Dynamics Flashcards
Define space for Newtonian dynamics
Space is Euclidean R^3
Define a reference frame for Newtonian dynamics
A choice of origin, O, with orthogonal, right - handed 3D axes at O.
Define a point particle
An idealised object modelled as being at position r(t) relative to a reference frame, where t is time.
Define velocity
The first derivative of position
Define speed
The magnitude of velocity
Define acceleration
The second derivative of position
Define linear momentum
p = mv
Give Newton’s First Law
In an inertial reference frame, a point particle moves with constant momentum, unless acted on by a non-zero total external force.
Give Newton’s Second Law
In an inertial reference frame, the dynamics of a point particle satisfies F = dp/dt, where F is the total external force.
Give Newton’s Third Law
If particle A exerts a force F on particle B, then particle B exerts the force -F on particle A.
Define the Galilean group
The group of combinations of the following transformations on a reference frame:
Translations: r’ = r - x, where x is constant
Rotations: r’ = Rr, where R is a rotation matrix
Galilean boosts: r’ = r - ut, where u is a constant velocity
Give the formula for the Gravitational force near the Earth’s surface
F = mg towards the Earth
Give the formula for the Gravitational force between two celestial objects at large distance.
F = -G(m_1)(m_2)/(r^2) in the direction towards the other body, where m_1 and m_2 are the masses of the two objects
Give the formula for fluid drag in a viscous fluid
F = -6πµRv, where µ is the viscosity, R is the radius of the sphere and v is the velocity.
Give the formula for fluid drag on an aerofoil in air
F = - D|v|v, where D is the drag coefficient and v is the velocity.
Give the formula for the spring force
F = - k(x - L), where k is the spring constant, x is the length of the spring and L is the natural length of the spring, acting in the direction to return the spring to its natural length.
Give the fomula for the force on a charged particle in electric and magnetic fields.
F = qE + q(v^B), where q is charge, m is mass, v is velocity, E is the electric field and B is the magnetic field.
Define the kinetic energy of a point particle
T = 1/2.m.(dr/dt)^2
Define the potential energy of a point particle
V(x) = - the integral of F(s)ds, from an arbtritary point to the particle’s position.
Define an equilibrium solution of Newton II
A position x = x_e for all time, so F(x_e) = 0 for all t.
Define an equilibrium point
A position (x, y, z) = (x_e, y_e, z_e) for all time so F(x, y, z) = G(x_e, y_e, z_e) = 0 for all t.
Define a stable equilibrium
If m*(d^2ξ/dt^2) = ξF’(x_e), where ξ = x - x_e. The equilibrium at x_e is stable if the solutions to the differential equation are complex. Stable equilibriums are local minimums of the potential.
Define the work done by a force
The work done by a force F in moving a particle from r_1 to r_2 is the integral of F.dr from r_1 to r_2.
Define a conservative force.
A force F is conservative if there exists V such that F = −∇V
Define a central force
A force, F, that is proportional to r, where r is the position vector of a particle relative to the origin of an inertial reference frame.
Define angular momentum
The angular momentum of a particle about a point P is L_p := (r - x)∧m(dr/dt)
Define torque
The torque of a force F about a point P with position vector x is τ_p = (r-x)^F, where r is the position of the particle.
Give the inverse square law
V(r) = -κ/r and F = -κ/r^2(e_r)
Give Newton’s Law of Universal Gravitation
F = -(G_N)(m_1)(m_2)/|(r_1)-(r_2)| in the direction between the particles.
Give Kepler’s 1st Law
The path of a planet is an ellipse.
Give Kepler’s 2nd Law
A straight line joining the Sun and a planet sweeps out equal areas in equal time.
Give Kepler’s 3rd Law
(Period of an orbit)^2/(Semi major axis of the ellipse)^3 is constant for all planets.
Define the centre of mass
The point with position vector 1/M times the sum of (m_i)(r_i).
Define the total momentum
The total momentum of a system of particles is P = M(dR/dt), where R is the centre of mass.
Define a closed system
A system where the net external force is 0.
Define a centre of mass reference frame
A reference frame with the origin as the centre of mass.
Define the total angular momentum about P.
Sum of (r-x)^m(dr/dt), where x is the position of P.
Give the strong form of Netwon’s Third Law
If particle A exerts a force F on particle B, then particle B exerts the force -F on particle A. Additionally, this force acts along the vector (r_A - r_B).
Define the angular velocity of a reference frame.
If e_i(dot) = ω ^ e_i, then ω is the angular velocity of the reference frame.
Define a rigid body
An object where the distribution of mass between any two points is fixed.
Define the rest frame
The reference frame where r_I are at rest, so [d(r_I)/dt]S = 0.
Define the inertia tensor
I_ij = Sum of: (m_I)[(r_I.r_I)(δ_ij) - (r_Ii)(r_Ij)]
Define rotational kinetic energy
The rotational kinetic energy about G is (1/2)ω.(L_G)
Define the moment of inertia
n^T.I.n, where I is the inertia tensor.
Define the principle moments of inertia
The eigenvalues of the inertia tensor.
