Physics 151 Equations

Name

Equations

Units (if specified)

Extra Notes

Average speed

\overline{v}=\dfrac{d}{\Delta t}

m/s

Average velocity

\overline{v}=\dfrac{\Delta x}{\Delta t}

m/s

2D kinematics: replace x with r

Instantaneous velocity

\overrightarrow{v}=\lim _{\Delta t\rightarrow 0}\dfrac{\Delta \overrightarrow{x}}{\Delta t}

m/s

2D kinematics: replace x with r

Average acceleration

\overrightarrow{\overline{a}}=\dfrac{\Delta \overrightarrow{v}}{\Delta t}

m/s^{2}

Displacement

\Delta x=x-x_{0}

m

2D kinematics: replace x with r

Slope

\dfrac{\Delta y}{\Delta x}

\dfrac{\Delta x}{\Delta t} or \dfrac{\Delta v}{\Delta x}

Kinematics (1D)

v=v_{0}+at

x=\dfrac{1}{2}\left( v_{0}+v\right) t

v^{2}=v_{0}^{2}+2ax

x=v_{0}t+\dfrac{1}{2}at^{2}

Kinematics (2D)

v_{?}=v_{0?}+a_{?}t

?=v_{0?}t+\dfrac{1}{2}a_{?}t^{2}

?=\dfrac{1}{2}\left( v_{0?}+v_{?}\right) t

v_{?}^{2}=v_{0?}^{2}+2a_{?}?

? means replace with x OR y

Projectile motion:

a_{y}=-9.80m/s^{2}

a_{x}=0

Relative velocity

v_{ab}=v_{b}-v_{a}

m/s

Newton’s 2nd Law

\Sigma F=ma

N

2D: \sum F_{?}=ma_{?}

Gravitational force

F=G\dfrac{m_{1}m_{2}}{r^{2}}

\dfrac{N\cdot m^{2}}{kg^{2}}

G =6.673\times 10^{-11}

N\cdot m^{2}/kg^{2}

Weight

W=mg

N

Always acts down

Normal force

F_{N}=mg

N

Perpendicular force

Static friction

f_{s}^{MAX}=\mu_s \cdot F_{N}

Force before breakaway

Kinetic friction

f_{\text{k}} = \mu_k \cdot F_{N}

Moving surfaces

Velocity in circular motion

v_{c}=\dfrac{2\pi r}{T}

m/s

Changes direction, not constant

Period (T)

T=\dfrac{1}{frequency}

s

To make one revolution

Centripetal acceleration

a_{c}=\dfrac{v^{2}}{r}

m/s^{2}

Direction towards the center

Centripetal force

F_{c}=\dfrac{mv^{2}}{r} or F_{c}=ma_{c}

N

Always directed towards the center, changes direction

Speed of a satellite

v=\sqrt{\dfrac{GM_{E}}{r}}

m/s

Period of a satellite

T=\dfrac{2\pi r^{\dfrac{3}{2}}}{\sqrt{GM_{E}}}

Work

W=Fs or

W=Fcos\theta s

J

cos0° = 1 (F)

cos90° = 0

cos180° = -1 (-F)

Kinetic energy

KE=\dfrac{1}{2}mv^{2}

J

Always positive

Work-energy theorem

W=KE_{f}-KE_{0}

J

Potential energy and gravitational PE

PE=mgh

W_{gravity}=mg\left(h_{0}-h_{f}\right)

J

Stored energy

PE max = KE is 0

PE is 0 = KE max

Work for closed path

W_{gravity}=0 J

J

Conservative and Nonconservative forces acting together

W=W_{c}+W_{nc}

J

Both forces act on an object at the same time

Conservation of energy

E_{f}=E_{0}

KE_{f}+PE_{f}=KE_{0}+PE_{0}

J

Work energy theorem (nonconservative)

W_{nc}=E_{f}-E_{0} or

\left( PE_{f}+KE_{f}\right) -\left( PE_{0}+KE_{0}\right)

J

Total mechanical energy

E=KE+PE

J

Average power

\overline{P}=\dfrac{W}{t}

W

Impulse

\overrightarrow{J}=F\Delta t

N\cdot s

change in momentum

same direction as avg. force

Linear momentum

\overrightarrow{p}=m\overrightarrow{v}

kg\cdot m/s

mass in motion

Impulse-momentum theorem

\sum F\Delta t=p_{f}-p_{0}

when a net force acts on an object, the impulse force is equal to the change in the momentum

Principle of conservation of linear momentum

p_{f}=p_{0}

Collisions in 1D

p_{f1}+p_{f2}=p_{01}+p_{02}

Collisions in 2D

p_{f1?}+p_{f2?}=p_{01?}+p_{02?}

? means x or y

Center of mass (CoM)

x_{cm}=\dfrac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}

Velocity of CoM

v_{cm}=\dfrac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}

Angular displacement

\Delta \theta =\theta -\theta _{0}

Angular displacement in radians

\theta =\dfrac{s}{r}

Arc length

s=r\theta

Full revolution

2\pi rad=360°

Angular velocity

\overline{\omega }=\dfrac{\Delta \theta }{\Delta t}

Angular acceleration

\overline{\alpha }=\dfrac{\Delta \omega }{\Delta t}

Rotational kinematics

\omega =\omega _{0}+\alpha t

\theta =\dfrac{1}{2}\left( \omega _{0}+\omega \right) t

\theta =\omega _{0}t+\dfrac{1}{2}\alpha t^{2}

\omega ^{2}=\omega _{0}^{2}+2\alpha \theta

Tangential velocity

v_{T}=r\omega

Tangential acceleration

a_{T}=r\alpha

Centripetal acceleration

a_{c}=r\omega ^{2}

Pythagorean theorem

(acceleration)

\overrightarrow{a}=\sqrt{\overrightarrow{a}_{c}^{2}+\overrightarrow{a_{T}}^{2}}

Translational velocity

v=rw

Translational acceleration

a=r\alpha

Torque

\tau =Fl

Equilibrium conditions

\Sigma F_{x}=0

\Sigma F_{y}=0

\Sigma \tau =0

Center of gravity

x_{cg}=\dfrac{W_{1}x_{1}+W_{2}x_{2}}{W_{1}+W_{2}}

Tangential force

F_{T}=ma_{T}

w/ sub \tau =\left( mr^{2}\right) \alpha

Inertia

I=mr^{2}

Moment of inertia

\Sigma \tau=\sum \left( mr^{2}\right) \alpha

\tau1+\tau2+\tau3

Net external force

\Sigma \tau =I\alpha

Rotational work

W_{R}=\tau \theta

Rotational KE

KE_{R}=\dfrac{1}{2}I\omega ^{2}

Total KE

KE_{tota1}=\dfrac{1}{2}I\omega ^{2}+\dfrac{1}{2}mv_{T}^{2}

Mechanical energy of a rotating object

E=\dfrac{1}{2}mv^{2}+\dfrac{1}{2}I\omega ^{2}+mgh

Energy conservation: E_{f}=E_{0}

Angular momentum

L=I\omega

Mass density

\rho =\dfrac{m}{V}

Pressure

P=\dfrac{F}{A}

Pressure and depth in static fluid

P_{2}=P_{1}+\rho gh

Pressure gauges

P_{atm}=\rho gh

Archimede’s principle

F_{B}=\rho Vg

buoyant force = mass of displaced fluid

Linear thermal expansion

\Delta L=\alpha L_{0}\Delta T

Volume thermal expansion

\Delta V=\beta V_{0}\Delta T

Specific heat capacity

Q=mc\Delta T

Latent heat

Q=mL

Restoring force produced by a spring

F_{x}=kx

Hooke’s law

F_{x}=-kx

displacement

x=A\cos \omega t

Frequency

f=\dfrac{1}{T}

Angular frequency

\omega =2\pi f or

\omega =\sqrt{\dfrac{k}{m}}

Max speed (shm)

v_{max}=Aw

Max acceleration (shm)

a_{max}=Aw^{2}

Work done by spring

W=\left( F\cos \theta \right) s

W=\dfrac{1}{2}kx_{0}^{2}-\dfrac{1}{2}kx_{f}^{2}

Mechanical energy

insert

PE elastic

\dfrac{1}{2}kx^{2}