5.4 Normal Forces

Normal Forces

  • Definition: Normal force is the upward force exerted by a surface, acting perpendicular to the contact surface.

  • Purpose: Prevents objects from penetrating the surface beneath them.

Example: Normal Force on a Book

  • Scenario: A 1.2 kg book rests on a table.

  • Additional Force: Pressing down with 15 N.

  • Weight Calculation: Weight $w = mg = 1.2 ext{ kg} imes 9.8 ext{ m/s}^2 = 12 ext{ N}$.

  • Normal Force Calculation: $N = f + w = 15 ext{ N} + 12 ext{ N} = 23 ext{ N}$.

  • Implication: The normal force is greater than the weight due to added downward force.

Forces on an Incline

  • Key Forces: Only gravity and normal force act on the object.

  • Orientation: Normal force is always perpendicular to the inclined surface.

  • Weight Components:

    • wy=−wimesextcos(heta)w_y = -w imes ext{cos}( heta) (vertical component)

    • wx=wimesextsin(heta)w_x = w imes ext{sin}( heta) (horizontal component)

Common Errors with Forces on Inclines

  • Normal force must be drawn perpendicular to the incline, not directly upward.

  • Weight always acts straight down (negative y-direction).

Example: Mountain Biker

  • Scenario: Cyclist on a 20° slope.

  • Normal force relates to the y-component of weight, which is less than the total weight under equilibrium conditions.

Example: Acceleration of a Downhill Skier

  • Scenario: Skier on a 27° slope, friction ignored.

  • Coordinate System: Align x-axis with slope direction.

  • Forces:

    • extSumoff<em>x=w</em>x+n<em>x=mgextsin(heta)=ma</em>xext{Sum of } f<em>x = w</em>x + n<em>x = mg ext{ sin}( heta) = ma</em>x

    • extSumoff<em>y=w</em>y+ny=−mgextcos(heta)+n=0ext{Sum of } f<em>y = w</em>y + n_y = -mg ext{ cos}( heta) + n = 0

  • Acceleration Formula:

    • ax=gextsin(heta)a_x = g ext{ sin}( heta)

    • Example: For θ = 27°, ax=9.8extm/s2extsin(27°)ext=4.4m/s2a_x = 9.8 ext{ m/s}^2 ext{ sin}(27°) ext{ = 4.4 m/s}^2

  • Implication: Mass cancels; independent of mass for acceleration on slope.