Pressure, Gravity, and Moments

Density

Mass per unit volume

SI Unit: kg m-3

Density=massvolumeDensity=\frac{mass}{volume}

ρ=mV\rho=\frac{m}{V}

Pressure

Force per unit area

Pressure=ForceAreaPressure=\frac{Force}{Area}

P=FAP=\frac{F}{A}

Pressure due to a liquid

P=ρghP=\rho gh

h = depth, g = gravity

Archimedes Principle

  • Upthrust of an object is equal to the weight of the fluid displaced.

Law of Floatation

  • Weight of a floating body is equal to the weight of the fluid it displaces.

Hydrometers - measure density

e.g. home brewing (alcohol percentage)

Boyle’s Law

  • For a fixed mass of gas at a constant temperature the volume is inversely proportional to the pressure.

Pα1VP\alpha\frac{1}{V} or PV=kPV=k

Newton’s Law of Universal Gravitation

  • States that any two point masses in the universe attract eachother with a force that is directly proportional to the product of their masses and inversely proportional to the sqaure of the distance between them.

Fαm1m2d2F\alpha\frac{m_1m_2}{d^2}

F=Gm1m2d2F=\frac{Gm_1m_2}{d^2}

F is the force of attraction, m1 and m2 are the masses, d is the distance between the masses, G is the universal gravitation constant.

  • Gravitational force is always attractive.

  • The size of the force on each mass is the same.

  • The distance between bodies is the distance between their centres.

G = 6.7 ×10-11 N m2 kg-2

  

Moment of a Force

  • Magnitude of the force multiplied by the perpendicular distance from the axis to the force.

M=FdM=Fd

  • Scalar quantity

SI Unit: newton metre (N m)

Co-Planar Forces

  • Forces acting in the same plane.

Equilibrium

When the body as a whole is not accelerating and the body is not rotating with angular acceleration.

A body will be in equilibrium if:

  • The vector sum of the forces in any direction is zero.

  • The sum of the moments about any point is zero.

Static vs Dynamic Equilibrium

Static: When a body is not moving in any way it is in static equilibrium.

Dynamic: When a body is moving in a straight line at a steady speed and is not rotating with constant angular velocity.

Couple

Two parallel forces with the same magnitude acting in opposite directions

  • Resultant of the two forces of a couple is zero.

  • A couple therefore has only a turning effect, so it has a turning moment.

  • The moment of a couple about any point in its plane is the same.

  • Moment of a couple also known as torque of a couple.

TorqueTorque =T=Fd= T = Fd