An Overview of Kinetics
Kinetics studies the cause of motion. In general the causes of motion can be explained by the action of forces or the transfer of energy and momentum. Force, energy, momentum, and power are all related, similar to position, velocity, and acceleration in kinematics.
Momentum
\dfrac{d}{dt}(m\bm{\overrightarrow{v}})=\overrightarrow{\bm{F}}
Force
\displaystyle \int\bm{\overrightarrow{F}}\cdot d\bm{\overrightarrow{r}}=U
Energy
\dfrac{dU}{dt}=P
Power
Forces
Forces are generally thought of as the main causes of motion and are measured in Newtons [N], which has base units of \left[\dfrac{kg\cdot m}{s^2}\right]. Newton defined forces by describing three fundamental laws of motion:
- Law of Inertia: An object in motion will tend to stay in motion, an object at rest will tend to stay at rest.
- Definition of Force: \displaystyle m\bm{\overrightarrow{a}}=\sum\bm{\overrightarrow{F}}
- Law of Action and Reaction: Every action has an equal and opposite reaction.
Force can be more generally defined as the time rate of change of momentum, \displaystyle\sum\bm{\overrightarrow{F}}=\dfrac{d}{dt}(m\bm{\overrightarrow{v}}).
There is a well defined approach to solving problems involving forces:
- Choose a body to isolate.
- Draw a free body diagram for the chosen body.
- Apply Newton’s second law in a convenient coordinate frame.
- Determine if the system can be solved by counting the number of equations and unknowns.
- Solve for the unknown values.
Cartesian Coordinates
\displaystyle \sum F_x=ma_x
\displaystyle \sum F_y=ma_y
\displaystyle \sum F_z=ma_z
Normal-Tangential Coordinates
\displaystyle \sum F_n=m\dfrac{v^2}{\rho}
\displaystyle \sum F_t=m\dfrac{dv}{dt}
Polar Coordinates
\displaystyle \sum F_r=m(\ddot{r}-r\dot{\theta}^2)
\displaystyle \sum F_\theta=m(r\ddot{\theta}+2\dot{r}\dot{\theta})
Momentum and Impulse
Momentum, p, is defined as the product of an objects mass and speed and is measure in Newton-seconds [N\cdot s], which has base units of \left[\dfrac{kg\cdot m}{s}\right].
\bm{\overrightarrow{p}}=m\bm{\overrightarrow{v}}
Momentum is a conserved property, just like mass and energy. This fact results in the following conservation equation:
\displaystyle\sum\bm{\overrightarrow{p}}_1=\sum\bm{\overrightarrow{p}}_2
Impulse, J, describes the change in momentum of an object:
\displaystyle\bm{\overrightarrow{J}}=\int\bm{\overrightarrow{F}}dt
\displaystyle\bm{\overrightarrow{J}}=\int\dfrac{d\bm{\overrightarrow{p}}}{dt}dt
\displaystyle\bm{\overrightarrow{J}}=\int d\bm{\overrightarrow{p}}
Angular momentum is associated with spinning objects and is defined using linear momentum.
\bm{\overrightarrow{L}}=\bm{\overrightarrow{r}}\times\bm{\overrightarrow{p}}
Angular momentum, like linear momentum, is also conserved.
\displaystyle\sum\bm{\overrightarrow{L}}_1=\sum\bm{\overrightarrow{L}}_2
Angular momentum and torque are related, just as linear momentum and force are.
\displaystyle\sum\bm{\overrightarrow{T}}=\dfrac{d\bm{\overrightarrow{L}}}{dt}=\sum(\bm{\overrightarrow{r}}\times\bm{\overrightarrow{F}})
Energy
Energy is a very common value to calculate when working on engineering projects. Typically efficiency is directly related to how well energy is being utilized in a system and so it is an important property to understand. Energy is measured in Joules [J], which has base SI units of \left[\dfrac{kg\cdot m^2}{s^2}\right].
Work
Work is the energy associated with moving an object with a force. It is the product of force and distance traveled. The change in work from state 1 to 2 is defined as follows:
\displaystyle U_{1\to 2}=\int\bm{\overrightarrow{F}}\cdot d\bm{\overrightarrow{r}}
- Under the action of a constant gravitational field: \displaystyle U_{1\to 2}=\int(mg)(-\bm{\hat{j}})\cdot(dx\bm{\hat{i}}+dy\bm{\hat{j}})=mg(y_1-y_2)
- Under the action of a variable gravitational field: \displaystyle U_{1\to 2}=\int -(G\dfrac{mm_e}{r^2})(\bm{\hat{e}}_r)\cdot(dr\bm{\hat{e}}_r)=Gmm_e\left(\dfrac{1}{r_2}-\dfrac{1}{r_1}\right)
- Spring force: \displaystyle U_{1\to 2}=\int kx(-\bm{\hat{i}})\cdot(dx\bm{\hat{i}})=\dfrac{1}{2}k(x_1^2-x_2^2)
Kinetic Energy
Kinetic energy is based on the state of a particle in motion, being defined as follows:
\displaystyle T=\int m\bm{\overrightarrow{v}}\cdot d\bm{\overrightarrow{v}}=\dfrac{1}{2}mv^2
Principle of Work-Kinetic Energy
Energy is a conserved property, just like momentum, and can therefore be neither created or destroyed.
\displaystyle\sum E_1=\sum E_2
\displaystyle T_1+U_{1\to 2}=T_2
Power
Power is the time rate of performing work. The measurement of power is useful when determining energy usage over a specified time frame. Power is measured in Watts [W], which has base SI units of \left[\dfrac{kg\cdot m^2}{s^3}\right].
P=\dfrac{dU}{dt}=\bm{\overrightarrow{F}}\cdot\bm{\overrightarrow{v}}