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Mechanical Work

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Core Definition

Mechanical Work (WW) is the scalar measure of energy transfer that occurs when an object is displaced by an external force:

W=F⃗⋅d⃗=∣F⃗∣∣d⃗∣cos⁡θW = \vec{F} \cdot \vec{d} = |\vec{F}| |\vec{d}| \cos\theta
where:

  • F⃗\vec{F} is the constant force vector (in Newtons, N\text{N}).
  • d⃗\vec{d} is the displacement vector (in meters, m\text{m}).
  • θ\theta is the angle between the force vector and the direction of displacement.
  • WW is the resulting work scalar, measured in Joules (1 J=1 N⋅m1\text{ J} = 1\text{ N}\cdot\text{m}).

Concept Explanation

Because work is defined via the Dot Product, it is strictly a scalar quantity:

  • Positive Work (W>0W > 0): The force aids motion (θ<90∘\theta < 90^\circ). Energy is added to the system.
  • Zero Work (W=0W = 0): The force acts perpendicular to motion (θ=90∘\theta = 90^\circ). No mechanical work is performed (e.g., centripetal force in uniform circular motion, normal force on a level track).
  • Negative Work (W<0W < 0): The force opposes motion (θ>90∘\theta > 90^\circ). Energy is removed from the system (e.g., kinetic friction, drag).

Straight-Line Displacement in 3D Space

If an object moves from initial point P(x1,y1,z1)P(x_1, y_1, z_1) to final point Q(x2,y2,z2)Q(x_2, y_2, z_2), the displacement vector is:

d⃗=PQ⃗=⟨x2−x1,y2−y1,z2−z1⟩\vec{d} = \vec{PQ} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle
The work done by a constant 3D force F⃗=⟨Fx,Fy,Fz⟩\vec{F} = \langle F_x, F_y, F_z \rangle is computed as:
W=F⃗⋅d⃗=Fx(x2−x1)+Fy(y2−y1)+Fz(z2−z1)W = \vec{F} \cdot \vec{d} = F_x(x_2 - x_1) + F_y(y_2 - y_1) + F_z(z_2 - z_1)

Engineering Applications

  • Mechanical Conveyors & Actuators: Sizing motor output requirements to move industrial payloads across defined assembly tracks.
  • Wind Turbines & Aerodynamics: Calculating the power delivered by aerodynamic wind lift forces across rotating turbine blades.
  • Civil Earthmoving: Computing energy expended by bulldozers and cranes moving soil and building materials over elevations.

Example Problems

Example 1: 3D Coordinate Displacement Work

A force F⃗=⟨4,−4,4⟩ N\vec{F} = \langle 4, -4, 4 \rangle\text{ N} acts on a component as it moves from P(−7,−2,5) mP(-7, -2, 5)\text{ m} to Q(8,−7,−5) mQ(8, -7, -5)\text{ m}. Compute the work done.

Solution
  1. Compute Displacement Vector d⃗=PQ⃗\vec{d} = \vec{PQ}:
    d⃗=⟨8−(−7),−7−(−2),−5−5⟩=⟨15,−5,−10⟩ m\vec{d} = \langle 8 - (-7), -7 - (-2), -5 - 5 \rangle = \langle 15, -5, -10 \rangle\text{ m}
  2. Compute Work via Dot Product:
    W=F⃗⋅d⃗=(4)(15)+(−4)(−5)+(4)(−10)=60+20−40=40 JW = \vec{F} \cdot \vec{d} = (4)(15) + (-4)(-5) + (4)(-10) = 60 + 20 - 40 = 40\text{ J}

Connections & References