IA: 1P1: Mechanics Flashcards
What is kinematics?
Describing the motion of objects, without reference to force
What is dynamics?
Predicting the motion of objects with mass when forces are applied
How are vectors represented in equations as opposed to scalar quantities?
Either:
* In bold
* Underlined
What are newtons laws?
- When all external influences and forces on a particle are removed, the particle will have a constant velocity
- The sum of forces acting on a particle of mass m is equal to the mass times the instantaneous acceleration ΣF = ma
- When two particles interact they exert equal and opposite forces on each other along a line parallel to the straight line joining the two particles
What is the energy principle?
The increase in the total energy of a body is equal to the work done by external forces:
What is the linear momentum principle?
Include the associated equation
The rate of increase of linear momentum of a body is equal to the total external force applied:
What is the angular momentum principle?
Include the associated equation
The rate of increase of angular momentum of a body about a fixed point is equal to the total external moment of force about the same point:
What are the 2 equations for the gravitational force?
What is the equation for drag?
D = 0.5ρCAv²
ρ = fluid density
C = drag coefficient
A = area
What is the method of polygon of forces?
A graphical method of determining the resultant force acting on a body by stacking vectors end to end. If the resultant force on a body is zero the forces will form a closed polygon. If the forces do not make a closed polygon, then there will be a resultant force.
What does J represent?
linear impulse
What does I represent?
mass moment of inertia
The subscript indicates about which point or axis
What does q₀ represent?
the moment of force about a point
The subscript indicates about which point or axis
What does Q₀ represent?
moment of force about an axis
The subscript indicates about which point or axis
Note: Q is not bold and is therefore scalar
What does Q₀ represent?
moment of impulse about a point
The subscript indicates about which point or axis
Note: Q is bold
What does h₀ represent?
angular momentum about a point
The subscript indicates about which point or axis
What does H₀ represent?
angular momentum about an axis
The subscript indicates about which point or axis
Note: H is not bold and is therefore scalar
What does T represent?
kinetic energy
What is a frame of reference?
A frame of reference is a coordinate system or a set of axes relative to which the position, motion, and other physical properties of objects can be measured
What is an inertial frame of refence?
A frame of reference where Newton’s laws can be applied. This is when the reference frame itself has no acceleration (and hence no angular velocity) with respect to some imagined fixed coordinate system describing the universe
What are 4 different coordinate systems?
- Cartesian coordinates
- Cylindrical polar coordinates
- Spherical polar coordinates
- Intrinsic coordinates
What are the coordinates of the cartesian coordinates system?
(x, y, z)
What are the unit vectors of the cartesian coordinates system?
i, j, k OR eₓ, eᵧ, ez
What is the position vector in cartesian coordinates?
r = xi + yj + zk
What are coordinates?
not referring to intrinsic coordinates
3 independent scalar variables that uniquely describe position in 3D space
What are unit vectors?
not referring to intrinsic coordinates
Vectors that define 3 orthogonal directions according to the coordinate system being used. They have a magnitude of 1.
How do you determine in which direction the unit vector should point in a coordinate system?
Consider increasing each coordinate by a small amount and think about which way the point moves, each coordinate change defines each unit vector.
What is a position vector?
The vector from getting from the origin to the point with general coordinates, using only the coordinates and unit vectors
What are the coordinates of the cylindrical polar coordinate system?
ρ, θ, z
What are the unit vectors of the cylindrical polar coordinate system?
What is the position vector in cyclindrical polar coordinates?
eθ is not required for the position vector as the unit vectors are aligned such that eρ is always points radially towards the position
Why is the right hand rule important when considering the coordinate system?
The unit vectors are always orthogonal, therefore given one unit vector’s direction the right hand rule alows you to determine the other 2
Determine the position vector in cylindrical polar coordinates
Determine the position vector in cylindrical polar coordinates
What are the coordinates of the spherical polar coordinate system?
r, θ, ψ
Note: θ is defined differently in the cylindrical system
What are the unit vectors of the spherical polar coordinate system?
What is the position vector in spherical polar coordinates?
In spherical polar coordinates, neither eθ or eψ are required for the position vector as the unit vector eᵣ always points directly at the position
What are the coordinates in the intrinsic coordinates system?
(s, ψ)
This is fundamentally different to the other coordinate systems as you can only define a point uniquely in space when a path has also been defined:
* s is the distance along a pre-specified path
* ψ is the angle of the path with respect to some fixed reference
What are the unit vectors in the intrinsic coordinates system?
eₜ, eₙ
Again, these are defined differently for the intrinsic coordinate system:
* eₜ is the tangent direction to the path
* eₙ points towards the centre of curvature of the path
What is the position vector in intrinsic coordinates?
This is not usually defined neatly in intrinsic coordinates, rather it is usually used to describe velocity and acceleration vectors
It can be neatly defined in circular motion as eₙ will always point towards a fixed point
Why are the vectors for velocity and acceleration more complicated in polar coordinates?
The unit vectors depend on changing position, therefore when differentiating position with respect to time the unit vectors themselves must also be differentiated
For the position vector below, find the expression for velocity and acceleration
When differentiating a vector in the with respect to time what is important to remember about the coefficient of the unit vector?
Use r = R sinθ i as an example
If you are differentiating the vector with respect to time you must be careful if the coefficient of the unit vector contains another variable (often this is θ) you must use the chain rule. For example:
For the following position, determine the expression for the acceleration of the particle:
What happens when you differentiate a vector in polar coordinates with respect to time?
The unit vectors will rotate, they are always unit length but their direction can change. This is because the direction of the unit vectors depends on positon
For cylindrical polar coordinates, what is the derivative of eᵣ with respect to time?
For cylindrical polar coordinates, what is the derivative of eθ with respect to time?
Do not forget the minus sign!
How can angular velocity be represented?
If rotating in the 2D plane
- Magnitude: equal to the rotation rate
- Direction: parallel to the axis of rotation adopting the right hand grip rule
What is the general result for differentiating a unit vector in cylindrical polar coordinates?
VERY IMPORTANT
ω = ωk = (first derivative of θ) k
If a plane is 2D and is in a polar coordinate system (not specified as to spherical or cylindrical), which definition of θ do you use?
Cylindrical
Determine the expression for acceleration in a 2d plane in a polar coordinates system
Use r to represent the coefficient of the unit vector and assume it is a variable (i.e is a function of θ)
What is the expression for velocity in intrinsic coordinates?
This is essentially the starting point as there are only very specific circumstances where you can determine the position in intrinsic coordinates (usually circular motion)
What is the expression for acceleration in intrinsic coordinates?
For intrinsic coordinates, what is the derivative of eₜ with respect to time?
For intrinsic coordinates, what is the derivative of eₙ with respect to time?
What is the relationship between the 2 coordinates (s and ψ) in an intrinsic coordinate system when a short section of path is being approximated as a section of a circle with radius of curvature R?
s = Rψ
This is only true for a circular path where R is constant
How can you determine velocity and acceleration when you do not have an analytic expression for position, but rather measured data
They can be estimated by numerical differentiation
What are the 2 methods of numerical differentiation?
- Single-sided estimate
- Central difference
How can you use a single-sided estimate to perform numerical differentiation?
Using the difference in position between each step and the discrete time interval, you can estimate the velocity by dividing the difference in position by the time step. This makes each velocity estimate halfway between time-samples. You can then assign the velocities to the “left” or “right” samples
How can you use the central difference method to perform numerical differentiation?
To perform the central difference method, you must first perform a single sided estimate and then take the average between each of the values obtained in the single sided estimate. This can be done for all the “middle values” where there is a (k+1) and a (k-1) term, but at the ends you must use a single-sided estimate
What is the equation for numerical differentiatation using a single-sided estimate?
What is the equation for numerical differentiatation using a central difference method?
How can a numerical approximation for differentiation become more accurate?
Refer in terms of differentiation with respect to time
As the time step (Δt) decreases, the numerical approximation becomes more accurate
What is the equation for newtons second law?
What is the equation of motion of a system?
An equation that determines the motion of the particle. It defines the rules that the motion is governed by, but doesnt tell us what the actual motion is. It is derived through newton’s second law
How can you apply the “D’Alembert Forces” approach to mechanics problems?
Newtons second law can be rearranged to give: ΣFᵢ - ma = 0. Therefore we can think of ma as an “interial force” and so the sum of all forces on the body is zero and we can apply the principles of static equilibrium on the body. Therefore to apply this approach you just need to add the interial force onto the free body diagram, ensuring it acts in the opposite direction to the acceleration
What are the 2 approaches for integrating equations of motion?
- Displacement integrals
- Time integrals
Given the following equation for motion, find a general solution by integrating with respect to time
Given the following equation for motion, find the solution by integrating with respect to displacement
Use final velocity = 0
What happens when you apply a time or displacement integral to newtons second law?
- A time integral results in a momentum perspective
- A displacement integral results in an energy perspective
What happens when an equation of motion is too complex and cannot be solved analytically?
You can solve them using numerical integration
What are the 2 main steps of numerical integration for systems containing particles or rigid bodies?
- Approximate the derivatives in the equations of motion using discrete time steps
- Solve the discrete equation at every time step to calculate the answer
What are the 2 methods for performing numerical integration?
- Euler’s method
- Euler-Cromer / semi-implicit method
How does the Euler method work?
The Euler method is based on the Taylor series expansion of v(t₀ + δt):
READ FULLY
How does the Euler-Cromer / semi-implicit method work?
READ FULLY: ESPECIALLY THE DISPLACEMENT STEP
What is the difference between the Euler method and the Euler-cromer / semi-implicit method?
- Euler Method: Updates both position and velocity using the same time step.
- Euler-Cromer / semi-implicit Method: Updates velocity first, then position using the updated velocity, which can provide better stability for certain physical systems.
What type of numerical integration methods are the Euler Method and the semi-implicit method?
First order - they only use the first term in the Taylor Series expansion
How can you check if the numerical integration method is producing an accurate answer?
- Check a case / aspects of the simulation where you have a known answer
- Check convergence: make Δt smaller until the answer stops changing
What is the identity for:
What is the energy principle?
Over a given interval, the increase in kinetic energy is equal to the total work done by the applied forces
When is the total work done between positions A and B the same regardless of the route?
When the force field is conservative
What is the principle of conservation of energy?
when a particle moves in a conservative force field, the sum of the kinetic and potential energy remains constant:
T + V = E
For this scalar equation, what would the vector form be?
Unit vector would be eᵣ
What is the general 3D vector equation for a conservative force field and the potential energy within it?
Derive the equation for the gravitational potential energy of a particle starting with the equation for the gravitational force:
What are 3 examples of where conservative forces arise?
- Gravitational fields
- Electrostatic fields
- Elastic forces in springs (without hysteresis)
What are examples where forces are NOT conservative?
Most cases depend on motion or velocity, and so the forces are not independent of the path taken:
* Springs with hysteresis
* Drag forces / air resistance
How is this equation relevant if there are both conservative and non-conservative forces acting on a particle?
You can split the forces into conservative (Fc) and non-conservative (Fₙ) components:
How can you determine if a particle in a conservative system is in equilibria?
Consider the potential energy as a function of displacement: V(x)
A particle is in equilibrium when:
dV/dx = 0
V’ = 0
What are the 2 types of equilibrium?
- Stable equilibrium: When slightly displaced from its equilibrium position, it experiences a force that returns it to the equilibrium position
- Unstable equilibrium: When slightly displaced form its equilibrium position, it experiences a force moves it further away from the equilibrium position
How can you determine if a particle in a conservative system is in stable equilibria?
Consider the potential energy as a function of displacement: V(x)
A particle is in stable equilibrium when V(x) is at a minimum:
V’’ >0
d²V/dx² > 0
How can you determine if a particle in a conservative system is in unstable equilibria?
Consider the potential energy as a function of displacement: V(x)
A particle is in unstable equilibrium when V(x) is at a maximum:
V’’ < 0
d²V/dx² < 0
How can Newton’s second law be written in terms of linear momentum?
What is the linear momentum principle?
The rate of increase of linear momentum is equal to the total external force applied. Equivalently: the net increase of linear momentum is equal to the total external impulse applied.
What can be said for a perfectly elastic collision?
since kinetic energy is conserved:
What is the coefficient of restitution, e?
The ratio of relative velocities before and after an impact:
For a body expelling a stream of particles at a speed u, and a mass dm within a short time interval dt, what is the equation for the force acting on these particles?
What is the force acting on the body expelling the particles?
For a body expelling or collecting mass with a constant relative stream velocity what is the general equation for the force acting on the body?
- m dot represents the rate of increase in mass
- Vp/b is the non-zero velocity of the particles relative to the body
A chain of total length L and mass m is lying on a table before being pulled up vertically at a speed of v by a force P. Determine an expression for how the force P varies with the height above the surface x
What is the rocket equation?
u = constant relative velocity between gas stream and rocket
m₀ = initial mass
Derive the rocket equation:
u = constant relative velocity between gas stream and rocket
m₀ = initial mass
m dot = dm/dt = rate of increase in mass
What is the equation for the moment of a force about a point, q₀?
You can use the right hand grip rule to determine the direction of the moment from the resulting vector
What is the expanded form of this vector equation?
What is torque?
The moment of a force
Where does the moment vector point?
parallel to the axis of the moment or torque applied by F, therefore the right hand grip rule can be applied
What is the equation for the moment of a force about an axis (Q₀) in the direction with the unit vector n?
Explain what this equation actually does:
It is the moment of F about an axis. This is the shortest distance d from the axis to the particle multiplied by the component of the force that is perpendicular to both the axis and the shortest-distance line.
Determine the moment of force about the centre of the loop:
What is the moment of momentum?
Angular momentum
What is the angular momentum of a particle about a point O defined as?
What is the rate of increase of angular momentum of a particle P about a point O?
It is equal to the moment of the force about O
What is the equation for the angular momentum of a particle about an axis in the direction with the unit vector n?
What happens if you differentiate H₀ with respect to time?
When is angular momentum conserved about a point?
When there is no external moment:
* If the moment of force about that point is zero
* If the line of action of the net force passes through the point O (and hence F is parallel to r)
When is angular momentum conserved about an axis?
- If the moment of force about that axis is zero
- If the line of action of the net force passes through (or is parallel to) the axis
Show that angular momentum is conserved in planetary motion and determine the expression for h₀:
just read through the workings rather than actually showing it!
Prove angular momentum about O is constant
For a satellite orbiting a body, what is the expression for angular velocity in terms of angular momentum?
What is the general equation of motion for a satellite orbiting a body?
How would you begin to solve the following equation?
Substitute u = 1/r:
This will then produce a linear second order differential equation which can be solved with ease
What is the equation for the shape of a satellites orbit about a body?
What is the significance of e in this equation?
The constant e determines the shape of the orbit:
* e = 0: Circle (constant radius)
* e < 1: Ellipse
* e = 1: Parabola
* e > 1: Hyperbola
Note: Only in the first 2 cases is the satellite in a periodic orbit
How can you analyse a satellites orbit when e = 0?
You do not need to reference the general solution as it corresponds to circular motion and so can be analysed as such
(However the general solution can still be used)
Label each component of this ellipse:
What is the equation for the radius of the perigee of the orbit?
What is the equation for the radius of the apogee of the orbit?
What is the equation for distance c?
c = ea
Why is the satellites angular momentum easiest to find at the perigee and apogee?
The velocity of the satellite at these positions is orthogonal to the position vector from the Earth
What is the ratio of the velocity at the perigee to the velocity at the apogee?
Vₚ/Vₐ = rₐ/rₚ = (1-e)/(1+e)
Determine an expression for “a” the equation from the shape of a satellites orbit about a body
Where is the radius measured from in an ellipse?
From one of the foci, NOT the centre
What is the expression for a?
What is the expression for b?
For a particle moving in a periodic ellipitcal orbit, what does it mean when at one specific point the angular momentum is given by mv₁r₁ and the kinetic energy is given by 0.5mv₁². Key point: v₁ is the same in both expressions
It means that the velocity at this point is orthogonal to the position vector from one of the foci. This is because T uses total velocity whereas angular velocity uses the velocity perpedicular to r only.
What is specific angular momentum?
angular momentum per unit mass
What does this mean?
The displacement of B from A
What is a rigid body?
An object with continuously ditributed mass that cannot deform: it can be thought of as a collection of particles that have a fixed spacing between them.
For a fixed body, what can be said about the components of Va and Vb along e?
They are equal, this is because by definition r is constant. Therefore in the rb/a direction, Va and Vb must be the same, but can be different in the perpendicular direction.
What can be said about relative velocities within a rigid body?
Relative velocities within a rigid body:
* Are orthogonal to their relative displacement
* It is due to rotation alone
Why are the general expressions for velocity and acceleration within a rigid body so much simpler than free particles?
r is fixed, therefore if r is differentiated it becomes zero
For the following position vector of B within a rigid body, determine the expression for velocity and accceleration
For a general 3D case, how many variables are required to define the motion of all the points?
Six variables as it has six “degrees of freedom”:
1. Vₓ
2. Vᵧ
3. Vz
4. ωₓ
5. ωᵧ
6. ωz
For a general 2D case, how many variables are required to define the motion of all the points?
Three variables as it has three “degrees of freedom”:
1. Vₓ
2. Vᵧ
3. ω
What is an instantaneous centre?
For 2D planar motion, there is a point somewhere in the plane of a lamina that has zero velocity. At a given instant, the lamina behaves as if it were rotating with angular velocity ω about the instantaneous centre
When can you find the instantaneous centre?
If you know either:
* The directions of velocities at two point of the lamina
* The direction and magnitude of velocity at one point on the lamina
How can you determine the position of the instantaneous centre if you know the direction of velocities at two points on the lamina?
Key equation: Vx = ωρx
Vx = Velocity of a point in the rigid body
ω = Angular velocity
ρx = Displacement of point from the instantaneous centre
How can you determine the position of the instantaneous centre if you know the firection and magnitude of velocity at one point on the lamina?
Key equation: Vx = ωρx
Vx = Velocity of a point in the rigid body
ω = Angular velocity
ρx = Displacement of point from the instantaneous centre
In general, is the instantaneous centre at the centre of curvature of the paths of the lamina?
NO! The instantanous centre moves from one moment to the next and so in general it is not at the centre of curvature.
What is the centre of mass/gravity of a rigid body?
It represents a balance point where if a rigid body were simply supported at this point, then it would not rotate.
What is the equation for the moment about G (centre of mass)?
This discete form can prove useful!
rG = position vector of centre of mass
ri = position vector of particle
Since a rigid body is a collection of infinitesimally small masses, what can this be simplified to?
rG = position vector of centre of mass
ri = position vector of particle
rG = position vector of centre of mass
ri = position vector of particle
M = total mass
How can you apply this formula if given a body of uniform density, ρ?
Use the substitution:
Determine the position of the centre of mass of this rigid body:
How can you obtain the combined centre of mass for a rigid body which is made of two components?
Obtain the centre of mass of the two components individually and then treat them as particles at their centre of masses. You can then determine the combined centre of mass:
Why is knowing the centre of mass of a rigid body so important?
Newtons second law can be used to describe the motion of the centre of mass of any rigid body
What does J stand for?
The mass moment of inertia / polar moment of inertia
What is the total moment (or torque) of a rotating rigid body about a fixed axis given by?
It is equal to the mass moment of inertia multiplied by the angular acceleration
What is the mass moment of inertia about the axis of rotation given by?
r = position vector relative to the fixed axis of rotation
What does the mass moment of inertia represent?
It represents the resistance of a rigid body to changing angular velocity. (it is analagous to mass in the equation F = ma)
What is the equation of motion of the pendulum of a clock, assuming that the pendulum is a rigid uniform bar of length L and mass m?
What is the mass moment of inertia about the x axis (Ixx) given by?
Note: when the system is a lamina it has zero depth and so the z² term can be neglected
What is the mass moment of inertia about the y axis (Iyy) given by?
Note: when the system is a lamina it has zero depth and so the z² term can be neglected
What is the mass moment of inertia about the z axis (Izz) given by?
What is the perpendicular axis theorem?
This is valid for a lamina only
When can the perpendicular axis theorem be used?
ONLY for lamina’s
What is the parallel axis theorem?
The moment of inertia about an axis passing through O, is equal to the moment of inertia about a parallel axis passing through G (centre of mass), plus a correction term: Mr².
What is the radius of gyration, k?
The radius of gyration describes how the mass of a rigid body is distributed relative to its axis of rotation
What is the mass moment of inertia (k) in terms of the radius of gyration?
I = mk²
The radius of gyration for common shapes can be found in the data book
How can you determine the mass moment of inertia for a composite body?
The moment of inertia about an axis if the sum of the moments of inertia of the individual components about the same axis (via the parallel axis theorem). Therefore, to calculate the moments of inertia of composite bodies you need to:
1. Find IG of each component
2. Use the parallel axis theorem to find Iₒ of each component
3. Sum the contributions from each component
What is the total moment of all the forces about the centre of mass for a planar rigid body (a lamina)
This is the same result for a general body that is constrained to rotate about a fixed axis, this result is for a lamina that can translate and rotate about its centre of gravity. An example of this could be a cylinder rolling along a flat surface
How can D’Alembert forces be applied to the dynamics of rigid bodies in planar motion?
You can use D’Alembert forces for the moments, it is important that these are applied with respect to the centre of mass
For a circle rolling along flat surface with no slip, what can be said about the linear and angular acceleration?
What is the reactio force at the pivot of a pendulum of a clock, assuming that the pendulum is a rigid uniform bar of length L and mass m?
D’Alembert approach:
Note: The D’Alembert forces and moments have been applied relative to the centre of mass, not P
What does “keep your I on G” mean?
When using a D’Alembert approach to solving a rotational planar motion of a rigid body question, you should apply the apply the mass moment of inertia about the axes centred on the centre of mass.
What is the angular momentum principle about a fixed point?
When does the angular momentum principle hold when O is a moving point rather than a fixed point?
If the angular momentum and moments are taken about the centre of mass “G”
What is the angular momentum principle about G?
What is the total kinetic energy of a rigid body that is body translating and rotating?
How can impulse be applied to angular momentum?
The moment of the impulse about the fixed point O or the moving centre of mass G is equal to the corresponding change in angular momentum
What assumptions are made during a collision event?
- The impulse is infinitesimally short with an infinite force
- The direction of J is the same as the force F and does not change during the impact
- During the impact the bodies do not change position significantly due to a short contact time
- All other non-impulsive forces are negligible (such as self weight)
What is the general method for a collision event question?
- Draw three diagrams for “before”, “during”, and “after” the impact:
* Each diagram should show the system in the same configuration
* The “before” and “after” diagrams should define the “before” and “after velocities”
* The “during” diagram is like a free body diagram, but showing impulses and ignoring non-impulsive forces - Equate linear momentum change to impulsive forces (as vectors or for each component)
- Equate angular momentum change to impulsive moments (planar motion so only one component)
For a collision question when can you use Io?
Only if the body is rotating about a fixed axis, if it is not rotating about a fixed axis you must use IG
What is the the equation for the angular momentum of a rigid about a fixed point O in terms of the angular momentum about the centre of mass?
Determins hc for each of these scenarios in terms of IG
What is the most important step of every single mechanics problem?
Define your coordinate system and the direction of the unit vectors!!!!
What is the equation for the angular momentum of a body in terms of angular velocity?
h = Iω
