1P1 Mechanics Flashcards
What are the unit vectors for cartesian, polar and intrinsic coordinates?
Cartesian: e_i, e_j
Polar: e_r, e_theta
Intrinsic: e_t, e_s
Differentiate unit vectors in polar coordinates.
e_rdot = e_theta x thetadot
e_thetadot = - e_r x thetadot
What is Euler’s method of numerical integration?
vn = v_n-1 + a_n-1 x t
xn = x_n-1 + v_n-1 x t
What is the Euler-Cromer/Semi-implicit method of integration?
v_n = v_n-1 + a_n-1 x t
x_n = x_n-1 + v_n x t
How do the Euler and semi-implicit method of integration differ in terms of energy?
Euler method tends to overestimate energy, growth in energy.
Whereas, the semi-implicit tends to underestimate energy.
Integral form of the change in kinetic energy:
Tb-Ta = integral(F.dr)[a->b] = integral(F.v dt) [a->b]
When is energy conservation true for the particles total energy?
When only conservative forces act on the particle
What is the gain in total energy of a particle equal to?
The work done by non-conservative forces acting on it.
How do you determine equilibrium positions from potential energy?
With conservative forces, the minima and maxima of the potential energy curves are equilibrium position.
How is the rate of change of linear momentum determined?
The total external force applied
How is the total change of momentum determined?
The total external impulse applied
What is the vector form of the moment of a force?
q = r x F
What is the moment of force about an axis?
Q = (r x F).n
Vector form of angular momentum?
h = r x p = r x (mv)
How is angular momentum and moment of force related
q = dh/dt
When is angular momentum conserved around an axis?
When all forces applied are parallel to the axis, or pass through the axis
When is a force conservative?
When the work done in moving a particle from A -> B is independent of the path taken and is fully reversible.
How do you derive the eq of motion of a satellite?
Using energy conservation
T+V = E where V is GPE, and T is kinetic energy v^2 = 0.5 m (rdot^2 + (r thetadot)^2)
And then differentiate since E is constant so goes to 0.
Using Newtons second law (slighlty easier)
Define specific angular moment h = ho/m = r^2 thetadot
How to find a solution of the equation of motion of a satelite?
Define u = 1/r and substitute in.
Velocity of point b on a rigid body in terms of velocity and position of point a.
vb =va + omega x r_b/a
What is always true for two points on a rigid body.
v_b/a .e= 0 where a is in the direction r_b/a
Equation for instantaeneous centres.
v = omega x instantaeneous radius
Integral equation for centre of mass.
1/M integral(rdm) where r is a position vector
Newton’s second law for rigid bodies.
F = M rddot_g
Integral form of polar mass moment of inertia.
integral(r^2 dm)
Perpendicular axis theorem and when is it valid.
For lamina it is true such that Izz = Ixx +Iyy
Mass Moment of inertia about a certain axis.
Ixx = integral(y^2 + z^2 dm)
Parallel axis theorem
Io = Ig + rg^2 * M.
The moment of inertia about an axis passing through O is equal to the mass moment of inertia about a parallel axis passing through the centre of gravity plus the correction term. Only works with the centre of mass.
Relationship between total moment of the forces and angular acceleration.
For a planar body: q = Ig * omegadot
