6.3 + 6.4 - moments

how do lever / spanners work?

by using a force to turn an object about a pivot, making an object turn about its wheel axis

what does the effect of the force of the lever / spanner depend on?

how far the force is applied from the axle - the longer the spanner, the less force needed

the longer the spanner, the less or more force needed?

the longer the spanner, the less force needed

the shorter the spanner, the less or more force needed?

the more force needed

what happens if the spanner is too long and the nut (for instance) is too tight?

the spanner could snap if too much force is applied to it

when could a spanner snap?

if the spanner is too long and the nut (for example) is too tight

is a tight nut (lol) easier to loosen with a long or short spanner?

long

why is a long spanner better to loosen a tight nut?

because the effect of the force needed to loosen the nut depends on how far a force is applied to loosen it (the further it is, the easier it is to loosen)

what is the moment of a force about any point?

• moment of a force = the force x perpendicular distance from the line of action to the point

• (i.e.,), moment = force x distance

here

what is the perpendicular distance of a force?

here

what is the moment of a turning force?

here

what is the unit for a moment?

newton metre (Nm)

what is the symbol for a moment?

it doesn’t have one …

what makes the moment increase?

increased distance

does an increase of distance or force make the moment increase?

increase of distance

why does the moment increase with increase of distance?

because moment = force x distance

does moment increase with increased force?

no

why doesn’t moment increase with increased force?

because the force needed to rotate an object is a constant for that object

what is a body?

an object that is not a point object

when does a body turn?

when a force is applied to it anywhere other than through its centre of mass

will a body turn if a force is applied directly through its centre of mass?

no

will a body turn if a force is applied anywhere other than its centre of mass?

yes

why does a body turn when a force is applied anywhere but its centre of mass, and not when the force is applied directly through its centre of mass?

because the body pivots around its centre of mass

when is a body, with two or more forces acting on it, in equilibrium?

• when the turning effects of the forces balance out

• sum of clockwise moments = sum of anticlockwise moments (i.e., the principle of moments)

what is the principle of moments?

sum of clockwise moments = sum of anticlockwise moments

where do you take moments about?

the pivot

where do you take the moments about when there are several unknown forces?

take moments about point through which one of the unknown forces acts, since the moment will be zero here

when taking a moment about an unknown force, what is the moment of that force?

zero

when taking a moment about an unknown force, why is the moment of that force zero?

• moment = force x distance

• the distance from the pivot, when taking moments about the unknown force, is zero

• moment = force x (0)

• moment = 0

what’s the difference between the moment of a body and the moment of a point object?

here

where does a ruler balance when placed on the tip of your finger?

the centre of the ruler

why does a ruler balance when placed at its centre on the tip of your finger?

• the ruler’s centre of mass is at its centre

• therefore the point of support force (from your finger) is directly under the centre of mass

• there is no turning force when a force is applied through the centre of mass

• therefore the ruler does not fall

why does a ruler tip and fall when when it is balanced on a finger about a point that’s not its centre of mass?

when the ruler is balanced at its centre, the support force on the ruler from the finger is equal the weight of the ruler, therefore the ruler is in vertical equilibrium. there is a turning force when force is applied at any point but the centre of mass, therefore the ruler tips and falls

how do you find the centre of mass of a triangular card?

• suspend the piece of card and a plumb on a clamp stand

• draw pencil lines along the plumb line

• centre of mass is where the lines drawn on the card cross

here

finding centre of mass of triangle card here

finding the centre of mass of a triangle card

what is a plumb line?

a plumb bob / plummet (weight) suspended from a string that’s used as a perfectly vertical reference line

when finding the centre of mass of a triangle card, what is the plumb line parallel to?

the pencil line from the top of the triangle card, where the card is suspended

what is the centre of mass?

• the central point of a body where all the mass is concentrated

• the point through a single force on the body has no turning effect

where is the mass of a body concentrated?

at the centre of mass, at the centre of the body

what is the point of the body through which a single force on the body has no turning effect?

the centre of mass

how does a tightrope walker stay on the rope?

• holding a horizontal pole to ensure their centre of mass is always directly above the rope

• the support force from the rope then acts upwards through the centre of mass of the walker

why does a tightrope walker use a horizontal pole?

to ensure their centre of mass is always directly above the rope, so the support force from the rope then acts upwards through the centre of mass of the walker

what are assumptions we make when finding moments?

• that all the mass is concentrated in the centre of a body, so the weight of the body acts at the centre of mass (i.e., the body is uniform)

• the pivot is knife-edged, so the body is balancing off a single point

how can we make our moment models more accurate?

• by not assuming the weight of the body acts at the centre of mass, and rather is distributed across the body (i.e., the body is not uniform)

• by not assuming the pivot is knife-edged, and that the the body is not balancing off a single point

where do we assume the weight of the body is?

at the centre of the body, at its centre of mass

when taking moments of a long rectangular body, how do we describe the pivot?

knife-edge

what does it mean for the pivot to be ‘knife-edged’?

the body is balancing on a single point

what does ‘uniform’ mean?

the centre of mass of a body is exactly in the middle

what kind of body has a centre of mass exactly in the middle?

a uniform one

how do we calculate the weight of a metre ruler?

1. locate centre of mass by balancing it horizontally on a horizontal knife-edge

2. note the position of the centre of mass. if its uniform, this will be at the ruler’s centre

3. balance the metre rule off-centre on a knife-edge using a known weight

4. the position of the known weight needs to be adjusted gradually until the ruler is horizontal

5. anticlockwise moment = clockwise moment

6. measure the distance between the pivot and the known weight, and the pivot and the centre of mass

7. w₀d₀ = w₁d₁

8. rearrange for weight, w₀ = w₁d₁ / d₀

here

how do we denote weight of the body?

w₀

what does d₀ mean?

the distance from the centre of mass to an off-centre pivot

what does w₀ mean?

the weight, at the centre of mass, of a body

finding weight of metre rule here

calculating the weight of a metre ruler

what is a single support object?

• the body is balanced on a single pivot

• S = W₀ + W₁ + W₂

here

where is the support force on a single support problem?

at the (knife-edged) pivot

what is always the structure of an object when at equilibrium?

supported at one point only

here

what is the support force of a single support object at equilibrium?

• S = W₀ + W₁ + W₂

• support force equal to the total downwards weight

here

what are the moments about a single support object?

w₁d_{1 =} w₂d₂

here

why are the moments about a single support object w₁d_{1 =} w₂d₂ ?

derivation here

why do we take moments about unknown forces (in terms of equation)?

derivation here

• taking moments around unknown sources gives the same equation as taking moments about the original pivot, w₁d_{1 =} w₂d₂

• therefore moments can be taken about any point

• therefore moments should be taken around unknown forces as the moment of these forces is zero at this point

for a single support object, can moments be taken at any point?

yes

for a single support object, why can moments be taken at any point?

derivation of equation here

where is the weight acting on a single support object?

• at the body’s centre of mass

• if the body is uniform, this is at the body’s centre

single support object here

moments about a single support object

what is a two support object?

here

• body supported by two pivots (usually pillars)

two support object here

moments about a two support object

where is the weight on a two support object?

• if the centre of mass is midway between the two pillars, weight is distributed equally between the two pillars, therefore the support force from each pillar is half the weight

• if the centre of mass is at different distances to each pillar, take moments about both pillars to to find the support force from each pillar, then add them to find the weight

where is the weight on a two support object when the centre of mass is between the two pillars?

weight is distributed equally between the two pillars

what is the support force of each pillar when the centre of mass of a two support object is between the two pillars?

both are equal to half the weight, since weight is distributed equally between the pillars in this case

for a two support object, when is weight equally distributed between the two pillars?

when the centre of mass is midway between the two pillars

for a two support object, when is the support force from each pillar equal to half the weight?

when the centre of mass is midway between the two pillars

how do you find the weight using support forces?

• total weight = total support force

• i.e., add all the support forces together

how do you find the weight of a two support object when the centre of mass is at different distances to each pillar

take moments about both pillars to to find the support force from each pillar, then add them to find the weight

how do you find the support forces from each pillar of a two support object when the centre of mass is at different distances to each pillar?

take moments about both pillars to to find the support force from each pillar, then add them to find the weight

for a two support object where the centre of mass is at different distances to each pillar, which pillar has a higher support force?

the one with the shortest distance to the centre of mass

for a two support object where the centre of mass is at different distances to each pillar, why does the pillar with the shortest distance to the centre of mass have the larger support force?

idk smth to do with equations or idk

what is a couples object?

pair of balanced (equal and opposite) forces acting on a body, yet not along the same line

here

couples object here

couples object

what is the moment of a couple?

moment of a couple = force x perpendicular distance between the lines of action of the forces

here

what is the perpendicular distance between the lines of action of the forces involved in a couple?

idk

derive why the moment of the couple is what it is

is the total moment of a couple constant or does it vary?

it is the same (constant), regardless of the point about which the moments are taken

what is the total moment of a couple constant regardless of?

regardless of the point about which moments are taken