Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

University > Principles of Dynamics > Flashcards

Principles of Dynamics Flashcards

1
Q

What is index notation used for?

A

To express and manipulate multi-dimensional equations

Index notation simplifies mathematical expressions involving vectors and tensors.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Define a scalar.

A

A magnitude that does not change with a rotation of axes.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Define a vector.

A

Associates a scalar with a direction.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Define a tensor.

A

Associates a vector (or tensor) with a direction.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

What does the unit vector in index notation represent?

A

The unit vector along the th direction in Cartesian coordinates.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is a free index in index notation?

A

Labels the component/element of the equation and must appear on both sides of the equation.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

What is a dummy index in index notation?

A

Summed over, usually from 1 to 3, and does not have to appear on both sides of the equation.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What is the Kronecker delta?

A

A rank 2 tensor defined as δ_ij, which is the identity matrix in matrix form.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

What does the Levi-Civita tensor represent?

A

A rank 3 tensor that is equal to 1 for even permutations and -1 for odd permutations.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

How is the dot product expressed in index notation?

A

a_i * b_i = a · b.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

How is the cross product expressed in index notation?

A

a × b = ε_ijk a_j b_k e_i.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

What is a symmetric tensor?

A

A tensor that is equal to its transpose.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

What is an anti-symmetric tensor?

A

A tensor that is equal to the negative of its transpose.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

What are the three laws of Newton?

A
  • N1: An object will stay at rest or in constant motion unless an external force is applied.
  • N2: Force is directly proportional to the rate of change of momentum.
  • N3: Every action has an equal and opposite reaction.
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

What are Kepler’s three laws?

A
  • K1: The motion of a planet is an ellipse with the sun at the focus.
  • K2: Each planet sweeps out equal areas in equal time intervals.
  • K3: The square of the time period is proportional to the semi-major axis cubed.
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

What does the conservation of energy derive from?

A

Newton’s Second Law (N2L) and dotting both sides with the velocity vector.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

What does the conservation of angular momentum derive from?

A

Crossing both sides of Newton’s Second Law (N2L) with the position vector.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

What is the Lagrangian?

A

A function that describes the dynamics of a system, depending on time, position, and velocity.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

What is Hamilton’s Principle?

A

The correct path of motion corresponds to a stationary path of the action.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

What is the two-body problem in dynamics?

A

A system where two bodies interact through gravitational force, with the center of mass as an important coordinate.

21
Q

What is a rigid body?

A

A system of particles whose relative positions to one another are fixed.

22
Q

State the Perpendicular Axis Theorem.

A

The moment of inertia about an axis perpendicular to a lamina is the sum of the moments of inertia about two perpendicular axes in the plane of the lamina.

23
Q

What is the angular momentum of a particle?

A

L = r × p, where r is the position vector and p is the linear momentum.

24
Q

Define kinetic energy for a rigid body.

A

The sum of the kinetic energies of all constituent particles.

25
Q

What is Chasles’ Theorem?

A

The most general rigid body displacement can be produced by a translation along a line followed by a rotation about an axis parallel to that line.

26
Q

What is the axis of the moment of inertia through?

A

The pivot and perpendicular to the plane of the lamina.

27
Q

What does the Euler-Lagrange Equation describe?

A

The motion of a dynamical system.

28
Q

Chasles’ theorem states that the most general rigid body displacement can be produced by what?

A

A translation along a line followed by a rotation about an axis parallel to that line.

29
Q

How can the kinetic energy of a rigid body be split up?

A

Into the KE of the centre of mass and KE of the rotations relative to the centre of mass.

30
Q

What are holonomic constraints?

A

Relations between position variables that can be expressed as equations.

31
Q

What is an example of a holonomic constraint in a simple pendulum?

A

A fixed string length introduces a tension in the rope as a constraint force.

32
Q

How are holonomic constraints introduced into the Lagrangian?

A

By introducing new variables called Lagrange Multipliers.

33
Q

What does the LHS of the full Euler-Lagrange equation describe?

A

The motion of the unconstrained system.

34
Q

What can be done in a constrained system regarding the coordinate system?

A

A coordinate system can be chosen to avoid constraints in the Lagrangian.

35
Q

What is Hamilton’s Principle?

A

The path of least action is found by solving the Euler Lagrange equations.

36
Q

Name two types of conserved quantities.

A
  • Conserved momentum
  • Conserved energy
37
Q

What is defined as an ignorable coordinate?

A

A coordinate for which the Lagrangian is independent.

38
Q

What does conserved momentum depend on?

A

Whether the Lagrangian is independent of a particular coordinate.

39
Q

What is required for conserved energy in terms of the Lagrangian?

A

The Lagrangian must not explicitly depend on time.

40
Q

What is the Lorentz Force related to?

A

The motion of a charged particle in an electromagnetic field.

41
Q

What is the Hamiltonian function?

A

A function that describes the total energy of a system.

42
Q

How do Hamilton’s equations differ from Euler-Lagrange equations?

A

Hamilton’s equations form a set of 2n 1st order differential equations.

43
Q

Under what condition is energy conserved in Hamiltonian systems?

A

When the kinetic energy is explicitly independent of time.

44
Q

What does a spinning top exhibit in terms of motion?

A

Rotating about two axes.

45
Q

What is the potential energy of the spinning top given by?

A

A specific function of its parameters.

46
Q

What is the significance of ignorable coordinates in the Lagrangian of a spinning top?

A

They indicate that the Lagrangian is independent of them.

47
Q

What does it mean for a system to be reduced to 1 degree of freedom?

A

The Hamiltonian is independent of certain ignorable coordinates.

48
Q

What is the equation for the motion of a system derived from Hamilton’s equations?

A

A differential equation that describes the dynamics.

University (7 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2025 Bold Learning Solutions. Terms and Conditions