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Turning ForcesThe moment of a force is the product of the force and the perpendicular distance from the line of action of the force to the pivot. The unit of a moment is Nm. It is a vector quantity.

(a)

Moment of force about P is Moment = F × d

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(b)

Moment of force about P is

Moment = F × x

Moment = F × d sin α

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Note: The unit of a moment is Nm but not Joules because a moment is

force × perpendicular distance, but work is force × parallel distance.

For perpendicular vectors, cross-product is used which results in a vector

quantity.

For parallel vectors, dot product is used which results in a scalar

quantity.

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Principle of Moments

If a system/body is in equilibrium, then the sum of the clockwise

moments about any point is equal to the sum of the anticlockwise

moments about the same point.

e.g. 1.

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Find T1 and T2. Taking moments about the 80 cm mark,

Anticlockwise moments = Clockwise moments

Taking moments about the 10 cm mark, Anticlockwise moments = Clockwise moments

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e.g. 2. A uniform metre rule of weight 5 N is balanced at the 40 cm

mark by hanging a weight at the 10 cm mark. Find the weight that must

be hung at the 10 cm mark. Draw a diagram, putting in all the forces

acting, and find the reaction at the pivot.

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Taking moments about the 40 cm mark, Anticlockwise moments = Clockwise moments

Taking moments about the 10 cm mark, Anticlockwise moments = Clockwise moments

or

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Sum of upward forces = Sum of downward forces

R = 5 + 1.67 R = 6.67 N

1.67 N must be hung at the 10 cm mark. The reaction at the pivot is 6.67 N.

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Worked Example 1

A drawbridge is held in place by a chain as shown. The distance between

the point of attachment of the chain on the drawbridge and the hinge is

2.0 m. The tension in the chain is 500 N and the chain makes 50° with the

drawbridge (horizontal). What is the moment produced by the tension in

the chain about the hinge of the drawbridge?

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Solution

The line of action of the tension in the chain is simply along the chain

itself. So, the perpendicular distance between the line of action of the

tension and the hinge is shown as the dotted line above, and is equal to

(2.0 sin 50°) m.

So, the moment due to the tension about the hinge

= 500 × (2.0 sin 50°)

= 770 Nm

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Worked Example 2

A picture of weight 10 N is hung by a rope, such that both sides of the

rope make an angle of 20° with the horizontal.

What is the tension in the rope?

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Solution First, we should draw in the forces acting on the picture:

In the upward direction, we have the vertical components of the tensions, each of which is T sin 20°. Since there are two tensions acting on the picture (one on each side of the rope),

the total upward force acting on the picture is 2T sin 20°.

The only downward force acting on the picture is its weight W.

So, 2T sin 20° = W

Tension T = 14.6 N

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Worked Example 3 A ladder of weight 20 N is leaning against a smooth wall, such that the ladder makes an angle of 50° with the ground. The length of the ladder is 2.0 m long.

(i) What is the normal reaction from the wall acting on the ladder? (ii) Find the normal reaction and the friction from the ground which are acting on the ladder.

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SOLUTION

First, we draw all the forces acting on the ladder. We shall use the

following symbols to represent the various quantities:

W: weight of ladder L: length of ladder Nw: normal reaction from the wall Ng: normal reaction from the ground F: friction from the ground

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(i) We will use the principle of moments to find Nw. We will choose the point of contact between the ladder and the ground as the pivot. Then the forces Ng

and F can be ignored, as their moments about the chosen pivot is zero.

Then, anticlockwise moment = clockwise moment

Nw L sin 50° = W cos 50°

Nw = 8.4 N

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(ii) Now that we have the value of NW , notice that horizontally, the resultant force acting on the ladder must be zero.

So, F = NW

= 8.4 N

Similarly, vertically, the resultant force acting on the ladder must also be zero.

So, Ng = W

= 20 N

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Torque and couple

How is it possible for two forces of equal magnitude acting in opposite directions still not give rise to the body being in equilibrium? (Assume no other forces act)

This can happen if the two forces do not act through the same point. They will give rise to a turning effect.

A couple is two forces equal in magnitude, opposite in direction acting on a body but not acting through a common point. This will give rise to a turning effect or a torque(t).

The torque of a couple is the product of one of the forces and the perpendicular distance between them.

In the diagram, Torque of the couple = F × 2r

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Archemede’s Principle Notice how different floating objects have different proportions of their volumes jutting out of the water? What determines the proportion of the object that sticks out of the water?

It was the ancient Greek mathematician, Archimedes, who discovered the principle that is named after him:

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Archemede’s Principle states that if a body is wholly or partially immersed is a fluid, it will experience an up thrust which is equal to the weight of the fluid displaced.

As an object is lowered into water, it will experience more and more upthrust as it displaces more and more water. If the object is less dense than water, before it becomes totally immersed in water, it will have displaced an amount of water so that the weight of the displaced water is numerically equal to the weight of the object. When that happens, the upthrust acting on the object balances its weight, and it floats.

On the other hand, if the object is more dense than water, by the time it is fully immersed in water, it still has not displaced enough water to provide the required upthrust to balance its weight. Since the weight is more than the upthrust, the object sinks.

Archemede’s principle also applies for an object floating in a gas, such as a helium balloon floating in air.

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Origin of Upthrust Force Archemede’s Principle states that if a body is wholly or partially immersed in a fluid, it will experience an upthrust, which is equal to the weight of fluid displaced.

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Take the density of the fluid to be ρ. Recall Pressure at a depth in a fluid is given by P = h ρ g The pressure at the top surface = h1 ρ g

Therefore force on top surface = h1 ρ g A

Similarly, the force pushing upwards on the bottom surface = h2 ρ g A

So net force pushing upwards (since h2 > h1) is given by:

Net force up = (h2 − h1) ρ g A

= h ρ g A But h ρ g A is the weight of fluid displaced.

Hence upthrust = weight of fluid displaced

Conclusion: The upthrust force comes about because the pressure in a fluid increases with depth. That is the greater the pressure below, the greater the force from below, hence the greater the net force from below. This leads to the upthrust.

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The Law of Floatation If a body is floating in a fluid, then the weight of fluid displaced is equal to the weight of the body.

Note: 1. If W > U then the body sinks (moves downwards) 2. If W < U then the body rises (moves upwards) 3. If W = U then the body floats at whatever depth it is (The body does not move vertically)

For a body floating (no vertical motion)