4. Mechanics & Materials Flashcards
What is a vector?
A physical quantity that has a direction as well as a magnitude
Examples of vectors?
- field strength
- force
- momentum
- weight
- velocity
- acceleration
- displacement
What is a scalar?
A physical quantity that is not directional
Examples of scalars?
- density
- charge
- resistance
- work done
- energy
How can a vector be represented?
By an arrow
What does the length of an arrow representing a vector determine?
The magnitude of the vector quantity
How to resolve vectors into their horizontal and vertical components?
- use angle of vector (Θ)
- for sinΘ
- then for cosΘ
If two forces act on an object, when is equilibrium achieved?
When the two forces are equal and opposite
If three forces act on an object, when is equilibrium achieved?
When the resultant of any two of the forces is equal and opposite to the third force
What are the ways to work out forces on an object when it is in equilibrium?
- vector triangle
* resolve the components
What is the centre of gravity of an object?
The point through which the entire weight of the object may be considered to act
Word equation to calculate the moment of a force?
Force x Perpendicular distance from the line of action of the force to the pivot
What is the principle of moments?
For any object that is in equilibrium, the sum of the clockwise moments about a point is equal to the sum of the anticlockwise moments about that same point
What is a couple?
A pair of equal and opposite forces, not acting in the same straight line
What is the turning effect that couples create called?
A torque
How to calculate the torque of a couple?
torque = one of the forces x perpendicular distance between them
What are the two conditions for equilibrium?
- there is no net (resultant force)
2. there is no turning effect (moment) about any point
What is the centre of mass of a body?
The point on an object where the mass may thought to be concentrated
How to find COM of symmetrical objects?
Along the line of symmetry, or where multiple lines of symmetry intersect
How to find COM of non-symmetrical objects?
If an object swings freely, when it stops the COM is on a vertical line passing through the pivot
When an object is in equilibrium, what is true of the vertical forces?
They are equal in magnitude and opposite in direction
When an object is in equilibrium, what is true of the horizontal forces?
They are equal in magnitude and opposite in direction
When an object in equilibrium is supported at one point, what is the support force equal to?
It is equal and opposite to the total downward force acting on the object
How to calculate average speed?
distance / time
How to calculate acceleration?
change in velocity / time
How to calculate average velocity?
displacement / time
What does velocity measure?
The rate of change of displacement
What does acceleration measure?
The rate of change of velocity
What is the gradient in a distance-time graph?
Speed
What is the gradient in a displacement-time graph?
Velocity
What is the gradient in a velocity-time graph?
Acceleration
What is the gradient in an acceleration-time graph?
Rate of change of acceleration
What is the area underneath a velocity-time graph?
Displacement
What is the area underneath an acceleration-time graph?
Velocity
What do negative values of displacement or velocity on a graph indicate?
Motion in the opposite direction
On a velocity-time graph, what does a straight upwards line represent?
Constant acceleration
On a velocity-time graph, what does a straight horizontal line represent?
Constant velocity
On a velocity-time graph, what does a straight downwards line represent?
Constant deceleration
On an acceleration-time graph, what does a horizontal line represent?
An object travelling at constant acceleration - or no acceleration so constant velocity
On an acceleration-time graph, what does a straight line downards represent?
Acceleration is speeding up, but at a slowing rate
On an acceleration-time graph, what does a straight downwards line, below the zero line, represent?
Negative acceleration - the deceleration is slowing down, and area below zero line so negative change in velocity
What is the equation for an object’s speed, moving at constant speed on a circle?
v = 2πr / T
where T = time to move round once
What are suvat equations?
Equations describing motion
When can suvat equations be used?
When acceleration is constant
What do the letters in suvat stand for?
- s - displacement
- u - initial velocity
- v - final velocity
- a - acceleration
- t - time
How is the equation a=(v-u)/t derived using a graph?
- velocity-time graph
- gradient = acceleration
- increase in y = v-u
- over increase in x = t
How is the equation s=(u+v)t÷2 derived using a graph?
- velocity-time graph
- area underneath = displacement
- area of trapezium = 1/2(a+b)h
- s = 1/2(u+v)t = (u+v)t÷2
What is the suvat equation that doesn’t include s?
v=u+at
What is the suvat equation that doesn’t include u?
s=vt-1/2at²
What is the suvat equation that doesn’t include v?
s=ut+1/2at²
What is the suvat equation that doesn’t include a?
s=[(u+v)/2]t
What is the suvat equation that doesn’t include t?
v²=u²+2as
What experiment can be used to measure g?
- drop an object from a height - measure using meter rule
- use light gates to measure time taken for object to fall
- s=1/2at² ∴ s=1/2gt²
What is a projectile?
An object that is projected or thrown through the air at an angle
What can be done with a projectile’s initial velocity?
Can be separated into components
When calculating projectiles, what force is ignored?
Air resistance
Ignoring air resistance, what is the only force acting on a projectile?
Gravity
What does gravity acting objects mean for the components of projectiles?
- gives a downward acceleration - so affects vertical component of velocity
- horizontal component of velocity remains constant
Which component of acceleration of a projectile is affected by gravity?
The vertical component
Which value is the same for both components when calculating projectiles?
Time
What is a resultant force?
A single force that has the same effect as all the forces combined
What is Newton’s 1st law?
An object will remain at rest or continue to move with a constant velocity as long as the forces acting on it are balanced
When is an object’s momentum not constant?
When there is a resultant unbalanced force acting on it
What is inertia?
The resistance to change velocity
What does the inertia of an object depend on?
Its mass
How does a bigger mass affect inertia?
Bigger mass = bigger force to overcome its inertia and change its motion
When travelling in a car and decelerating, what keeps you moving?
Your inertia - until something stops you, hopefully your seatbelt
Word equation for momentum?
Mass x velocity
Symbol equation for momentum?
ρ = mv
What is the unit for momentum?
kgms⁻¹ or Ns
What do objects have when stationary, in terms of momentum and inertia?
They have no momentum, but still have inertia
Is inertia a scalar or a vector?
Scalar
Is momentum a scalar or a vector?
Vector
What is Newton’s 2nd law?
The rate of change of momentum of an object ∝ to resultant force acting on it. Change in momentum takes place in the direction of the force
If mass is constant, how can change in momentum be calculated?
Δρ = mv - mu
What can F ∝ Δρ/Δt be simplfied to? And under what condition?
F = m [(v-u)/Δt]
if the mass of the object doesn’t change
What is tension equal to (in terms of mass and force)?
T = ma + mg
What is force equal to (in terms of tension and weight)?
F = T - mg
What happens in terms of forces when two objects interact?
They exert equal and opposite forces on eachother
What is true about pairs of forces acting on different objects?
They are always the same type of force
What is Newton’s 3rd law?
If object A exerts a force on object B, then object B exerts an equal and opposite force of A
Do forces act in isolation?
No - they act in pairs
Why do pairs of forces not cancel each other out?
They act on different objects
How can Newton’s laws show conservation of momentum?
- 3rd says F₁=-F₂
- 2nd says F=ma so ΔPB/ΔTB = ΔPA/ΔTA
- ΔTB=ΔTA ∴ ΔPB=-ΔPA
- ∴ ΔPB+ΔPA = 0
What is the principle of conservation of momentum?
When bodies in a system interact, the total momentum remains constant - provided no external force acts on the system
What is the principle of conservation of momentum - in simple terms?
Total momentum before = total momentum after
provided no external forces act
What happens to momentum when objects collide?
Momentum is conserved
What is assumed when two objects collide and momentum is conserved?
There are no external forces acting
During a collision, will energy and momentum be conserved if there are no external forces acting?
- momentum will be conserved
* energy might be conserved
How is the kinetic energy before and after an elastic collision?
Ek before = Ek after
What are the types of collision?
- elastic
* inelastic
Are collisions between snooker balls elastic or inelastic?
Very nearly elastic
What is the difference between elastic and inelastic collisions?
- elastic - Ek before = Ek after
* inelastic - Ek before > Ek after
Are collisions between molecules in a gas elastic or inelastic?
Elastic
What would it mean if collisions between molecules in a gas were inelastic instead of elastic?
Repeated collisions would slow gas molecules down, and would eventually settle at the bottom of a container
Are most collisions elastic or inelastic?
Inelastic
What happens to the kinetic energy that is ‘lost’ during an inelastic collision?
It is transferred to other forms
What is the ‘lost’ energy usually converted to during inelastic collisions?
Internal (heat) energy
What type of collision is undergone in crash barriers and crumple zones of cars?
Inelastic
Why are crash barriers and crumple zones designed to collide inelastically?
To absorb the kinetic energy in a crash
Which principle can be applied to explosions?
The principle of conservation of momentum
What can the principle of conservation of momentum be used for?
- collisions
* explosions
In exam questions, what is the movement of all objects involved in explosions?
Initially, they are all stationary
What is the initial momentum of explosions when the objects are both stationary?
Total initial momentum is zero
What is impulse equal to?
The change in momentum
What is the equation to show impulse?
Δp = FΔt
What is the advantage of ‘following through’ in sports?
- keeps force acting on ball for longer
* refer to F = Δp/Δt
What is the advantage of drawing your hands back when catching a ball?
- reduces sting
- as Δp happens over a longer time
- reducing force on hands
What happens to Δp when there is a greater force on an object, acting for longer?
Greater change in the object’s momentum
What is the area underneath a force-time graph?
Impulse
What is the area underneath a force-time graph NOT referred to as?
Change in momentum
What is the area underneath a force-time graph?
Impulse (kgms⁻¹)
What is the equation for work done that includes displacement?
W = Fx
What does the x stand for in the equation W=Fx?
Displacement
For equation W=Fx, what must be done if force and displacement are not in the same direction?
The force needs to be resolved to find the component acting in the direction of the displacement
What can work done also be referred to as?
Energy transferred
What is the definition of 1J
The work done when a force of 1N moves through a distance of 1m (in the direction of the positive force)
What is power?
The rate at which work is done
What is the symbol equation for power that involves work done?
P = ΔW / Δt
What is 1W in derived units?
1 Js-1
What equation can be used to calculate power that involves velocity?
P = Fv
What equation for power might not be on the data sheet?
P = Fv
What is true for the velocity in P=Fv?
It is constant
What are the types of energy?
- heat
- light
- chemical
- sound
- electrical
- nuclear
- elastic potential
- kinetic
- gravitational potential
How is the equation for gravitational potential energy derived?
- W = Fx
- = Weight x Δh
- Work done = Ep gained
- so Ep = weight x change in height
- Ep = mgh
How is the equation for elastic potential energy derived?
- F = kx
- when a spring is stretched, work is done
- (area under f-x graph) = 1/2kx
- W = Fx
- W = 1/2 kx x x
- W = 1/2ke²
How is the equation for kinetic energy derived?
- an object gains Ek if a force does work on it
- W = Fx
- & F=ma → so W=mas
- suvat to find F → s=[(u+v)/2]/t & a=(v-u)/t
- subs a and s into W=mas to get W=1/2m(v²-u²)
- so Ek = 1/2mv²-1/2mu²
What is the principle of conservation of energy?
Energy can be transferred from one form to another, but it cannot be created or destroyed
What is true of the total energy when using the law of conservation of energy?
The total amount of energy always stays the same
Equation for efficiency?
Efficiency = useful energy output / total energy output
What is the unit for efficiency?
No unit
What does it mean when energy is ‘wasted’?
It is transferred to internal (heat) energy
What is density defined as?
A substance’s mass per unit volume
Equation for density?
ρ = m/v
Method for finding density of a regular solid?
- ruler - measure width, length and height
- top-pan balance - measure mass
- use ρ = m/v
Method for finding density of an irregular solid?
- top-pan balance - measure mass
- use a eureka can to measure water displaced by object
- volume of water = volume of object
- use ρ = m/v
What type of force is acting when a spring is squashed?
Compressive
What type of force is acting when a spring is lengthened?
Tensile
What is Hooke’s law?
The force needed to stretch a string is proportional to the extension of the spring from its natural length, provided the elastic limit isn’t exceeded
What is the equation for Hooke’s law?
F = kΔL
What does it mean for the stiffness of the spring when spring constant is larger?
It is a stiffer spring
What happens to a spring when it is stretched beyond its elastic limit?
It doesn’t regain its initial length when the force applied is removed
How can the spring constant of springs ‘in parallel’ be calculated?
Multiply the spring constants together of the original springs
How can the spring constant of two springs ‘in series’ be calculated?
1/kₜₒₜₐₗ = 1/kₑₐ𝒸ₕ x number of springs
What is the area underneath a force-extension graph equal to?
Strain energy (as work done = force x distance)
Equation for strain energy?
Eₑ = 1/2kΔL²
When does the equation Eₑ = 1/2kΔL² apply?
Only if the spring obeys Hooke’s law
In materials, what does the stiffness depend on?
Material, length and cross-sectional area
What is the equation for Young Modulus?
k = EA / l
What does the ‘E’ mean in k = EA / l?
Young Modulus
What is the definition of stress on a material?
The force acting per unit cross sectional area
Equation for stress?
Force / cross sectional area (σ = F/A)
What is stress measured in?
pascals (Pa)
What is the breaking stress of a material?
The maximum stress it can withstand without fracture
What can breaking stress also be referred to as?
Ultimate tensile stress
What happens when materials get a thinner section when they are stretched?
They break here as stress is increased here
What is strain defined as?
The extension produced per unit length
Equation for strain?
Extension / length (ε = x / l)
What is strain measured in?
Has no units
Equation for Young Modulus, E?
Tensile stress, σ / Tensile strain, ε
How can the two separate equations for stress and strain be simplified?
- σ=F/A divided by ε=x/l
- F/A x L/ΔL = FL/AΔL
- F/ΔL is spring constant
- so E = kL/A
On a stress-strain graph showing a stiff and a flexible material, which material has the line with the steepest gradient?
The stiff material
What are materials that permanently deform described as?
Plastic
What are materials that return to their original shape after the stretching force is removed called?
Elastic
What can plastic materials also be described as?
- ductile - can be drawn into wires
* malleable - they can be hammered into sheets
Describe the force-extension graph of a metal wire.
- loading - the line starts straight, and curves as it surpasses the limit of elasticity
- unloading - the line doesn’t come back along the same line as when loading
- difference between loading and unloading lines = permanent extension of wire
Describe the force-extension graph of a rubber band.
- loading - the line is curved
- unloading - the line is curved, but doesn’t follow the same curve as the loading line
- unloading line finishes at the origin - rubber returns to its original shape
What is the opposite of a tough material?
A brittle material
What happens when you try to deform a malleable material e.g. lead?
It deforms plastically - gives way gradually, absorbing a lot of energy before it snaps
Do brittle materials deform plastically?
No
Do brittle materials absorb much energy before they break?
No, unlike plastic materials
What are force-extension graphs used for, vs stress-strain graphs?
- force-extension → usually for objects e.g. particular spring
- stress-strain → usually for materials (of any size)
What does the gradient represent on force-extension and stress-strain graphs?
- force-extension → spring constant (Nm⁻¹)
* stress-strain → Young Modulus, E (Nm⁻² or Pa)
What is the Young Modulus measured in?
Nm⁻² or Pa
What does the area represent on force-extension and stress-strain graphs?
- force-extension → work done (1/2kΔL²) in J
* stress-strain → work done per unit length (W/V) in Jm⁻³
What is work done per unit length measured in?
Jm⁻³
On a force-extension graph, what does it mean if the area of the unloading graph is smaller than that of the loading graph?
Some energy has been transferred
What is the reason for energy transference on a force-extension graph?
Some energy stored in the object (e.g. rubber band) becomes the internal energy of the molecules when the rubber band unstretches
On a force-extension graph, what does the area between the loading and unloading curve represent?
Difference between energy stored in the object when it is stretched and the useful energy recovered from it when it is unstretched
Brief explanation of experiment to find the Young Modulus of a wire?
- stress → wire with mass attached - measure mass using top-pan balance and use W=mg. measure diameter of wire using micrometer, then calculate area
- then stress = F/A
- strain → measure extension by measuring distance marker moves from original position, and length of wire. calculate strain
- vary mass for range of values - plot stress-strain graph
How to improve accuracy in the experiment to calculate the Young Modulus of a wire?
- use long thin wire and heavier weights → greater Δl so smaller % uncertainty
- measure diameter accurately using micrometer
- measure wire by holding ruler as close to the wire as possible
In an experiment to calculate the Young Modulus of a wire, how can kinks in the wire be avoided?
Weights are added at the beginning, before length measured
In an experiment to calculate the Young Modulus of a wire, how can we make sure there is no thermal expansion?
By comparing the test wire to a control wire
What is the elastic limit?
The point beyond which a wire will not return to its original length after weight has been added and then remove
