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What is the equation for final velocity if you are given distance, gravity, and starting velocity.
v^{2} = v_{0}^{2} + 2a(x − x_{0})

What is the standard equation for oscillatory motion?
where a and b are constants determined by initial conditions.

How do you determine the natural frequency of a simple mass spring oscillation, ignoring damping?
 ω_{n }=√(k/m) = 2 * pi * f_{n} = (2*pi)/T_{n}
 ω_{n} is natural freq in rad/s
 f_{n} is nat frequency in cycles/s or hz
 T_{n} is period of natural oscillation in s

According to Hookes law, what is the force equation of a simple, linear, and massless spring?
 F=kx
 k is stiffness in N/m or pound/inch
 x is deflection distance

What is the standard differential equation of simple oscillatory motion in a simple springmass system?
X″ + ω_{n}^{2}x=0
Remember ω_{n}^{2}=k/m

In a simple pendulum, what is the restoring force when the pendulum is displaced by a small angle. (Hint, F_{net })
F_{net} = mg sin(Θ)

What is the differential equation for simple pendulum motion?
Θ"+(g/L)Θ=0
 L is pendulum length.
 This is for small angles since sin(Θ)≈Θ at small theta.

What is the standard equation for simple pendulum oscillation?
Θ=Θ_{max} sin{√(g/L)t}

What is the moment of inertia about the geometric axis of a disc or cylinder?
I=1/2 MR^{2}

What is the Inertia of a body from on axis to another (Parallel axis theorem)
I=I_{cm} + md^{2
Icm is the inertia about the center massd is the distance between parallel axes.
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What is the moment of inertia of a hoop about it's axis of symmetry?
I=MR^{2}

What is the Moment of inertia of a solid sphere?
I=2/5 MR^{2}

What is the moment of inertia of a rod about it's center
I=1/12ML^{2}

What is the equation of motion for a simple mass spring system if the motion starts with zero initial velocity, but with an initial displacement?
x(t)=x_{0} cos(ω_{n}t)

What is the equation of motion if initial displacement is 0, but there is an initial velocity (like a hammer impact).
x(t)=(x_{0}'/ω_{n})*sin(ω_{n}t)

What is the differential equation of a shaft disk system?
Θ''+ω_{n}^{2} Θ=0 (1)
Derivation I*Θ'' = k_{t}Θ, so I*Θ''+k_{t}Θ = 0. This gives Θ''+(k_{t}/I)Θ. And since ω_{n}^{2}=k_{t}/I, we get the top equation (1).

What is k_{t} of a shaft disk system?
k _{t}=(G*pi*d ^{4})/(L*32)
 G= shaft modulus of elasticity
 d= shaft diameter
 L= shaft lenght
Bad mother.. shut yo mouth = Shaft.

For a mass spring system what is the kinetic energy?
T=1/2m*x'^{2}^{}
(1/2mv^{2})

In a mass spring system what is the gravitational potential energy?
V_{g}=mg(Δ+x)

In a mass spring system what is the elastic potential energy?
V_{e}=1/2 k(Δ+x)^{2}

In a mass spring system, what is equation for conservation of energy?
U_{NonConservative}=ΔT + ΔV_{g} + ΔV_{e}

What is the law of conservation of linear momentum?
M_{1}V_{01}+M_{2}V_{02}=M_{1}V_{f1}+M_{2}V_{f2}

What is the effective mass of a spring?
 1/3 * m_{s}_{ }
 This is because we take the derivative of y^{2} to determine the KE of a spring.

What is the natural frequency considering the mass of the spring in a simple massspring system.
ω_{n}=√(k/[m+{m_{s}/3}])

What is the equation for springs in series?
1/k_{eq} = 1/k_{1} + 1/k_{2}

What is the total spring rate for spring in parallel?
k_{eq}=k_{1}+ k_{2}

What is the equation for simple harmonic motion?
x(t) = X sin(ω_{n}t)

The amplitude of the oscillation of a simple massspring system depends on what?
The initial conditions (velocity and displacement)

The period of natural frequency of a simple springmass system depends on what?
The parameters of the sytsem (specifically, the mass and spring rate)

What is equation to find the roots of ax^{2}+bx+c=0?

What is the solution to the IDE d^{2}x/dt^{2}+ζωn dx/dt+ω_{n}^{2}=0
 x(t)=e2ω_{n}t [Ae^{ωn√(ζ^21)}+Be^{ωn√(ζ^21)}]^{}

What is Eulers identity?
e^{ix}=cos(x)+i*sin(x)

What is the critical damping ratio for a simple massspringdashpot system?
2*√(km)

What is the logarithmic decrement for vibration?
δ=ln(x1/x2)= (2Πζ)/√(1ζ^{2})

What is the damped natural frequency?
ω_{d}=ω_{n}*√(1ζ^{2})=2pi/τ_{d}

What is the damping ratio ζ?
ζ=c/c_{cr}

What is the moment of inertia for a rod, from the end of the rod?
1/3 mL^{2}

For a given force, how do you convert that to the moment about some point?
Multiply the force times the distance to the point.

How do you solve this mx''+cx'+kx=0?
 Assume the solution is in the form on e^{st}.
 Substitute for (ms^{2}+cs+k)e^{st}=0.
 Use quadratic to find roots of ms^{2}+cs+k=0.
 The solution is

How do you determine c_{cr}_{} from the equation of motion?
Take equation of motion, and substitute e^{st}, this will replace x'' with s^{2}. Take this equation, use the quadratic to find the roots. Set the portion under the square root to 0. Solve for c, this is c_{cr}.

What is the general steps to solve a coulomb damping problem?
 Solve just like an oscillation problem except you have to consider when X' is + or .
 Solve for one half of motion, then you need to determine the initial conditions for the second half of the motion.
 The second half starts when ω_{n}=pi

What is the general equation for sinusoidal excitation?
 It is a second order ordinary differential nonhomogeneous equation in the form:
 x''+cx'+kx=F_{0}*sin(ωt)

Find zeta from logarithmic decrement.
ζ=δ/√((2pi)^{2}+δ^{2})

