Physics 1111: Conservation of Energy Study Notes

Work Equation

$W = extbf{F} ullet extbf{A} = F A ext{ cos}( heta)$

Total Work Sum Equation

$W*{ ext{total}} = extstyle extsum W$

Total Work Net Force Equation

$W{ ext{total}} = extbf{F}{ ext{total}} ullet extbf{A} ext{ cos}( heta)$

Kinetic Energy Equation

$K = rac{1}{2}mv^2$

Work-Kinetic Energy Theorem Equation

$W{ ext{total}} = riangle K = rac{1}{2}mvf^2 - rac{1}{2}mv_i^2$

Work Done by Springs Equation

$W{ ext{sp}} = - rac{1}{2}k xf^2 + rac{1}{2}k x_i^2$

Potential Energy Change Equation

$ riangle U = -W_{ ext{cons}}$

Work Done by Gravity Equation

$W_g = -mg riangle y$

Gravitational Potential Energy Equation

$U_g = mg riangle y$

Spring Potential Energy Equation

$U_s = rac{1}{2}kx^2$

Total Mechanical Energy Equation

$E_{ ext{mech}} = K + U$

Change in Mechanical Energy Equation

$ riangle E_{ ext{mech}} = riangle K + riangle U = 0$

Conservation of Mechanical Energy Equation

$K{ ext{before}} + U{ ext{before}} = K{ ext{after}} + U{ ext{after}}$

Work (W)

The net effect of a force $ \mathbf{F} $ on an object as it undergoes a displacement $ \mathbf{A} $.

How is work defined for a constant force?

Work is defined as $W = \mathbf{F} \cdot \mathbf{A} = F A \cos(\theta)$, where $ \mathbf{F} $ is the magnitude of the force, $ \mathbf{A} $ is the magnitude of the displacement, and $ \theta $ is the angle between $ \mathbf{F} $ and $ \mathbf{A} $.

How is total work calculated for multiple forces?

The total work is the sum of the work done by individual forces ($ \displaystyle \sum W$) or can be expressed using the net force: $W{\text{total}} = \mathbf{F}{\text{total}} \cdot \mathbf{A} \cos(\theta)$.

Kinetic Energy (K)

The energy of moving objects. It is a scalar quantity, calculated as $K = \frac{1}{2}mv^2$ where $m$ is mass and $v$ is speed.

Work-Kinetic Energy Theorem

States that when forces act on an object that displaces it, the net work done results in a change in kinetic energy: $W_{\text{total}} = \Delta K$.

How is work done by a spring determined?

Work done by a spring is determined solely by the starting and ending positions (spring lengths) and is path-independent. Springs represent a conservative force.

Conservative Force

A force is considered conservative if the work done depends only on initial and final positions, independent of the path taken.

Potential Energy (PE)

For conservative forces, it is possible to define a potential energy such that the change in potential energy equals the negative work done by the force: $ \Delta U = -W_{\text{cons}} $.

How is work done by gravity (Wg) defined?

The work done by gravity is defined by the height difference between starting and ending points: $W_g = -mg \Delta y$. It is positive when moving downward and negative when moving upward, and is independent of the path taken.

Gravitational Potential Energy (Ug)

A type of potential energy defined as $U_g = mg \Delta y$, where $m$ is mass, $g$ is the acceleration due to gravity, and $ \Delta y $ is the change in height. A reference point where $U=0$ can be chosen arbitrarily.

Spring Potential Energy (Us)

A type of potential energy defined as $U_s = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.

Total Mechanical Energy (Emech)

The sum of kinetic energy and potential energy ($E_{\text{mech}} = K + U$). This includes gravitational potential energy ($Ug$) and spring potential energy ($Us$). It remains constant if only conservative forces do work.

Conservation of Mechanical Energy

States that if only conservative forces do work, the total mechanical energy of a system remains constant ($ \Delta E{\text{mech}} = 0$). This means $K{\text{before}} + U{\text{before}} = K{\text{after}} + U_{\text{after}} $.

What is the first step in solving problems using energy conservation?

Identify the system by drawing a picture.

What must be identified to solve problems using energy conservation?

The object(s) in motion and the forces acting in the system.

How can one determine if forces are conservative?

By observing if the work done reveals changes in potential energy, as $ \Delta U = -W $.

What are the mechanical energy conservation conditions?

$ \Delta E_{\text{mech}} = 0 = \Delta K + \Delta U $.

What is the final step in solving problems using energy conservation?

Confirm that the changes in kinetic energy align with expectations (validation step).