region where mass experiences gravitational force.
  • Field strength
    • G = Fg/m.

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