Energy & Work

• The Work-Energy Theorem states that the net work done by all forces acting on an object is equal to the change in the object's Kinetic Energy ($K$). This principle links the dynamics of force and motion to the energy state of the object.

• Mathematically, this is expressed as $W_{net} = \Delta K = K_f - K_i$, where $K = \frac{1}{2}mv^2$. This relationship is particularly useful because it allows for the calculation of final speeds without needing to know the details of the acceleration over time.

• This theorem applies to both constant and variable forces. For variable forces, work is calculated as the integral of the force over the displacement: $W = \int_{x_1}^{x_2} F(x) dx$.

• Potential Energy ($U$) is the energy stored within a system due to its configuration or position. Common forms include Gravitational Potential Energy ($U_g = mgh$) and Elastic Potential Energy ($U_s = \frac{1}{2}kx^2$ for a spring).

• A force is considered conservative if the work it does on an object moving between two points is independent of the path taken. Gravity and electrostatic forces are conservative, meaning the energy spent moving against them can be fully recovered.

• For conservative forces, the work done is equal to the negative change in potential energy: $W_c = -\Delta U$. This allows us to define a potential energy function for any conservative force field.

• In an isolated system where only conservative forces perform work, the total Mechanical Energy ($E_{mech}$) remains constant. Mechanical energy is the sum of kinetic and potential energies: $E_{mech} = K + U$.

• The principle of conservation is expressed as $K_i + U_i = K_f + U_f$. This is a powerful tool for solving problems involving motion where forces like friction are negligible, such as a pendulum swinging or a satellite in orbit.

• When non-conservative forces (like friction or air resistance) are present, they do work that transforms mechanical energy into other forms, such as thermal energy. In these cases, $W_{nc} = \Delta E_{mech}$.

• Power ($P$) is defined as the rate at which work is done or energy is transferred over time. It measures how quickly a task is completed rather than the total amount of energy used.

• The average power is calculated as $P = \frac{W}{\Delta t}$, while instantaneous power can be expressed as the dot product of force and velocity: $P = \vec{F} \cdot \vec{v} = Fv \cos \theta$.

• The standard unit of power is the Watt (W), where $1 \text{ W} = 1 \text{ J/s}$. In engineering contexts, horsepower (hp) is also frequently used, where $1 \text{ hp} \approx 746 \text{ W}$.

Comparison of Force Types

• Understanding the difference between conservative and non-conservative forces is vital for determining whether energy is conserved in a system.

• Check the Angle: Always identify the angle between the force and displacement vectors. A common mistake is using the angle of an incline instead of the angle between $\vec{F}$ and $\vec{d}$.

• Define the System: Clearly identify which objects are part of your system. Potential energy belongs to the system (e.g., Earth-object system), not just the object itself.

• Sign Conventions: Be meticulous with signs. Work done by the system is usually positive in energy transfer contexts, but work done against a conservative force increases potential energy ($W_{ext} = \Delta U$).

• Units Consistency: Ensure all mass is in kg, distance in meters, and time in seconds to yield results in Joules and Watts. Mixing units is a frequent source of calculation errors.