Rigid Body Mechanics Flashcards
What is the prefix for each of the following:
- x10^12
- x10^9
- x10^6
- x10^3
- x10^2
- x10
x10^12 = T (tera) x10^9 = G (giga) x10^6 = M (mega) x10³ = k (kilo) x10² = h (hecto) x10 = da (deca)
What is the prefix for each of the following:
- x10^-1
- x10^-2
- x10^-3
- x10^-6
- x10^-9
- x1-^-12
x10⁻¹ = d (deci) x10^-2 = c (centi) x10^-3 = m (milli) x10^-6 = μ (micro) x10^-9 = n (nano) x1-^-12 = p (pico)
SI base unit for:
- length
- time
- weight
- angle
- temperature
- amount of substance
- electrical current
- luminosity
length - m time - s weight - kg angle - radian temperature - k amount of substance - mol electrical current - A luminosity - cd (luminous intensity)
7 scalar quantities?
distance speed angle rotation rate mass temperature energy
7 vector quantities?
displacement velocity angular displacement angular velocity force momentum acceleration
How to combine vectors by resolving?
Split into component parts of ‘up/down’ and ‘across’, then use pythagoras to work out the magnitude of the resultant vector, and trig to work out the angle
Why is a cartesian system said to be orthogonal?
The axes are all independent of one another - you can travel along one axis without changing position in the others
What is a plane?
A flat 2D surface, with zero thickness, encompassing 2 axes of a 3D co-ordinate system
3 rules about planes?
2 planes can be at right angles to each other
3 planes, but no more, can mutually be at right angles (e.g. corner of a cube)
A straight line is formed when 2 planes cross each other
What is the difference between cartesian and polar coordinates?
Cartesian uses 3 axes to describe the position of a point in 3D, so direction is not required as it is implied by the position already
(x, y, z)
Polar coordinates uses a distance and either 1 or 2 angles depending on whether it is describing a position in 2D or 3D
(r, θ, ϕ)
What is kinematics?
The study of motion, concerned only with the ways in which the object moves, not the cause of the movement (i.e. the force)
Explain degrees of freedom?
What is constrained movement?
A free object can move along 3 axes independently, called 3 translational degrees of freedom. It can also rotate around each of the 3 axes, termed 3 rotational degrees of freedom. Therefore, a free object is said to have 6 degrees of freedom as it can move in any way it chooses.
An object is said to have constrained movement if one or more of these degrees of freedom is restricted. For example, a ball on a pool table can rotate around any 3 axes, but can only move along 2 ( it can’t move up and down off the table), therefore it only has 2 translational degrees of freedom
How to work out average velocity?
How to work out instantaneous velocity?
Av vel = change in displacement/time
Instantaneous = gradient at that point in a displacement-time graph
How to work out average acceleration?
How to work out instantaneous acceleration?
Av acc = change in velocity /time
Instantaneous = gradient at that point in a velocity-time graph
What does the area under a velocity-time graph show?
Displacement
Symbol for:
Anglar acceleration?
Angular velocity?
Angular displacement?
α - angular acceleration
ω - angular velocity
θ - angular displacement
What is angular velocity?
Unit of angular velocity?
The angular displacement travelled by an object per second
Units = rad/s
How to work out average angular velocity?
How to work out instantaneous angular velocity?
Change in angular displacement/time
ω=Δθ/t
OR plot an angle-time graph and calculate the gradient
Instantaneous = plot an angle-time graph, work out the gradient of the tangent at that point
If plotting an angle-time graph, what value is given by:
- gradient?
- gradient of tangent to the curve
gradient = average angular velocity
gradient of tangent = instantaneous angular velocity
What is angular acceleration?
Units?
How to calculate average angular acceleration?
The rate of change of angular velocity
Its are rad/s²
Average = change in angular velocity/time α = Δω/t
Instantaneous = gradient of ‘angular velocity-time’ graph
What is statics?
Branch of mechanics dealing with forces and moments acting on bodies which are at rest, or moving with constant velocity - this may be constant angular and/or linear velocity
2 effects a force may have on a body?
A force can:
1 - alter the body’s state of motion
2 - deform the body’s shape
Simple device to measure force and how it works?
A spring balance may be used to measure the force exerted by a body.
It consists of a loaded spring inside a holder, with a calibrated scale printed on it. A hook at one end of the spring enables the application of a force, and this stretches the spring proportional to the magnitude of the force. The spring balance’s scale shows the magnitude of force acting parallel to the spring. The scale is usually in Newtons (N)
What is mass?
What is weight?
Mass = quantity of matter of which a body is composed (kg)
Weight = force of gravity acting on a body (N)
What is density?
Formula?
Units
Density is the mass in a body per unit volume
ρ=m/V
kg/m³
What is gravity?
Formula for weight?
Gravity is the acceleration due to gravitational attraction between 2 bodies. The gravitational attraction increases as the mass of the bodies increases. This is what results in a body having weight.
W=mg
What is the centre of mass?
What is the centre of gravity?
Centre of mass is a point where all the mass of the body can be assumed to act. In a completely symmetrical structure with uniform density this will be in the geometric centre of the object. However, in a complex system such as the human body, each ‘segment’ has a centre of mass, so the whole body centre of mass moves accordingly in relation to the segments, meaning sometimes it lies outside of the body itself.
Centre of gravity is a point where all the weight in the object can be assumed to act, therefore only exists within a gravitational field. If it exists, it is always coincidental with the centre of mass.
What is friction?
Unit?
Friction is the force arising between 2 surfaces when they rub against each other. It tends to oppose motion (or impending motion) and acts at a tangent to the surfaces
Unit = N (since it is a force)
What is the maximum magnitude of friction force between two surfaces dependent on?
Dependent on the texture of the surfaces, lubrication, and the magnitude of force pushing the 2 surfaces together. More friction exists between dry, rough surfaces than wet, smooth ones, and more friction also exists if the magnitude of force pushing the surfaces together is greater
What is the coefficient of friction?
The maximum friction force that can exist between 2 surfaces - it is the ratio of the friction force to the force acting normally (perpendicular) to press the surfaces together
Formula for coefficient of friction?
Units?
μ = F/N
μ - coefficient of friction
F - friction force
N - force acting normally to the surfaces
No units, since it is a ratio
3 types of friction?
Static friction
Sliding friction
Rolling friction
What is static friction?
This only exists when motion is about to occur between 2 surfaces - the static friction present will be just sufficient to oppose the applied force that is trying to move the 2 surfaces over one another. Once the maximum static friction force is exceeded, motion will begin
What is the coefficient of static friction?
Why is it useful?
The ratio of the maximum static friction force to the force acting normally to the 2 surfaces.
If it is known, then it is possible to calculate the force needed to move the 2 surfaces over one another
What is sliding friction?
What is the measure of this?
Friction which exists only when sliding occurs between 2 surfaces.
The coefficient of sliding friction - as with static friction, the coefficient is the ratio of the friction force to the force acting normally to the surfaces
What is rolling friction?
Why does it occur?
Measure?
How does it compare to other types of friction?
How does lubrication affect rolling friction?
Friction which arises between and object and the surface over which it is rolling.
It arises because of the deformation of the 2 surfaces caused by the force acting normally to the 2 surfaces.
The measure is the coefficient of rolling friction
The friction force on a rolling object is much lower than that of a sliding one - hence the use of ball bearings in machinery
Use of lubrication does not lower the rolling friction, however it may reduce wear
If a person is trying to push a cupboard over a wooden floor, what formula could be used to calculate the maximum static friction force between the 2 surfaces (therefore the force that must be exceeded to move the object)?
Fmax = Wμ
Fmax = max possible static friction W = weight of the wardrobe μ = coefficient of static friction
What is pressure?
Unit?
Is it scalar or vector?
Formula?
The force exerted per unit area on a surface.
Unit = Pascal (N/m²)
Scalar
p=F/A
What is Newton’s 3rd law of motion?
The law of reaction - to every action there is an equal and opposite reaction
When can it be difficult to resolve forces graphically?
If one of the forces is much smaller than the others, or there is a particularly acute angle involved, then it can be very difficult to sketch an accurate diagram to scale and it is easier to resolve them mathematically
What is kinetics?
Study of how forces produce change in the state of motion of bodies. It is also called dynamics.
What is Newton’s 1st Law of Motion?
Law of Inertia - every body remains at rest or moving at constant velocity unless it is acted upon by a resultant force
What is Newton’s 2nd Law of Motion?
Law of Acceleration - the acceleration of a body is proportional to the applied force and inversely proportional to its mass
Which 2 laws govern motion due to forces?
Newton’s first and second laws
What is inertia?
The property of a body by which it continues to move/rest in its current state of motion unless acted upon by a force.
i.e. a body’s reluctance to accelerate - it is represented by its mass
Express Newton’s 2nd Law in an equation?
F=ma
What is dynamic equilibrium?
When the sum of all external forces acting on a body does not equal zero, and is equal to the resultant force
What is momentum?
Momentum is an expression of a body’s persistence to continue in its present state of motion. It incorporates a body’s resistance to change motion (inertial properties) and its velocity
Equation for momentum?
Units?
p=mv
kg m s⁻¹ (N s)
How is momentum incorporated into Newton’s 2nd law?
Express this as an equation
The law of acceleration, also called the law of momentum:
The rate of change of linear momentum is proportional to the applied force
F=(mv-mu)/t
(take out m as a common factor and you have (u-v)/t=a, hence F=ma
Express Newton’s 1st law in terms of momentum?
What is another name for this?
A body will continue to move with constant momentum unless an external force acts to change that momentum
This is the principle of conservation of momentum
What does the principle of conservation of momentum mean in real life application?
In a closed system, a number of bodies can interact by colliding and the total momentum in the system will remain constant
What is a moment?
A moment of a force is the tendency of a force to produce a rotation about an axis. It is also called torque.
Formula for a moment?
Units?
M=Fd
M = moment F = Force d= length of lever arm
Units = N m
How to determine if a moment is acting in a +ve or -ve direction?
Right hand grip rule - by ‘gripping’ the arrow shaft of the axis around which the moment is rotating, thumb pointing along the axis, extend the index finger and it will be pointing in the direction of the positive moment
What is the second condition of static equilibrium?
All sum of moments acting on a body must equal zero - if they are not then the body will angularly accelerate
Components of a lever system?
A Rigid Bar, which pivots around a Fulcrum, acted on by an Effort Force and a Resistance Force
What is mechanical advantage?
Formula?
Ratio of effort arm to resistance arm which is used to assess the performance of a lever system
MA = d(F)/d(R)
(mechanical advantage = distance of force / distance of resistance)
What does a high mechanical advantage suggest?
Low?
A high mechanical advantage means that the effort force can be lower than the resistance force and still work
A low mechanical advantage means that the effort force must be greater than the resistance force in order to work
Why are muscles in the human body often at a mechanical disadvantage?
Why is this the case?
The muscles insert close to the joint than the resistance force (the centre of mass for that limb segment), hence the force produced by the muscles is often much greater than the resistance just to move the limb.
There are 2 reasons for this - if a muscle inserted closer to a joint, then smaller movements of the muscles produce corresponding larger movements in the limb, and also the quicker the speed of movement. Therefore, there is a trade-off between strength and mobility
What is a first class lever system? 3 examples?
When the Fulcrum is located between the force and resistance
e.g. see-saw, scissors, crow bar
What is a second class lever system? 2 examples?
When the Resistance is located between the fulcrum and the force
e.g. wheel-barrow and nutcracker
What is a third class lever system? 3 examples? What isn't great about third class levers?
When the Effort is located between the fulcrum and resistance
e.g. fishing rod, tweezers, muscles e.g. biceps femurs
3rd class levers are always at a mechanical disadvantage, since the resistance will always be longer than the effort
Equation for tangential linear velocity of a rotating object?
v = rω
v = tangential linear velocity r = radius of circle ω = angular velocity
When considering angular acceleration, what 2 things must be considered?
Acceleration which acts to change the magnitude of velocity of the object
Acceleration which acts to change the direction of the object
Equation for tangential linear acceleration of a rotating object?
at = rα
at = tangential linear acceleration r = radius of circle α = angular acceleration
When is the tangential linear acceleration equal to zero?
During uniform angular motion
i.e. when the body rotates with a constant angular velocity
What is centripetal acceleration?
The same as radial acceleration - it accelerates a rotating body towards the centre of its rotation, maintaining it on a circular path
Equation for radial linear acceleration?
ar = v²/r = rω²
v = tangential linear velocity r = radius of circle ω = angular velocity
What is moment of inertia?
A body’s internal resistance to rotation
it is the sum of the products of the mass of each particle of the body and the square of its perpendicular distance from the axis of rotation
If there are two flywheels, one which is solid and one which is hollow, both with the same diameter and mass, rotating about their own central axis - which one will be harder to move and why?
The hollow one, since its mass is distributed further from its axis of rotation, so will have a great moment of inertia, making it more difficult to accelerate
Formula for angular torque?
M = Iα
Where I = mk² and k is radius of gyration
Formula for moment of inertia?
Units?
I = mk²
kg m²
What happens to the moment of inertia in a rigid and non-rigid body?
The moment of inertia (mk^2) remains constant in a rigid object
For a non-rigid object (e.g. jelly) its mass will move outwards from the axis of rotation when it is rotating (centrifugal), resulting in a change of its moment of inertia
What is the radius of gyration?
Relationship between radius of gyration and moment of inertia?
A particular radius from the axis of rotation where all the mass is concentrated
(similar to how mass is considered to be concentrated at the centre of mass of a body)
k = √ (I/m)
Formula for radius of gyration of a solid ring or spoked wheel? Solid disc (e.g. a flywheel)?
k = r
k² = 0.5(r²)
What is angular momentum?
An expression of a body’s persistence to maintain the same state of angular motion, incorporating its resistance to change its rotary motion (inertial properties) and its angular velocity
Formula for angular momentum?
Units?
L = Iω
L = angular momentum I = moment of inertia ω = angular velocity
Unit = kg m² rad s⁻¹
Does the principal of conservation of momentum apply to angular momentum?
Yes
What is anthropometry?
The study of human size and form
What are the 4 basic body segment parameters in anthropometry?
length
mass
centre of mass
radius of gyration
How can anthropometric predictions be made more accurate?
Use a more specific data set appropriate to the subject in question, based on their body dimensions, age, build, sex and racial origin
When are anthropometric data sets used?
Only to make predictions when it is not possible to make direct measurements
In anthropometry, what are body segment length given as?
A ratio of the entire height
e.g. foot length = 0.152H
In anthropometry, how is segment mass given?
How is centre of mass given?
Segment mass is given as a ratio of entire body mass e.g. forearm = 0.016M
Segment centre of mass is given as proximal and distal ratios of segment length e.g. forearm centre of mass = proximal 0.430segment length; distal 0.570segment length
(this means the centre of mass lies slightly proximal within the forearm)
Why does the centre of mass generally lie more proximally within segments?
In general, the diameter of each segment increases towards the proximal end
In anthropometry, why is the radius of gyration given around 3 different axes?
It is dependent on the distribution of mass relative to the axis of rotation, and will therefore be different for different axes of rotation
(it has values for centre of mass, proximal and distal)
What is Work?
Formula?
Unit?
The product of the applied force and the distance through which it moves
w=Fs
Unit = J
What is Power?
Formula?
Unit?
Power is the rate at which energy is expended, or the rate at which work is done
P=w/t
Unit = W
What is energy?
The property of a system that is the measure of its capacity to do work. It has several forms, e.g. kinetic and potential.
Formula for linear kinetic energy?
KE = 0.5mv²
Formula for rotary kinetic energy?
KE=0.5Iω²
What is the total kinetic energy in a system?
The sum of its linear (translational) and rotational kinetic energies
What is potential energy?
Formula?
The energy a body possesses by virtue of its position
PE=mgh OR PE=Wh
What is the law of conservation of energy?
How does this work for a falling object?
Energy can be redistributed or changed into another form, but cannot be created or destroyed.
In other words, the total energy in a closed system remains constant
Before an object falls from a height, it has no KE but it has PE. As it falls, it loses PE as its height decreases, but it gains KE as it accelerates towards the ground. Throughout this time, the whole energy stays the same as PE+KE=E
