Mechanical Work and Energy Principles in Physics

Mechanical Work (W)

Mechanical Work (W): Relates to the force applied on a body that moves the body a certain distance. Work can be positive, negative, or zero.

Work Calculation

Work (W): Applying a force and measuring the displacement in the direction of the force (measured in Newtons (N) for force).

Formula: W = F * D

Work in Joules (J)

• Work (W) in Joules (J) equals the amount of energy transferred to the body by the force (a scalar magnitude).
• Force in Newtons (N) (Vector).
• Distance in meters (Vector).

Energy and Work Relationship

Energy: Directly relates to the capacity to perform work (W).

Sign of Work

• W positive: When the force acts in the same direction as the displacement ($\theta = 0^{\circ}$), so $W > 0$.
• W negative: When the force acts in the opposite direction to the displacement ($\theta = 180^{\circ}$), so $W < 0$.
• W zero: When the force acting on the body is perpendicular to the displacement ($\theta = 90^{\circ}$), so $W = 0$.

Many forces can act on an object, but not all contribute to the work done in a specific context.

Types of Mechanical Energy

Kinetic Energy (E_k)

Kinetic Energy (E_k): Is proportional to the mass and the square of the speed. This energy depends on how fast a body moves and its mass.

Translational Formula: E_k = 1/2 M V^2

• $E_k$ = Joules (J)
• Mass ($M$) = kilograms (kg)
• Velocity ($V$) = meters per second (m/s)

Rotational Kinetic Energy: E_{kr} = 1/2 I \omega^2

• $I$ = Moment of inertia
• $\omega$ = Angular velocity

Potential Energy (E_p)

Potential Energy (E_p): Depends on the position of the body within a force field.

Elastic Potential Energy

Formula: E_{pe} = 1/2 K X^2

Gravitational Potential Energy

Formula: E_{pg} = mgh (Depends on the reference point)

Work done against gravity: $W = mgh = E_p$

Total Mechanical Energy

Mechanical Energy (E_m): E_m = E_p + E_k = mgh + 1/2 mv^2

Forces Affecting Mechanical Energy

Dissipative Forces: Forces that cause the total mechanical energy ($E_m$) to decrease because they are not constant.

Law of Conservation of Mechanical Energy

There is no variation in $E_k$ if the work involved is due only to conservative forces acting on the change of position.

The work done by conservative forces (like gravity) on an object depends only on the initial and final heights, resulting in $\Delta E_k = -\Delta E_p$.

Equilibrium Points

• Stable Equilibrium Point: Every body tends to move toward this position; if displaced, the body returns to the same place.
• Unstable Equilibrium Point: Tends to move away to opposite sides when slightly displaced.
• Turning Point: When an object moving upward momentarily stops (velocity is 0) and then returns due to gravity.

Conservative Forces (Examples)

Elastic potential energy, weight, and gravitational force.

Conditions for Applying the Law of Conservation of Energy

• The system must involve only Kinetic ($E_k$) and Potential ($E_p$) energy.
• Only conservative forces must be present.
• The initial and final positions must be defined without needing to track the entire trajectory.
• There should be no interaction with non-conservative (dissipative) forces ($F_{dissipative}$).