Forces and Free Body Diagrams

Topic 2.2: Forces and Free Body Diagrams

Overview

• Presenter: Kristen Gonzalez Vega, Centennial High School, Frisco, Texas.

• Focus: Drawing free body diagrams for forces at an angle, including forces on objects sliding up and down inclined planes with friction.

Key Concepts

Free Body Diagrams (FBD)

• Definition: A free body diagram visually represents all the forces acting on an object. It typically uses arrows to indicate the direction and magnitude of these forces.

General Rules for Drawing Free Body Diagrams

• Starting Point: Begin at the object’s center of mass, represented by a dot.

• Forces Representation:

  • Draw arrows away from the object, with the direction indicating the force.

  • The length of the arrows represents the magnitude of the forces—longer arrows indicate stronger forces.

Example: Cylinder Suspended by Strings

• Setup: Cylinder tied with two strings to different spring scales.

• Forces Acting on the Cylinder:

  • Gravitational Force: Acts downward toward the earth's center.

  • Tension Forces:

  • Tension to the left (horizontal).

  • Tension to the right (angled upwards).

• Note: In free body diagrams, all forces must be either parallel or perpendicular to each other.

Breaking Angled Forces into Components

• Avoid Drawing Components on FBD:

  • Components of forces should be calculated separately and not drawn on the FBD.

  • On a free body diagram, drawing components is considered incorrect as it leads to misrepresentation of forces.

Right Triangle for Components

• Use trigonometric functions to break down the tension at angle (B8).

  • Horizontal Component (X-axis):
    $FTR_{x} = FTR imes ext{cos}( heta)$

  • Vertical Component (Y-axis):
    $FTR_{y} = FTR imes ext{sin}( heta)$

Equilibrium Conditions

Case of the Cylinder

• Acceleration Analysis: Since the cylinder is not accelerating, the net force is zero in both the X and Y directions.

• X-Direction Equation:

  • Applied Forces: $FTL = FTR_{x}$

  • Which translates to:
    $FTL = FTR imes ext{cos}(40^ ext{o})$

• Y-Direction Equation:

  • Applied Forces:
    $mg = FTR_{y}$

  • Which translates to:
    $mg = FTR imes ext{sin}(40^ ext{o})$

• Given Data:

  • Mass of the cylinder: 500 grams or 0.5 kg.

  • Acceleration due to gravity: $g ext{ (approximately } 9.8 rac{m}{s^{2}})$.

• Tension Calculation:

  • Solving for vertical component:

  • $FTR = rac{mg}{ ext{sin}(40^ ext{o})} o FTR ext{ calculated} <br>ightarrow 7.7 ext{ Newtons}$

  • Solving for horizontal component:

  • $FTL = FTR imes ext{cos}(40^ ext{o}) o FTL ext{ calculated} <br>ightarrow 5.9 ext{ Newtons}$

Example: Block Sliding on an Inclined Plane

• Observation: Block pushed up the ramp, slows down, then slides back down.

• Forces Acting on The Block During Sliding:

  • Normal Force: Perpendicular to the ramp surface.

  • Gravitational Force: Always directed downwards towards center of the earth.

  • Friction Force: Opposes the direction of movement.

  • When sliding up, friction acts down the ramp (opposing motion).

  • When sliding down, friction acts up the ramp (opposing motion).

Redefining Axis for Inclined Motion

• Set the X-axis parallel to the ramp and the Y-axis perpendicular.

• Breaking Gravitational Force into Components:

  • Gravitational Force Parallel to the Ramp:
    $F<em>{g, ext{parallel}} = F</em>G imes ext{sin}( heta)$

  • Gravitational Force Perpendicular to the Ramp:
    $F<em>{g, ext{perpendicular}} = F</em>G imes ext{cos}( heta)$

Conclusion

• Importance of Free Body Diagrams:

  • Reinforces understanding of forces acting on objects and helps visualize the balance of forces.

  • Essential tool for applying Newton's second law to solve for acceleration and other variables.

• Key Takeaway: Always ensure that forces are drawn in a way that is parallel or perpendicular to the motion and break down any angled forces using trigonometric functions to facilitate calculations for real-world applications.