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**Moment of a force about an axis**

The moment of a force about an axis p is defined as the projection OB on p of the moment (see Fig. 3.5).

**Figure 3.5:** Moment of the force F about the axis p.

Denoting **Î»** the unit vector of p, the moment Mp of force **F** about an axis p can be expressed as scalar product (dot product)

**Mp = Î» . M _{o}** (3.22)

or as the mixed triple product of the unit vector **Î»** , the position vector **r,** and the force **F:**

Mp = **Î» **. ( r *F) (3.23)

Using the determinant form for the mixed triple product, we have the magnitude of the moment

Moment of a force about an axis

(3.24)

**Î» _{x},**

where** Î» _{y}, **are direction cosines of axis p

**Î» _{z}**

x, y, z are components of **r**

F_{x},

F_{y}, F_{z} are components of **F**

**Exercise 3.2.1**

Sample problem A force F = 40 N is applied at a point M(x_{M};y_{M};z_{M}) M(3;2;4) [m] (see Fig. 3.6). The line o_{F} of action of the force F is described by direction angles (Î±_{F} ;Î²_{F};C_{r} ) (80^{o};60^{o}; acute angle). An axis a which passes

through the origin O has direction angles (Î±a;Î²a;Î³a) â‰¡ (60^{o};100^{o}; acute angle).

**Determine:**

- The moment
**M**of the force F about O._{o} - The moments M
_{x}, M_{y}, and M_{z}of the force F about axes x, y, and z, respectively. - The moment
**M**of the force F about a axis._{a }

Moment of a force about an axis

**Figure 3.6:** Exercise 3.2.1. Moment of the force F about the axis a.

Notice: An acute angle is such an angle for which the condition 0 < Î± < 90^{o} is valid.

**Solution**

We determine the angle C_{r} first. Since the expression

is valid for any set of direction cosines, we have

According to the definition of the moment M_{o} of the force F, we may write

Notice: Matlab works with radians and not with degrees.

The magnitude of **M _{o}** is

88.06 Nm

The direction cosines of **M _{o}** are

As the components M_{Ox}, M_{Oy}, and MOz of M_{o} are equal to the moments M_{x}, M_{y}, and M_{z} of F about x, y, and z axes respectively, the following is valid:

M_{x} = M_{Ox}, M_{y} = M_{Oy}, M_{z} = M_{Oz}

According to the definition of the moment M_{a} of the force F about an axis a, we have

**M _{a} = Î»^{T} M_{o}**

where the unit vector **Î» **of a is

**Î» = [Cos Î± _{a}, cosÎ²_{a}, cosÎ³_{a}]^{T}**

As

we conclude that

M_{a} = [0.5, 0.1736, 0.8484] = 45.92 Nm

Using MATLAB the solution of the problem is much more convenient. The program for the purpose is called smoment.m. It can be found in program package.

**Exercise 3.2.2 Moment of a force** A force F = 50 N is applied at a point M(8;4;4) [m] and its line of action is described by direction angles (60^{o}; 60^{o}; acute angle). Determine the moments M_{x}, M_{y}, and M_{z} of the force F about the axes x, y, and z respectively and the moment **M _{a}** about the axis a passing through the point A(0;2;2) and having direction angles (30

**Figure 3.7:** Exercise 3.2.2. Moment of the force F about the axis a.

**Solution**

M_{x} = 41.4 Nm, M_{y} = -182.8 Nm, M_{z} = 100 Nm, M_{a} = -98.47 Nm

**Exercise 3.2.3** Moment of a force A force F = 50 N is applied at a point M(2;3;1) [m] and its line of action is described by direction angles (60^{o}; 60^{o}; acute angle). The axis a passes through the origin O and lies in xy-plane having the angle Î´= 30^{o} with y axis. Determine

- The moment M
_{o}of the force F about the origin O. - The moments M
_{x}, M_{y}, and M_{z}of the force F about the axes x, y, and z respectively. - The moment M
_{a }of the force F about the axsis a.

**Figure 3.8**: Exercise 3.2.3. Moment of the force F about the axis a.

**Solution **

M_{x} = 81.05 Nm, M_{y} = -45.7 Nm, M_{z} = -25 N m, M_{O} = 96.34 Nm, M_{a} = -52.07 Nm

**Exercise 3.2.4** Moment of a force A force F = 40 N is applied at a point M(4;3;-2) [m] and its line of action is described by direction angles (30^{o}; 90^{o}; acute angle).

Determine the moment **M _{A} **of the force F about the point A(-1;3;6) [m] and the moment M

**Figure 3.9: **Exercise 3.2.4. Moment of the force F about the axis a.

**Solution**

M_{Ax} = 0 Nm, M_{Ay} = -377.13 Nm, M_{Az} = 0 Nm, M_{a} = -65.49 Nm

**Exercise 3.2.5 **Moment of a force A tube is welded to the vertical plate yz at the point A. The tube is loaded by a force** F** at the point D. The line of action of the force has an angle Î² with y-axis and an angle Î³ with z-axis. Determine the moment of the force

**F** about the point A. Further determine the moment M_{x} about x-axis and the moment M_{BC }about BC axis of the same force **F** . How many solutions does the task have? It is known that F = 200 N; Î² = 60^{o}; Î³= 80^{o};a = 0.2 m;b = 0.4 m;c = 0.8 m.

**Figure 3.10: **Exercise 3.2.5. Moment of the force F.

**Solution**

M_{A} = 152.2 Nm, M_{x} = - 66.1 Nm, M_{BC} = 135.75 Nm

**Exercise 3.2.6 **Moment of a force Determine the moment of the force F = 500 N about the axis AC. The angle Î± = 30^{o}, A(5;0;0), C(0;12;0), E(0;0;20), AB = 6.5, BD = 10 (all dimensions in m).

**Figure 3.11: **Exercise 3.2.6. Moment of a force F.

**Solution**

M_{AC} = 4510 Nm

**Exercise 3.2.7** Tangent force A force T = 60 N acts at a tangent of a helix (see Fig.3.12). Find a generic expression for moments M_{x}, M_{y}, and Mz of the force T about x, y, and z axes respectively as a function of angle . Numerically compute the Ï† Numerically compute the magnitudes of these moments for Ï† =750^{o} knowing that the radius of the helix is r =10 m and the pitch angle is Î± =30^{o}. Write a MATLAB program which can be used to calculate the above moments and use it for creating plots of moments values versus angle 0 â‰¤ Ï† â‰¤ 6Ï€.

**Figure 3.12:** Exercise 3.2.7. Tangent force.

**Solution**

M_{x} = 2222 Nm, M_{y} = -519.6 Nm, M_{z} = 3248 Nm

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