1P1 Mechanics

Question 1

What are the unit vectors for cartesian, polar and intrinsic coordinates?

Answer 1

Cartesian: e_i, e_j
Polar: e_r, e_theta
Intrinsic: e_t, e_s

Question 2

Differentiate unit vectors in polar coordinates.

Answer 2

e_rdot = e_theta x thetadot
e_thetadot = - e_r x thetadot

Question 3

What is Euler’s method of numerical integration?

Answer 3

vn = v_n-1 + a_n-1 x t
xn = x_n-1 + v_n-1 x t

Question 4

What is the Euler-Cromer/Semi-implicit method of integration?

Answer 4

v_n = v_n-1 + a_n-1 x t
x_n = x_n-1 + v_n x t

Question 5

How do the Euler and semi-implicit method of integration differ in terms of energy?

Answer 5

Euler method tends to overestimate energy, growth in energy.
Whereas, the semi-implicit tends to underestimate energy.

Question 6

Integral form of the change in kinetic energy:

Answer 6

Tb-Ta = integral(F.dr)[a->b] = integral(F.v dt) [a->b]

Question 7

When is energy conservation true for the particles total energy?

Answer 7

When only conservative forces act on the particle

Question 8

What is the gain in total energy of a particle equal to?

Answer 8

The work done by non-conservative forces acting on it.

Question 9

How do you determine equilibrium positions from potential energy?

Answer 9

With conservative forces, the minima and maxima of the potential energy curves are equilibrium position.

Question 10

How is the rate of change of linear momentum determined?

Answer 10

The total external force applied

Question 11

How is the total change of momentum determined?

Answer 11

The total external impulse applied

Question 12

What is the vector form of the moment of a force?

Answer 12

q = r x F

Question 13

What is the moment of force about an axis?

Answer 13

Q = (r x F).n

Question 14

Vector form of angular momentum?

Answer 14

h = r x p = r x (mv)

Question 15

How is angular momentum and moment of force related

Answer 15

q = dh/dt

Question 16

When is angular momentum conserved around an axis?

Answer 16

When all forces applied are parallel to the axis, or pass through the axis

Question 17

When is a force conservative?

Answer 17

When the work done in moving a particle from A -> B is independent of the path taken and is fully reversible.

Question 18

How do you derive the eq of motion of a satellite?

Answer 18

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

Question 19

How to find a solution of the equation of motion of a satelite?

Answer 19

Define u = 1/r and substitute in.

Question 20

Velocity of point b on a rigid body in terms of velocity and position of point a.

Answer 20

vb =va + omega x r_b/a

Question 21

What is always true for two points on a rigid body.

Answer 21

v_b/a .e= 0 where a is in the direction r_b/a

Question 21

Equation for instantaeneous centres.

Answer 21

v = omega x instantaeneous radius

Question 22

Integral equation for centre of mass.

Answer 22

1/M integral(rdm) where r is a position vector

Question 23

Newton’s second law for rigid bodies.

Answer 23

F = M rddot_g

Question 24

Integral form of polar mass moment of inertia.

Answer 24

integral(r^2 dm)

Question 25

Perpendicular axis theorem and when is it valid.

Answer 25

For lamina it is true such that Izz = Ixx +Iyy

Question 26

Mass Moment of inertia about a certain axis.

Answer 26

Ixx = integral(y^2 + z^2 dm)

Question 27

Parallel axis theorem

Answer 27

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.

Question 28

Relationship between total moment of the forces and angular acceleration.

Answer 28

For a planar body: q = Ig * omegadot