IB mechanics ALL

When can you use the perpendicular axis theorem Izz = Ixx+Iyy

For lamina only

What is the radius of gyration

The radius of a hoop that has the same moment of inertia as the lamina

Should you use parallel axis theorem when finding kinetic energy

No, always use velocity of G and rotation around G for kinetic energy

What is the hard GPE equation

V = - GMm/r

How do you do dAlambert

Put a force ma and a torque Jw on your free body diagram (in the opposite direction to the accelerations) and then do the force balance as if it were at rest

What are the equations for linear and angular momentum

P=mv and h= Iwk (moment of inertia x ang velocity x vector by right hand screw rule)

How does an impulse affect a body

Impulse is change in linear momentum, moment of impulse is change in angular momentum

What is a useful result for yddot

yddot = d/dy( ½ ydot²)

In polar coords what 3 dalamberts do you add

The two perpendicular accelerations from the databook and a I x wdot

How do you draw a velocity diagram

Do instantaneous centres to find unknown angular velocities and therefore velocities. Corners are points, lines are relative velocities. Use lower case letters and don’t put arrowheads

What is the velocity image theorem

Points fixed distances apart will have the same ratio of distances on the velocity diagram. On a velocity diagram, triangle abc is similar to ABC but rotated 90 in the direction of w

What is the virtual power method

Work done by all external forces and torques must add to zero. F dot v = Tw

What do you do about sliders and pivots in virtual work

Slider or pivot friction, use relative velocity.

How does acceleration image vary from velocity image

It is NOT rotated 90. May be rotated any amount

How do you find the torque needed to resist inertia forces

Apply dalambert forces and couples like external forces and couples

How does force on a beam create shear

Force per unit length = dS/dx

How are shear and moment related

S = dM/dx

How do you find shears and moments due to inertia

Chop end of beam off, do FBD with dalambert, remember to do I about new G, balance forces

What is the formula for the gyroscopic couple

Q = Jw(Omega) = polar moment of inertia x ang v x ( ang v because of spinning) Capital Omega is the “gyroscopic precession”. Q is the rate of change of angular momentum

How to find the direction of gyroscopic precession

Q is in the Omega x omega (slow x fast) direction

What is the Lagrangian

Kinetic energy T - potential energy V

What does it mean for a system to be holonomic

The state can be fully described by displacements and velocities

How do you get the equation of motion from a lagrangian

P= dL/dqdot, F= dL/dq, F=dP/dt

How many equations of motion can you make

One per degree of freedom

How do you find the combined Lagrangian of independent systems

Just add the sub lagrangians together

What do we get by integrating the equation of motion

A statement of energy conservation

What does a phase portrait plot

qdot against q

How do you draw a phase portrait

Do dots at stable equilibria and crosses at unstable equilibria and then draw weird eye shapes (lines do not cross)

What does a phase portrait do

Summarises all possible motions

What is the effective potential strategy

Get to the EoM as usual. Exploit conservation laws to get rid of all but one coordinate. Rewrite EoM as

m qddot = -V’(q) then integrate to get V(q)

When does central potential arise naturally

When the only force is towards the origin

For a satellite, what EoM substitution would let you use effective potential

thetadot = h/mr² (if h is conserved)

How can you tell if a system is stable using Veff

By finding the second derivative. If positive, it is a minimum and therefore stable

How do you find M and K matrices

From the lagrangian, look at ½ ___v² bits. They go on the diagonal. For q1q2 things, you need to halve them. These matrices are symmetrical

How do you find eigenvalues and eigenvectors

Eigenvalues- det(A-LI) =0 and solve for L.

Eigenvectors- (A-LI)X =0 and solve. Remember magnitude not relevant to set one to 1

What is important when doing small amplitude vibrations from the Lagrangian

Keep cos = 1 - theta² / 2 because you lose a power when you differentiate

How do you get frequencies from your matrix equations with M and K

(-omega² M + K)A =0 so det(-omega² M + K)=0

What does it mean if M is a multiple of the identity matrix

The mode shapes are orthogonal, so each mode is independent

What does it mean if an equilibrium point is a maximum or a saddle

The frequency w will be imaginary. This means the solution is exponentially growing or decaying

What is the error when approximating an integral as rectangles (all under curve) (left or right reimann sum)

Proportional to width

What is the error when approximating an integral as rectangles with the centre of the middle one on the curve (mid point Reimann sum)

Proportional to width squared

What is the error for computing an integral with quadratic approximation between points (simpsons rule)

Proportional to width ^4

If a integration method has error order t, how do you reduce error by a factor of 100

Reduce time interval by a factor of 100^(1/t)

What is the only numerical way you are expected to calculate an integral

Forward Euler, using left rectangle method

What is a Poincaré section

A higher dimensional phase portrait

How do you get a poincare section

By samping q and qdot and plotting points

What is THE most important thing to remember about time varying systems

Energy is not conserved

How do you make a time varying Lagrangian if there is an external force

Add a force x displacement term on the end

What is the EoM for harmonic excitation and how is it solved

M qddot + K q = f cos(wt)

Made complex- get to (-w²M + K)Q =F