What are moments and shear in classical mechanics?

[r16]

I have heard the term moment being used quite often in physics (like a moment in the distributions of mass within closed systems, moments of inerta, as well as torque), but it has never been offically explained to me.

I was attempting to do a problem about the bending of a pliabile beam to one side under the load of a vertical force, and they started throwing around terms like shear and moment which I have no idea what they are (especially shear)

Anyone have any good explanations of these concepts?

 

[berkeman]

wikipedia.org is generally a good source of basic info:

http://en.wikipedia.org/wiki/Moment_(physics)

http://en.wikipedia.org/wiki/Shear_stress

 

[Cyrus]

Shear is a force per unit area that acts tangentially.

Moment is a rotation that is equal to the Force x distance.

 

[Astronuc]

Elaborating on Cyrus's comments, in shear, the force is acting parallel with the surface.

Moment involves a force acting perpendicular to a moment arm (distance), as opposed to parallel with the orientation of the moment arm (direction) which would act to increase or decrease length of a structural member. The moment causes bending or rotation.

 

[TuviaDaCat]

now, i have a question bothering me for a long while... how the hack is this moment crap increasind or decreasing force!
im sooo sure that its not ampirical... i just don't get it... what is the more foundamental explanation...?

 

[radou]

Greater moment implies greater force, if the distance is constant.

 

[TuviaDaCat]

Greater moment implies greater force, if the distance is constant.

thats far from being an answer for my question. i do not want an answer using the word moment... there must be a more fundamental principle based on matter properties, and forces...

 

[radou]

thats far from being an answer for my question. i do not want an answer using the word moment... there must be a more fundamental principle based on matter properties, and forces...

More fundamental? I wouldn't know. You can think of a concentrated moment as two parallel forces of equal sizes, but different direction, which are infinitely close to each other. Imagine a device of a T-form which is plugged into some rigid body and rotated around with the 'handle' (the hrizontal line on the letter T). That device is producing a concentrated moment in the point on the rigid body where it is plugged into. Pretty much of an idealisation, but I hope it makes things clear.

 

[Gokul43201]

I have heard the term moment being used quite often in physics (like a moment in the distributions of mass within closed systems, moments of inerta, as well as torque), but it has never been offically explained to me.

The mathematical object that connects all these moments is called the n'th moment of a distribution about a point. The n'th moment of the distribution (in a single variable, for simplicity) $f(x)$, about the point $x_0$, is defined as:

$$\mu_n(x_0) = \langle (x-x_0)^n \rangle = \sum f(x) (x-x_0)^n$$

So, a torque is nothing but the first moment of forces about a chosen point (axis). The first moment of a distribution of masses is called the center of mass, and their second moment is called the moment of inertia.

A shear is a linear transformation defined on a vector space that fixes all vectors in a chosen subspace of this space and translates all other vectors in the space along a direction parallel to the fixed subspace. In the space R^2, a shear fixes all points on a chosen line and translates all other points in the plane parallel to this line. The physical concept of a shear stress is the stress tensor that produces a shear transformation on a set of points in a 3-dimensional object (the vector space), typically fixing a plane (the fixed subspace) within the object.

PS : more here: http://en.wikipedia.org/wiki/Moment_(mathematics)
http://en.wikipedia.org/wiki/Shear_(mathematics)

 

[berkeman]

Great stuff, Gokul. Thanks.

 

[leright]

When an engineer uses the term "moment", he typically is referring to a torque. When an engineer uses the term "moment of inertia", he typically is referring to the ratio of the torque to the angular acceleration.

Torque, as you are probably aware, is the cross product of r (distance vector from the center of mass to the point a force is acting on) with force. So, the torque (or moment, if you will) is proportional to the magnitude of the force, the distance of r, and the sine of the angle between the two vectors. This is quite intuitively pleasing.

Basically, a moment is a torque. Physics and engineering professors for some reason fail to point this out. Well, more generally, in mathematics, a moment is the cross product of r with ANY vector, which doesn't necessarily have to be a force, but in physics, the "moment" usually refers exclusively to torque.

Simple ideas are made too complicated sometimes. 75% of the stuff in university that seem very complicated and abstract is really quite simple...it is just never explained in a really clear way. Kinda disappointing.

 

[masudr]

In classical mechanics:

The torque, $\vec{\tau}$, due to a force, $\vec{F}$, about at a point $\vec{r}$ is defined as

$$\vec{\tau} = \vec{r} \times \vec{F}$$

The effect of a torque is to cause rotation about the point $\vec{r}$ considered above, such that:

$$\vec{\tau} = \frac{d\vec{L}}{dt}$$

where $\vec{L}$ is the angular momentum:

$$\vec{L}=m\vec{r} \times \frac{d\vec{r}}{dt}.$$

For rigid bodies, we can re-write this as:

$$\tau=I\ddot{\theta}$$

where $\theta$ is the angular displacement, and $I$ is the moment of inertia of that rigid body, about the axis which contains the point $\vec{r}$, given by

$$I=\int_V \rho(r)r^2 dv$$

where $r$ is the perpendicular distance from that axis, and $\rho(r)$ is the mass density at each point.

In general, the motion of a rigid body due to a torque can be split into translational motion of the center of mass of the body, and rotation about an axis containing the center of mass.

Sometimes, the term moment is given a special definition. A moment is a combination of two forces that act on two different points on a rigid body, that have a resultant force equal to zero but a resultant non-zero torque: i.e. rotation without any translational motion.