Force

• Inertia is the tendency of an object to resist changes in its state of motion.

• Mass is a measure of the inertia of an object.

• A force is an interaction between two objects or between an object and its environment.

• A force is a vector quantity.

• ${\vec{\mathbf{F}}}_{\text{G}}=m\vec{\mathbf{g}}$ is the gravitational force (in $\mathsf{N}$) acting on an object with mass $m$ in a gravitational field with gravitational acceleration $\vec{\mathbf{g}}$.

  • $W=mg$ is the weight (in $\mathsf{N}$) of an object with mass $m$ in a gravitational field with gravitational acceleration $g$.
  • (Some define weight as a vector quantity, the gravitational force acting on an object, while others define it as a scalar quantity, the magnitude of the gravitational force)

• The normal force ${\vec{F}}_{\text{N}}$ is the force exerted by a surface perpendicular to the object in contact with it.

• The friction force ${\vec{F}}_{\text{fr}}$ is the force exerted by a surface parallel to the object in contact with it, opposing its motion.

  • $F_{\text{fr}}={\mu }_k{F}_{\text{N}}$ is the kinetic friction force (in $\mathsf{N}$) acting on an object moving on a surface with normal force $F_{\text{N}}$ and coefficient of kinetic friction ${\mu }_k$.
  • $F_{\text{fr}}={\mu }_s{F}_{\text{N}}$ is the static friction force (in $\mathsf{N}$) acting on an object at rest on a surface with normal force $F_{\text{N}}$ and coefficient of static friction ${\mu }_s$.

Newton’s Laws of Motion

1st Law

A body remains at rest, or in motion at a constant speed in a straight line, except insofar as it is acted upon by a force.

2nd Law

The acceleration of a body is directly proportional to the net force acting on it, and inversely proportional to its mass.

• $F=ma$
• $\vec{\mathbf{F}}=m\vec{\mathbf{a}}$
• $\vec{\mathbf{F}}=\frac{d\vec{\mathbf{p}}}{dt}$
• (rotational form) $\tau =I\alpha =\frac{dL}{dt}$
  • $\tau$ is the torque
  • $I$ is the moment of inertia
  • $\alpha$ is the angular acceleration
  • $L$ is the angular momentum

3rd Law

For every action force ${\vec{F}}_{1\to 2}$ exerted by object 1 on object 2, there is an equal in magnitude and opposite in direction reaction force ${\vec{F}}_{2\to 1}$ exerted by object 2 on object 1. ${\vec{F}}_{1\to 2}=-{\vec{F}}_{2\to 1}$

Gravity

• $G=6.67430(15)\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ is the gravitational constant (dim. ${\mathsf{M}}^{-1}{\mathsf{L}}^3{\mathsf{T}}^{-2}$)
• $g=\frac{F}{m}$ is the gravitational acceleration (in $m/{\mathsf{s}}^2$) of a free-falling object of mass $m$ in a gravitational field with gravitational force $F$ acting on it
• $g=G\frac{M}{r^2}$ is the gravitational acceleration (in $m/{\mathsf{s}}^2$) at a point a gravitational field due to a mass $M$, where $r$ is the distance from the center of mass of $M$ to the point
  • If $M$ is the Earth, then $g=G{M}_{\text{E}}{r}^2\approx 9.81m/{\mathrm{s}}^2$ (Gravity on Earth)

Newton’s Law of Universal Gravitation

$F=G\frac{m_1{m}_2}{r^2}$

• ${\mathbf{F}}_{12}=-F{\hat{\mathbf{r}}}_{12}=F{\hat{\mathbf{r}}}_{21}=-{\mathbf{F}}_{21}\text{(vector form)}$
• $F=|{\mathbf{F}}_{12}|=|{\mathbf{F}}_{21}|$ is the magnitude of gravitational force (${\mathbf{F}}_{12}$ is the force exerted by $m_1$ on $m_2$, and vice versa)
• $m_1$ and $m_2$ are the (center of) mass of the two objects
• ${\mathbf{r}}_{12}=-{\mathbf{r}}_{21}$ is the position vector from $m_1$ to $m_2$,
  • $r=|{\mathbf{r}}_{12}|=|{\mathbf{r}}_{21}|$ the distance between the two objects
• ${\hat{\mathbf{r}}}_{12}=\frac{{\mathbf{r}}_{12}}{r}$ is the unit vector in the direction of ${\mathbf{r}}_{12}$ (and vice versa)

Gravitational Potential Energy

• $U_g=-G\frac{m_1{m}_2}{r}$
• $\Delta U_g=mg\Delta h$

Mechanical Advantage

• The mechanical advantage of a machine is the ratio of the output force to the input force $\mathrm{MA}=\frac{F_{\text{out}}}{F_{\text{in}}}$

• A simple machine is a mechanical device that changes the magnitude of a force (i.e. the $\mathrm{MA}$ is not $1$), or the direction of a force

• Lever

  • $F_{\text{effort}}{d}_{\text{effort}}=F_{\text{load}}{d}_{\text{load}}$ (Law of the Lever)
  • $F_{\text{effort}}$ and $F_{\text{load}}$ are the effort and load forces
  • $d_{\text{effort}}$ and $d_{\text{load}}$ are the effort and load distances from the fulcrum
  • Class 1 Lever: fulcrum between the effort and load
  • Class 2 Lever: load between the fulcrum and the effort
  • Class 3 Lever: effort between the fulcrum and the load

• MA of bicycletodos

  • $\text{MA}=\frac{d_{\text{in}}}{d_{\text{out}}}=\frac{F_{\text{out}}}{F_{\text{in}}}=\frac{r_{\text{sprocket}}}{r_{\text{chainring}}}\cdot \frac{r_{\text{crank}}}{r_{\text{wheel}}}$

Density

• $\rho =\frac{m}{V}$ is the density (in $kg/{\mathsf{m}}^3$)

Mechanical Equilibrium

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