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Physics GRE - Classical Mechanics
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Gravity
Terms in this set (54)
Blocks on Ramps: Parallel Force
Fg = mgsin(θ)
Blocks on Ramps: Perpendicular Force
Fg = mgcos(θ)
Frictional Force
F = µmgcos(θ) = µN
Find coefficient of friction
µ = tan(θ)
Tension
T-mg=0
Work Done by a Force
W = ∫F * dr
Two Blocks on Top of Each other
µ = F / (M+m)g
Kinematics
Potential Energy and Force
F = -dU/dx
U = - ∫F*dr
Centripetal Acceleration
a = v^2/r
Centripetal Force
F = mv^2/r
Translational Kinetic Energy
1/2mv^2
Rotational Kinetic Energy
1/2 Iω^2
Gravitational Potential
mgh
Spring Potential Energy
1/2kx^2
Work Energy Theorem
Angular Momentum
L = r x p
Angular Moment of an Extended Body
L = Iω
Torque (Rotational)
τ = r x F = rFsin(θ)
Moment of Inertia
I =mr^2
Moment of Inertia for Extended Bodies
I = ∫r^2 dm
dm = pdV
Parallel Axis Theorem
I = I_CM + Mr^2
Center of Mass
r_CM = ∫r dm/M_tot = ∑r_i m_i/M_tot
Lagrangian
L = T-U
Euler-Lagrangian Equation
d/dt partial L/partial q_dot = partial L/ partial q
conserved if RHS=0
Hamiltonian
H = ∑p q_dot-L
Hamiltonian
H = T+U
Hamiltonian's Equations
p_dot = = partial H/ partial q
q_dot = partial H/ partial p
reduced mass
µ = m_1m_2/(m_1 + m_2)
Total Energy of Orbit
E = T+ V = 1/2mr_dot^2 + l^2/2mr^2 + U(r)
Classification of Orbits
E > 0: Hyperbolic Orbit
E = 0 : Parabolic Orbit
E < 0: Elliptical Orbit
Hooke's Law
F = -kx
Angular Frequency
Equation of Motion for an Oscillator with Damping
mx^** + bx^* + kx = 0
β = b/2m
Underdamped Solution
β^2 < ω_0^2
ω_1^2 = ω_0^2 - β^2
Overdamped Solution
β^2 > ω_0^2
Equation of Motion of Damped with External Driving Force
x^** + 2βx^* + ω_0^2x = A cos(ωt)
Resonant Frequency (Driven Oscillation)
ω_R^2 = ω_0^2 = 2β^2
Amplitude and ω
D = 1/(ω_0^2 - ω^2)
Pendulums: Equation of Motion
mLθ^** = -mgsin(θ)
Pendulum: Angular Frequency
ω = sqrt(mgR/I)
Effective Spring Constant in Parallel
k_tot = k_1 + k_2
Effective Spring Constant in Series
1/k_tot = 1/k_1 + 1/k_2
Bernoulli's Principle
v^2/2 + gz + P/p_ro
Fluid Conservation Equation
p_ro v_1 A_1 ∆t = p_ro v_2 A_2 ∆t
Buoyant Force
Period
T = 1/ƒ = 2πr/v
Escape Speed
v = sqrt(2GM/R)
Minimum Speed to Orbit Planet
v = sqrt(gr)
Velocity of an Orbiting Planet
v = sqrt(GM/r)
Elastic Collision
KE_1,i + KE_2,i = KE_1,f + KE_2,f
Inelastic Collision
m_1v_1,i + m_2v_2,i = (m_1 + m_2)v_f
2D Collision
x-component: m_1v_1i = m_1v_1f cos(θ_1) + m_2v_2f cos(θ_2)
y- component: m_2v_2i = m_1v_1f sin(θ_1) + m_2v_2f sin(θ_2)
rocket equation of motion for a constant fuel escape; speed u (relative to the rocket) and external force F(ext)
m*dv/dt+udm/dt=F(ext)
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