springs

Springs

  • Forces and Movement:
    • Springs can stretch, bend, and compress.
    • A single force applied to an object will result in movement in the direction of that force.
    • When forces are applied inwards from opposite sides, the spring will compress.
    • If one end of a spring is fixed and a force is applied to the other end, a reaction force from the fixed point means that more than one force is acting on the spring.

Deformation

  • Definition of Deformation:
    • Deformation refers to the change in shape of an object when a force is applied.
  • Types of Deformation:
    • Elastic Deformation:
    • The object returns to its original shape once the load is removed.
    • Example: An elastic band exhibits elastic deformation.
    • Plastic Deformation:
    • The object does not return to its original shape after the load is removed.
    • Example: A spring that has been stretched beyond its limit.

Linear Elastic Distortion & Hooke’s Law

  • Hooke’s Law:
    • States that the extension of a spring is directly proportional to the force applied.
    • Mathematical representation:
    • F=kxF = kx
    • Where:
    • FF = force applied to the spring (in Newtons, N)
    • kk = spring constant (in Newtons per meter, Nm⁻¹)
    • xx = extension of the spring (in meters, m)
  • Force/Extension Graph:
    • The linear section of the graph indicates that it follows Hooke's Law, characterized by a constant gradient, kk, which represents the spring constant.
    • Elastic Limit:
    • The point at which the graph stops being linear, marking the end of elastic behavior and the onset of plastic deformation.
    • Non-Linear Section:
    • Indicates that the material is not behaving elastically and does not conform to Hooke’s Law anymore.
    • Plastic Deformation Characteristics:
      • If shallow, implies significant extension with minimal force.
      • A purely linear graph suggests the material is brittle, causing it to snap instead of stretch after reaching the elastic limit.
  • Work Done:
    • The area under the force/extension graph, calculated as:
    • extWorkDone=12kx2ext{Work Done} = \frac{1}{2} kx^2

Atmosphere and Pressure

  • Atmospheric Pressure:
    • Defined as the total weight of the air above a unit area at a specific altitude.
    • As altitude increases, atmospheric pressure decreases due to fewer air molecules above the unit area, resulting in diminished weight and thus, reduced pressure.
  • Fluid Pressure:
    • A fluid is any liquid or gas.
    • Pressure in a gas depends on atmospheric pressure.
    • Example: A balloon's gas exerts an outward force while the air exerts an inward force.
    • If the inward force is greater, the balloon collapses.
    • In the vacuum of space, the balloon can expand as the outward gas pressure exceeds the lower atmospheric pressure outside.
    • Increasing gas particles in the balloon creates more collisions with the balloon surface, increasing pressure and enabling expansion.
    • Fundamental Pressure Calculation:
    • Pressure is expressed as:
      • extPressure=extForceextAreaext{Pressure} = \frac{ ext{Force}}{ ext{Area}}

Pressure in Liquids

  • Pressure Variation with Depth:
    • In fluids, the pressure experienced increases with depth.
    • Greater depth means more fluid above the object, resulting in greater weight  hence, increased pressure.
  • Density and Pressure:
    • The greater the density of a fluid, the higher the pressure it exerts:
    • extmass=extdensityimesextvolumeext{mass} = ext{density} imes ext{volume}
    • A denser fluid has more weight, resulting in increased force on submerged objects, causing higher pressure.

Floating and Sinking

  • Principle of Floating:
    • An object will float if its weight is less than the weight of the water it displaces.
    • Example: A 1000 kg boat sinks until it displaces an equivalent weight of water (1000 kg) but maintains buoyancy as long as it does not submerge completely.
  • Buoyancy Force:
    • The buoyancy force is the upward force that counteracts the weight of the floating object.
    • This force equals the weight of the fluid displaced by the object.
  • Example of Floating:
    • A ping pong ball floats in water because its density is less than that of the water.
    • Thus, for the volume displaced by the ping pong ball, the weight of the displaced water is greater than that of the ball, resulting in a buoyant force that allows it to float.
  • Depth and Pressure:
    • As depth increases, the weight of the water above increases, leading to an increase in pressure.
    • Pressure Calculation in Fluids:
    • extPressureduetoacolumnofliquid=extheightofcolumnimesextdensityofliquidimesgext{Pressure due to a column of liquid} = ext{height of column} imes ext{density of liquid} imes g
    • Where gg is the acceleration due to gravity.