Dynamics

Question 1

Define space for Newtonian dynamics

Answer 1

Space is Euclidean R^3

Question 2

Define a reference frame for Newtonian dynamics

Answer 2

A choice of origin, O, with orthogonal, right - handed 3D axes at O.

Question 3

Define a point particle

Answer 3

An idealised object modelled as being at position r(t) relative to a reference frame, where t is time.

Question 4

Define velocity

Answer 4

The first derivative of position

Question 5

Define speed

Answer 5

The magnitude of velocity

Question 6

Define acceleration

Answer 6

The second derivative of position

Question 7

Define linear momentum

Answer 7

p = mv

Question 8

Give Newton’s First Law

Answer 8

In an inertial reference frame, a point particle moves with constant momentum, unless acted on by a non-zero total external force.

Question 9

Give Newton’s Second Law

Answer 9

In an inertial reference frame, the dynamics of a point particle satisfies F = dp/dt, where F is the total external force.

Question 10

Give Newton’s Third Law

Answer 10

If particle A exerts a force F on particle B, then particle B exerts the force -F on particle A.

Question 11

Define the Galilean group

Answer 11

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

Question 12

Give the formula for the Gravitational force near the Earth’s surface

Answer 12

F = mg towards the Earth

Question 13

Give the formula for the Gravitational force between two celestial objects at large distance.

Answer 13

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

Question 14

Give the formula for fluid drag in a viscous fluid

Answer 14

F = -6πµRv, where µ is the viscosity, R is the radius of the sphere and v is the velocity.

Question 15

Give the formula for fluid drag on an aerofoil in air

Answer 15

F = - D|v|v, where D is the drag coefficient and v is the velocity.

Question 16

Give the formula for the spring force

Answer 16

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.

Question 17

Give the fomula for the force on a charged particle in electric and magnetic fields.

Answer 17

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.

Question 18

Define the kinetic energy of a point particle

Answer 18

T = 1/2.m.(dr/dt)^2

Question 19

Define the potential energy of a point particle

Answer 19

V(x) = - the integral of F(s)ds, from an arbtritary point to the particle’s position.

Question 20

Define an equilibrium solution of Newton II

Answer 20

A position x = x_e for all time, so F(x_e) = 0 for all t.

Question 21

Define an equilibrium point

Answer 21

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.

Question 22

Define a stable equilibrium

Answer 22

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.

Question 23

Define the work done by a force

Answer 23

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.

Question 24

Define a conservative force.

Answer 24

A force F is conservative if there exists V such that F = −∇V

Question 25

Define a central force

Answer 25

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.

Question 26

Define angular momentum

Answer 26

The angular momentum of a particle about a point P is L_p := (r - x)∧m(dr/dt)

Question 27

Define torque

Answer 27

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.

Question 28

Give the inverse square law

Answer 28

V(r) = -κ/r and F = -κ/r^2(e_r)

Question 29

Give Newton’s Law of Universal Gravitation

Answer 29

F = -(G_N)(m_1)(m_2)/|(r_1)-(r_2)| in the direction between the particles.

Question 30

Give Kepler’s 1st Law

Answer 30

The path of a planet is an ellipse.

Question 31

Give Kepler’s 2nd Law

Answer 31

A straight line joining the Sun and a planet sweeps out equal areas in equal time.

Question 32

Give Kepler’s 3rd Law

Answer 32

(Period of an orbit)^2/(Semi major axis of the ellipse)^3 is constant for all planets.

Question 33

Define the centre of mass

Answer 33

The point with position vector 1/M times the sum of (m_i)(r_i).

Question 34

Define the total momentum

Answer 34

The total momentum of a system of particles is P = M(dR/dt), where R is the centre of mass.

Question 35

Define a closed system

Answer 35

A system where the net external force is 0.

Question 36

Define a centre of mass reference frame

Answer 36

A reference frame with the origin as the centre of mass.

Question 37

Define the total angular momentum about P.

Answer 37

Sum of (r-x)^m(dr/dt), where x is the position of P.

Question 38

Give the strong form of Netwon’s Third Law

Answer 38

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).

Question 39

Define the angular velocity of a reference frame.

Answer 39

If e_i(dot) = ω ^ e_i, then ω is the angular velocity of the reference frame.

Question 40

Define a rigid body

Answer 40

An object where the distribution of mass between any two points is fixed.

Question 41

Define the rest frame

Answer 41

The reference frame where r_I are at rest, so [d(r_I)/dt]S = 0.

Question 42

Define the inertia tensor

Answer 42

I_ij = Sum of: (m_I)[(r_I.r_I)(δ_ij) - (r_Ii)(r_Ij)]

Question 43

Define rotational kinetic energy

Answer 43

The rotational kinetic energy about G is (1/2)ω.(L_G)

Question 44

Define the moment of inertia

Answer 44

n^T.I.n, where I is the inertia tensor.

Question 45

Define the principle moments of inertia

Answer 45

The eigenvalues of the inertia tensor.