Levers in everyday life We are familiar with levers in everyday life, they make our life easier..... GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH

Definitions The moment of a force about any point is defined as the force multiplied by the perpendicular distance from the line of action to the point Point Force f Perpendicular distance d Moment = f x d Units are Nm

The Principle of Moments 1 If a body is acted upon by more than one force and it is in equilibrium then the turning effects of the forces must also balance out:W 1 d 1 = W 2 d 2 W1W1 W2W2 d1d1 d2d2 Pivot or Fulcrum

The Principle of Moments 2 The sum of all the anticlockwise moments W1W1 W2W2 d1d1 d2d2 Pivot or Fulcrum The sum of all the clockwise moments =

The Principle of Moments 3 The sum of all the anticlockwise moments W1W1 W3W3 Pivot or Fulcrum The sum of all the clockwise moments = W2W2 W4W4 Even with more complex systems the same fundamental rule applies

Non-Perpendicular Forces When the force is not perpendicular we use our skills of resolving Point Force f Perpendicular distance d Moment = f cos (90 - θ) x d θ° 90-θ° Perpendicular Force

Our skills at resolving forces into components can be utilised to deal with forces which are not perpendicular to the distance from the pivot

Definitions The centre of mass of a body is the point through which a single force on the body has no turning effect Centre of Mass Force f No turning force Turning force

Support Forces (Single Support) W1W1 W2W2 d1d1 d2d2 Pivot or Fulcrum Weight of Ruler W 0 Reaction from pivot (support force S) Support Forces S = W 1 + W 0 + W 2 We know that taking moment about the pivot yields : W 1 d 1 = W 2 d 2

But what if we take moments about a different point? Consider the point where W 1 is attached Sum of clockwise moments = W 0 d 1 + W 2 (d 1 + d 2 ) Sum of anticlockwise moments = Sd 1 = (substituting for S) = (W 1 + W 0 + W 2 )d 1

(W 1 + W 0 + W 2 )d 1 = W 0 d 1 + W 2 (d 1 + d 2 ) Multiplying out... W 1 d 1 + W 0 d 1 + W 2 d 1 = W 0 d 1 + W 2 d 1 + W 2 d 2 Simplifying... W 1 d 1 = W 2 d 2 Exactly as before, no change

Taking moments about any point yields the same result... We can use this to our advantage if we have an unknown force in the system, resolving about the point where this unknown force removes it from the moments calculation since this unknown force has zero moment at this point

So equal forces in opposite directions doesn’t always mean equilibrium. There can be a net turning moment. Moments of a Force Decide which of these objects are in equilibrium? NO YES NO

So how do we make an object turn? Remember Newton’s third law – if the net force does not act through the centre of mass (or gravity) of an object it will be caused to turn. pair of equal and opposite forces notthrough it’s centre of masscouple If we have a pair of equal and opposite forces acting on an object, but not through it’s centre of mass, they create what we call a couple. one force more than two torque If is only one force or more than two forces causing the system to rotate it is referred to as a torque. You may have heard of a “torque wrench” that allows you to set how tight you are doing your nuts. The object may not necessarily be moving in one direction but will definitely be spinning. moment of the couple The moment of the couple is calculated from knowing that…

Be careful that you find the correct distance and in the case of a couple just use the size of ONE of the forces! The moment of a force is defined as the product of the force and it’s perpendicular distance from the point of rotation Moment = force * perpendicular distance Nm N m Nm N m The moment of a force is the turning effect of the force

Force applied from leg through pedal…… Causes a turning effect as the force doesn’t act through the pivot

A 2m ruler may be balanced at it’s mid point as shown! Calculate the moments to show whether it is or not. Remember that g = 9.81 ms -2 Mass = 5Kg 20 cm Mass = 1Kg 100 cm This process could be used to calculate the mass of an unknown object and is how old fashioned scales worked!

A 2m ruler with a mass of 200g may be balanced as shown! Calculate the moments to show whether it is or not. Remember that g = 9.81 ms -2 Mass = 1Kg 20 cm This method could be used to find out the mass of the bar. You know it will balance at it’s mid point and therefore can calculate the moments in a clockwise and anticlockwise direction. The only unknown will be the mass of the bar.

If a rigid body is in equilibrium, the sum of the clockwise moments will ALWAYS be equal to the sum of the anticlockwise moments. Principle of Moments Use the practical to confirm that the principle of moments is correct and then try the tank question on the next slide.

Now try this. g = 10 N/kg Mass of tank = 5 tonne Mass of trailer = 1 tonne Calculate the magnitude of the reactions R a and R b 20m BA 8m 10m W Tank W Trailer

So in fact we need not have taken moments around A at all. It was the only force left that we didn’t know. g = 10 N/kg Mass of tank = 5 tonne Mass of trailer = 1 tonne Calculate the magnitude of the reactions R a and R b Taking moments about A (5000x10 x 12) + (1000x10 x 10) = R b x N Rb = /20 = 35000N Taking moments about B (5000x10 x 8) + (1000x10 x 10) = R a x N Ra = /20 = 25000N Notice that the total upward force = = 60,000N Total downward force or weight = 1000x x10 = 60,000N BA 8m 10m W Tank W Trailer RaRa RbRb 20m

 Factsheet 04  Moments and Equilibrium Support  S&C Putting ones foot Down  Calculation sheet 4.7  Practical 4.5 – Investigating the Bridge Crane