formula.expert

Engineering Mechanics — formula sheet

Statics & dynamics · first-year engineering · 54 formulas · metric edition 1

The full booklet — learning zone, problems, answer key
Newton's Second Law
F=maF = m a
Weight (W = mg)
W=mgW = m g
Pressure (P = F/A)
P=FAP = \frac{F}{A}
Torque
τ=rFsin⁡θ\tau = r F \sin\theta
Weight Component Along an Incline (mg sin θ)
F∥=mgsin⁡θF_{\parallel} = m g \sin\theta
Normal Force on an Incline (N = mg cos θ)
N=mgcos⁡θN = m g \cos\theta
Maximum Static Friction (f = μₛN)
fs,max⁡=μsNf_{s,\max} = \mu_s N
Kinetic Friction Force (f = μₖN)
fk=μkNf_k = \mu_k N
Angle of Repose (μ = tan θ)
μs=tan⁡θ\mu_s = \tan\theta
Torque with a Lever Arm (τ = rF sin θ)
τ=rFsin⁡θ\tau = r F \sin\theta
Mechanical Advantage of a Lever
MA=dedlMA = \frac{d_e}{d_l}
Pulley System Effort Force
F=WnF = \frac{W}{n}
Machine Efficiency
η=WoutWin\eta = \frac{W_{out}}{W_{in}}
Support Reaction — Simple Beam, Off-Centre Point Load
RA=P(L−a)LR_A = \frac{P (L - a)}{L}
Max Moment — Simple Beam, Off-Centre Point Load
M=Pa(L−a)LM = \frac{P a (L - a)}{L}
Final Velocity (Uniform Acceleration)
v=v0+atv = v_0 + a t
Displacement (Uniform Acceleration)
d=v0t+12at2d = v_0 t + \tfrac{1}{2} a t^2
Velocity-Displacement Relation (v² = v₀² + 2ad)
v2=v02+2adv^2 = v_0^2 + 2 a d
Acceleration Down a Frictionless Incline
a=gsin⁡θa = g \sin\theta
Acceleration Down an Incline with Friction
a=g(sin⁡θ−μkcos⁡θ)a = g\left(\sin\theta - \mu_k \cos\theta\right)
Rope Tension When Lifting a Mass
T=m(g+a)T = m\left(g + a\right)
Atwood Machine Acceleration
a=(m1−m2)gm1+m2a = \frac{\left(m_1 - m_2\right) g}{m_1 + m_2}
Drag Force (F = ½CdρAv²)
FD=12CdρAv2F_D = \tfrac{1}{2} C_d \rho A v^{2}
Terminal Velocity
vt=2mgρACdv_t = \sqrt{\frac{2 m g}{\rho A C_d}}
Work (W = Fd cos θ)
W=Fdcos⁡θW = F d \cos\theta
Kinetic Energy
Ek=12mv2E_k = \tfrac{1}{2} m v^{2}
Gravitational Potential Energy (U = mgh)
U=mghU = m g h
Work–Energy Theorem
W=12m(v2−v02)W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)
Hooke's Law
F=kxF = k x
Elastic Potential Energy
U=12kx2U = \tfrac{1}{2} k x^{2}
Power (P = W/t)
P=WtP = \frac{W}{t}
Power from Force and Velocity (P = Fv)
P=FvP = F v
Linear Momentum (p = mv)
p=mvp = m v
Impulse (J = FΔt)
J=F ΔtJ = F \, \Delta t
Conservation of Momentum (Two Bodies)
m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2
Perfectly Inelastic Collision
v=m1u1+m2u2m1+m2v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}
Coefficient of Restitution
e=v2−v1u1−u2e = \frac{v_2 - v_1}{u_1 - u_2}
Angular Velocity (ω = θ/t)
ω=θt\omega = \frac{\theta}{t}
Angular Acceleration
α=ω−ω0t\alpha = \frac{\omega - \omega_0}{t}
Linear Speed from Rotation (v = ωr)
v=ωrv = \omega r
Angular Velocity from Period
ω=2πT\omega = \frac{2\pi}{T}
Speed in Circular Motion (v = 2πr/T)
v=2πrTv = \frac{2\pi r}{T}
Newton's Second Law for Rotation (τ = Iα)
τ=Iα\tau = I \alpha
Moment of Inertia: Solid Disk
I=12mr2I = \tfrac{1}{2} m r^{2}
Moment of Inertia: Point Mass
I=mr2I = m r^{2}
Centripetal Acceleration (a = v²/r)
ac=v2ra_c = \frac{v^2}{r}
Centripetal Force (F = mv²/r)
Fc=mv2rF_c = \frac{m v^2}{r}
Centripetal Acceleration (a = ω²r)
ac=ω2ra_c = \omega^{2} r
Rotational Kinetic Energy
KErot=12Iω2KE_{rot} = \tfrac{1}{2} I \omega^{2}
Angular Momentum (L = Iω)
L=IωL = I \omega
Rotational Power (P = τω)
P=τωP = \tau \omega
Shaft Torque from Power and Angular Speed
T=PωT = \frac{P}{\omega}
Undamped Natural Frequency
fn=12πkmf_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}}
Natural Frequency from Static Deflection
fn=12πgδstf_n = \frac{1}{2\pi} \sqrt{\frac{g}{\delta_{st}}}