6.6 - equilibrium rules

what is a free body force diagram?

a diagram showing only the forces acting on an object

why is a free body force diagram better than a diagram depicting all the forces acting in a system?

as objects exert equal and opposite forces on each other when they interact, a force diagram depicting all the forces would be complicated to read as it would also include the forces the object exerts on other objects as well. a free body diagram only shows forces acting on the object

what is weight (W)?

force resulting from the effect of gravity on a mass

what is support force (S)?

an upward, balanced force a surface exerts on an object

what is friction (F)?

resistance to motion due to contact forces

label the weight, support force, and friction on the free body force diagram here

here

where does the weight act on free body force diagram?

vertically down, through the object’s centre of mass

here

where does the support force act on a free body force diagram?

acts through the normal at the point where the lines of action act against the surface

here

where does the support force act on an object on a slope?

normal to the slope, through the point where the lines of action act against the surface

where does friction act on a free body force diagram?

up the slope / opposite to the direction of motion

here

when is a point object acted on by three forces in equilibrium?

• when the resultant force = zero

• i.e., when their vector sum, F₁ + F₂ + F₃ = 0

for a point object acted on by three forces, what does the sum of any two of the forces give?

the equal and opposite (balanced) magnitude of the third force (F₁ + F₂ = -F₃)

for a point object acted on by three forces, what does any one of the forces give?

the equal and opposite (balanced) resultant of the two other forces (F₃ = - (F₁ + F₂) )

what rules apply to a body in equilibrium acted on by three forces?

• the resultant of the three forces is zero, F₁ + F₂ + F₃ = 0

• the resultant of any two forces is equal and opposite to the third force, F₁ + F₂ = -F₃

• the lines of action must intersect at the same point otherwise the body can’t be in equilibrium, otherwise the forces will have a net turning effect

for a body in equilibrium acted on by three forces, what does it mean for the lines of action to intersect at the same point?

idk

for a body in equilibrium acted on by three forces, why must the lines of action intersect at the same point?

otherwise the forces will have a net turning effect, therefore the body won’t be in equilibrium

what is the triangle of forces for a point object?

here figure 2

what is the triangle of forces?

a triangle depicting three force vectors acting on an object / body at equilibrium

here figure 1 and 3

how do we draw a triangle of forces for a free body force diagram of an object on a slope?

here figure 3

where is the angle of a triangle of forces for a free body force diagram of an object on a slope?

here

the angle between the hypotenuse (weight) and adjacent side (friction)

where is the hypotenuse of a triangle of forces for a free body force diagram of an object on a slope?

here

the weight

where is the adjacent side on a triangle of forces for a free body force diagram of an object on a slope?

here

the friction

where is the opposite side of a triangle of forces for a free body force diagram of an object on a slope?

here

the support force

how do we find an unknown force or angle, given the other forces / angle?

by drawing a scale diagram of the triangle of forces

here

how do you draw a scale diagram of the triangle of forces?

1. draw one of the known force vectors (F₁) as the horizontal side of the force triangle

2. use a protractor and ruler to draw the other known force (F₂), at the correct angle to F₁, as the second side of the triangle

3. the third side (F₃) can then be drawn in to give the unknown force

4. resolve F₃ and F₂ perpendicular and parallel to the base F₁ (i.e., make the triangle into two right angled triangles)

5. use trig to find the perpendicular component of F₂ and F₂, then make them equal to each other OR use the sine rule

6. rearrange for F₃

here

how do you find an unknown force using a scale diagram (after drawing the scale diagram) using trig?

here

how do you find an unknown force using a scale diagram (after drawing the scale diagram) using the sine rule?

here

what is the movement of a body at equilibrium?

either at rest or moving with constant velocity, as long as there is no accelerating force and the forces are balanced

what is the moment of a body at equilibrium?

follows the principle of moments; clockwise moments = anticlockwise moments

what are the conditions for a body at equilibrium?

1. either at rest or moving with constant velocity

2. result force must be zero. if there are only three forces, they must form a closed triangle

3. the principle of moments must apply (clockwise moment = anticlockwise moment)

for a body acted on by three forces, what is the condition of equilibrium?

the three forces, as vectors, must join to make a closed triangle when in equilibrium

here

where does a spine pivot about?

the hip joints

what are the forces on the spine when lifting something the wrong way?

here figure 6

when lifting something, where does the force act on?

the spine

what forces are the spine acted upon when lifting?

• the tension T in the back muscles

• the weight W₀ of the upper part of the body

• the weight W of the suitcase

• the reaction force R from the hip

when lifting something, what is T?

the tension in the back muscles

when lifting something, what is W₀?

the weight of the upper part of the body

when lifting something, what is W?

the weight of what you’re lifting

when lifting something, what is R?

the reaction force from the hip

when lifting something, what is the pivot?

the hip joints

when lifting something, what is the reaction force from?

the hip

when lifting something, what is the tension?

the tension in the back muscles

when lifting something, what is the weight?

• the weight of what you’re lifting, W

• the weight of the upper part of the body, W₀

when lifting something, what is the distance between W₀ and the pivot?

d₀

when lifting something, what is the distance d₀ between?

W₀ and the pivot

when lifting something, what is the angle between W₀ and the pivot?

angle α

what lifting something, what is angle α?

the angle between W₀ and the pivot

when lifting something, what do we assume about W₀?

W₀ is acting on the spine at distance d₀ from the pivot

when lifting something, what is the angle between the spine and T?

θ

when lifting something, what is the angle θ?

the angle between the spine and T

when lifting something, what is the distance between W and the pivot?

d

when lifting something, what is the distance d between?

W and the pivot

when lifting something, what is the distance between T and the pivot?

z

when lifting something, what is the distance z between?

T and the pivot

what direction is the moment W₀ produces before an object is lifted?

here

anticlockwise

what is the anticlockwise moment produced by W₀ before an object is lifted?

here

anticlockwise moment = W₀ sin α d₀

why is the anticlockwise moment produced by W₀ before an object is lifted W₀ sin α d₀ ?

• full derivation here - including ;

• moment = force x distance

• = W₀ sin α x d₀

• = W₀ sin α d₀

W₀ sin α d₀

anticlockwise moment produced by W₀ before an object is lifted

what is the perpendicular component of the force W₀ on the spine before the object is lifted?

W₀ sin α

why is the perpendicular component of the force W₀ on the spine W₀ sin α before the object is lifted?

derivation here

W₀ sin α

the perpendicular component of the force W₀ on the spine before the object is lifted

before an object is lifted, what is the point of T₀?

to provide an equal and opposite (balanced) clockwise moment, T₀ x z, about the hip joints

what direction is the moment T₀ produces before an object is lifted?

clockwise

what is the clockwise moment produced by T₀ before an object is lifted?

T₀ z

derivation here

T₀ z

the clockwise moment produced by T₀ before an object is lifted

what is the purpose of the clockwise moment produced by T₀ before an object is lifted?

to balance the anticlockwise moment produced by W₀, W₀ d₀ sin α, so the moment of principles is followed and your body is at equilibrium, meaning you don’t fall when bending down to pick up the object

before an object is lifted, why don’t you fall over when bending down to pick up the object?

because the clockwise moment produced by T₀, T₀ z , is balanced to the anticlockwise moment produced by W₀, W₀ d₀ sin α. the principle of moments mean your body is in equilibrium, therefore you do not fall when bending down to pick up an object

what is the tension in the back muscles before an object is lifted?

T₀

what is T₀?

the tension in the back muscles before an object is lifted

what is the value of T₀ to the spine before an object is lifted?

T₀ = 10 W₀ sin α

why does T₀ = 10 W₀ sin α before an object is lifted?

• applying the principle of moments before the box is lifted gives T₀ z = W₀ d₀ sin α

• typically, d₀ = 10 z

• therefore T₀ z = W₀ 10 z sin α

• rearranging gives T₀ = 10 W₀ sin α

T₀ = 10 W₀ sin α

the tension in the back muscles before an object is lifted

what happens once an object is lifted off the ground?

• W creates an extra anticlockwise moment (W d sin α) about the hip joints

• extra tension (ΔT) is provide an equal and opposite clockwise moment (ΔT z)

where is the moment W produces about?

the hip joints, because that’s the pivot

what direction is the moment produced by W once the object is lifted?

anticlockwise

what is the anticlockwise moment produced by W once the object is lifted?

W d sin α

W d sin α - derive this anticlockwise moment produced by W once the object is lifted

derivation here

what happens to the moments when an object is lifted?

an extra anticlockwise moment (W d sin α) is produced by W about the hip joint

W d sin α

the anticlockwise moment produced by W when an object is lifted

what happens to the tension when an object is lifted?

extra tension from the back muscles is needed to provide an equal and opposite clockwise moment (ΔT z)

what direction is the moment produced by ΔT when an object is lifted?

clockwise

what is the clockwise moment produced by ΔT when an object is lifted?

ΔT z

ΔT z

clockwise moment produced by the change in tension of the back muscles when an object is lifted

why is a clockwise moment ΔT z produced when an object is lifted?

• because the clockwise moment produced by the extra tension is opposite and equal to the total anticlockwise moment produced by W and W₀

• the moments need to be balanced to ensure the principle of moments is followed and the person picking up the object doesn’t fall

when picking up an object, when doesn’t the person fall?

when the moment produced by the extra tension is equal and opposite (balanced) the anticlockwise moment produced by W and W₀, meaning the principle of moments is followed, and there is no rotational movement of the person falling (the person stays stable)

when picking up an object, when does the person fall?

when the extra tension is insufficient, and the moment produced by the extra tension is not equal and opposite (balanced) to the anticlockwise moment produced by W and W₀, meaning the principle of moments is not followed, and there is rotational movement of the person falling

what is ΔT when an object is lifted?

ΔT = 15 W sin α

why is ΔT = 15 W sin α when an object is lifted?

• applying the principle of moments when the box is lifted gives ΔT z = W d sin α

• typically, d = 15 z

• therefore ΔT z = W 15 z sin α

• rearranging gives ΔT = 15 W sin α

ΔT = 15 W sin α

change in tension of the back muscles when an object is lifted

what are the components parallel to the spine?

components of T, W₀, and W

here

what is the component of T parallel to the spine?

here

derive component of T parallel to the spine

here

component of T parallel to the spine here

component of T parallel to the spine

what is the component of W₀ parallel to the spine?

here

derive component of W₀ parallel to the spine

here

component of W₀ parallel to the spine here

component of W₀ parallel to the spine

what is the component of W parallel to the spine?

here

derive component of W parallel to the spine

here

component of W parallel to the spine here

component of W parallel to the spine

where do the parallel components to the spine (T, W₀, W) act?

they act down the spine