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Brown University

We have seen that a force acting on a rigid body has two effects: (i) it tends to move the body; and (ii) it tends to rotate the body.

A natural question arises – is there a way to rotate
a body without moving it? And is there
a kind of force that causes *only *rotation without translation?

The answer to both questions is yes.

A system of forces that exerts a resultant moment,
but no resultant force, is called a *force couple*.

The simplest example of a force couple consists of two equal and opposite forces and acting some distance apart. Suppose that the force acts at position while the force acts at position The resultant moment is

Of course, the vector is just the vector from the point where acts to the point where acts. This gives a quick way to calculate the moment induced by a force couple:

*The moment induced by two equal and opposite forces
is equal to the moment of one force about the point of action of the other*. It doesn’t matter which force you use to do this calculation.

(i) Has zero resultant force

(ii) Exerts the same resultant moment about all points.

Its effect is to induce rotation without translation.

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The effect of a force couple can therefore be
characterized by a single vector moment **M**. The physical significance of **M** is equivalent to the
physical significance of the moment of a force about some point. The direction of **M** specifies the axis
associated with the rotational force.
The magnitude of **M** specifies the intensity of the rotational
force.

1. The forces exerted by your hand on a screw-driver

2. The forces exerted by the tip of a screw-driver on the head of a screw

3. The forces exerted by one part of a constant velocity joint on another

4. Drag forces acting on a spinning propeller

**5.2 Pure moments, couples
and torques**.-
**Definition, Physical Interpretation, and Examples**

A *pure moment* is a rotational force. Its effect is to induce rotation, without
translation – just like a force couple.

Couples and torques are other names for a pure moment.

A pure moment is a vector quantity – it has magnitude and direction. The physical significance of the magnitude and direction of a pure moment are completely equivalent to the moment associated with a force couple system. The direction of a moment indicates the axis associated with its rotational force (following the right hand screw convention); the magnitude represents the intensity of the force. A moment is often denoted by a symbols

The concept of a pure moment takes some getting used to. Its physical effect can be visualized by thinking about our beam-balancing problem again.

The picture above shows the un-balanced beam. We saw earlier that we can balance the beam
again by adding a second force, which induces a moment equal and opposite to
that of the force *W*.

We can also balance the beam by applying a *pure
moment *to it. Since the moment of *W*
is ,
a moment applied anywhere on the beam would balance it.

You could even apply the moment
to the left of the beam – even right on top of the force *W* if you like!

**5.3 Units
and typical magnitudes of moments**

In the SI system, moments have units of Nm (Newton-meters).

In the US system, moments have units of ft-lb (foot pounds)

The conversion factor is 1 Nm = 0.738 ft lb; or 1 ft-lb = 1.356 Nm.

Typical magnitudes are:

· Max torque exerted by a small Lego motor: 0.1 Nm

· Typical torque output of a typical car engine 300-600 Nm

· Breaking torque of a human femur: 140Nm

Just as you can buy a force transducer to measure forces, you can buy a force transducer that measures moments. We showed an example of a force-transducer attached to the wheel of a car during our earlier discussion of force transducers.

Another common moment-measuring
system is a *torque-wrench*. (So then is Oprah a talk wench?) When you
tighten the bolts on a precision machine, it’s important to torque them
correctly. If you apply too much
torque, you will strip the thread. If
you don’t apply enough, the bolt will work itself loose during service.

You can buy a tool that
measures the moment that you apply to a bolt while tightening it. The device may be mechanical, or
electronic. An example (see **www.mac.ie/whatwedo/
torquestory.asp ) **is shown below.

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**5.5 Engineering systems that exert torques**

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There are many practical examples of moments, or torques, in engineering systems. For example,

(i) The driving axle on your car turns the wheels by exerting a moment on them.

(i) The drive shaft of any motor exerts a torque on whatever it’s connected to. In fact, motors are usually rated by their torque capacity.

(ii)
The purpose of a gearbox is to amplify or attenuate
torque. You apply a torque to the input
shaft, and get a bigger or smaller torque from the output shaft. To do this, the input and output shafts have
to rotate at different speeds. There
are also some clever gearboxes that allow you to *add* torques together – they are used in split-power variable speed
transmissions, for example.

(iii)
A *torque converter*
serves a similar purpose to a gearbox.
Unlike a gearbox, however, the input and output shafts don’t rotate at
the same speed. The output shaft can be
stationary, exerting a large torque, while the input shaft rotates quickly
under a modest torque. It is used as
part of an automatic transmission system in a car.

(iv)
Moments also appear as *reaction forces*. For example, the resistance you feel to
turning the steering wheel of your car is caused by moments acting on the
wheels where they touch the ground. The
rolling resistance you feel when you ride your bike over soft ground or grass
is also due to a moment acting where the wheel touches the ground.

(v)
Moments appear as *internal forces* in structural members
or components. For example, a beam will
bend because of an internal moment whose direction is transverse to the
direction of the beam. A shaft will
twist because of an internal moment whose direction is parallel to the
shaft. Just as an internal force causes
points in a solid to move relative to each other, an internal moment causes
points to *rotate* relative to each other.

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