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Dynamics: Motion, Forces, and Energy

This page contains formulas from Physics 1 that will be useful this semester.

See the Symbols, Units, and Constants page for help decoding mathematical operations or subscripts shown in this page.

Motion

These equations describe the one dimensional motion of an object traveling with constant acceleration $ a $ from initial position $ x_0 $ and having initial velocity $ v_0 $.

One-dimensional constant acceleration motion:

$$ \begin{aligned}&{\text{Position } x \text{ at time } t:}&& {x=x_0+v_0 t + \frac{1}{2}at^2} \\&{\text{Velocity } v \text{ at time } t:}&& v=at \\&{\text{Velocity } v \text{ after traveling } \Delta x :}&& v^2={v_0}^2+2a\Delta x \end{aligned} $$

Centripetal acceleration: Acceleration perpendicular to an object's velocity will cause it to travel in a circle of radius r:

$$ a_c=\frac{v^2}{r} $$

An object's momentum is a vector that points in the same direction as its velocity:

$$ \vec{p}=m\vec{v} $$

Forces

Forces are vector quantities. Any time you add forces together, you are adding vectors.

The total (vector sum) force acting on an object will cause its (vector) acceleration:

$$ \sum\vec{F}=m\vec{a} $$

We solve problems by applying Newton's 2nd Law one dimension at a time: All forces along one dimension cause acceleration along the same dimension

$$ \begin{aligned}&{\text{On the } x \text{ axis} :}&& F_{1x}+F_{2x}+\cdots=ma_x \\&{\text{On the } y \text{ axis}:}&& F_{1y}+F_{2y}+\cdots=ma_y \end{aligned} $$

Macroscopic force models

$$ \begin{aligned}&{\text{Gravity (near Earth's surface)}}&& F_g=mg \\&{\text{Spring force}}&& \vec{F}_s=-k\vec{x} \end{aligned} $$

A force applied perpendicular to a distance r from an object's rotation axis will create torque about that axis:

$$ \begin{aligned}\tau&=F_\perp r \\ &=F r_\perp \\ &=Fr\sin{\theta} \end{aligned} $$

Acceleration due to gravity (near the surface of Earth)

We use "standard gravity" for calculations:
g=9.81 m/s²

Work and Energy

Each force F does work (transfers energy) when an object is displaced distance d parallel to that force:

$$ \begin{aligned}W_F&=F_\parallel d \\ &=F d_\parallel \\ &=Fd\cos{\theta} \end{aligned} $$

Power is defined as the rate of energy transfer:

$$ P=\frac{W}{\Delta t} $$

Each force F pushing an object at velocity v parallel to the force has power (energy transfer rate):

$$ \begin{aligned}P_F&=F_\parallel v \\ &=F v_\parallel \\ &=Fv\cos{\theta} \end{aligned} $$

A moving object has kinetic energy:

$$ K=\frac{1}{2}mv^2 $$

Total work on an object changes that object's kinetic energy:

$$ W_{total}=\Delta K $$

Total mechanical energy of an object combines kinetic and potential energy:

$$ E=K+U $$

Conservative forces can store energy as Potential Energy (PE)

$$ \begin{aligned}&{\text{Conversion, Work to PE:}}&& W_c=-\Delta U \\&{\text{Gravitational Potential Energy:}}&& U_{g}=mgh \\&{\text{Spring Potential Energy:}}&& U_s=\frac{1}{2}kx^2 \end{aligned} $$

Nonconservative work changes the mechanical energy of an object:

$$ E_i+W_{nc}=E_f $$

Universal Gravitation

Two objects having masses m₁ and m₂ will exert a gravitational force on each other.

This force is always mutual and attractive (i.e., the masses pull toward each other).

Magnitude of gravitational force exerted on mass 2 by mass 1 when separated by distance r:

$$ |\vec{F}_{1\text{ on }2}|=G\frac{m_1 m_2}{{r}^2} $$

Gravitational potential energy stored between the two masses is:

$$ U_G=-G\frac{m_1 m_2}{{r}} $$

Warning: Negative potential energy #GPE

The negative sign is not a mistake in the formula sheet and should not be left out of your calculations. As the two masses get closer to each other, gravitational potential energy must decrease. Using the negative sign is the only way to make this possible.

Gravitational constant

G=6.67×10^-11 N m²/kg²