Petroleum and Mining DepartmentSecond Grade- Fall Semester

Statics-General Principles

(Lecture 1)& Vector principle of force

(Lecture 2)

Lecturer: Ms. Jwan Khaleel M. 1

Lecture content:• What is Mechanics of Engineering?

• Fundamental Concepts

• Units of Measurement

• Law of Gravitation

• Scalars and Vectors

• Vector Operations

• Vector Addition of Forces

• Addition of a System of Coplanar Forces

• Solving problems2Lecturer: Jwan Khaleel M.

Learning Outcomes:

At the end of this lecture you will be able to:

• Introduce the concept of engineering mechanics and general applications.

• Analyze the classification of engineering mechanics.

• Evaluate a statement of Newton’s Laws of Motion and Gravitation.

• Use the laws of gravitation.

• Analyze the system of forces according to components and resultants.

• Express vectors in terms of unit vectors and perpendicular components

• Perform vector addition and subtraction.

• Compute Parallelogram law and trigonometric rules.

• Solve related problems.

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Statics and Dynamics

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Whatis

Engineering Mechanics?

Engineering Mechanics

Statics Dynamics

Is concerned with the equilibrium of

bodies that is either at rest or moves

with constant velocity (no accelerated

motion).

Kinematics Kinetics

Is deal with the accelerated motion of a body

• Study of the relations existing between the

forces acting on a body, the mass of the

body, and the motion of the body.

• Kinetics is used to predict the motion

caused by given forces or to determine the

forces required to produce a given motion.

Study of motion without paying attention to the forces

causing it.

Or in other words

• Study of the geometry of motion.

• Relates displacement, velocity, acceleration, and

time without reference to the cause of motion.6

Lecturer: Jwan Khaleel M.

Mechanics is a branch of the physical sciences that is concerned with the state of rest

or motion of bodies that are subjected to the action of forces.

Applications To Mechanics:

1. Analysis and design of moving structures.

2. Fixed structures subject to shock loads

3. Robotic devices.

4. Machining of turbines and pumps.

5. Automatic control system

6. Rockets.

7. Missiles and spacecraft.

8. Ground and air transportation vehicles. And others.7

Lecturer: Jwan Khaleel M.

Basic Concept of Mechanics:

•Time: is a basic quantity in dynamics

•Mass: is a quantity of matter in a body (statics) measure of the inertia of a body,which its resistance to a change of velocity (dynamics)

•Force: push or pulls exerted one body to another.

•Particle: is a body of negligible dimensions.

•Rigid body:

•Deformable body:

•Length: is needed to locate the position of a point in space thereby

describe the size of a physical system.

•Space: is a geometric region occupied by bodies whose positions

are describes by linear and angular measurement relative to a

coordinate system two or three dimensions

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Newton’s laws:

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Units of Measurements:

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• Units Conversion:

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• Prefixes:

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Example 1//

What is the weight in both newton and pounds for 1500 kg beam?

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Example 2// Convert 2 km/h to m/s, How many ft/s is this?

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• Law of Gravitation:

The law of gravitation is expressed by the equation.

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Two uniform aluminum spheres are posted as shown. Determine the

gravitational force which sphere A exerts on sphere B. The value of R

is 50 mm. if you know that 𝐺 = 6.673 × 10−11𝑚3

𝑘𝑔.𝑠2assume the

sphere is Aluminum so , 𝜌 = 2690𝑘𝑔

𝑚3

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Example 3//

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SOLUTION

𝑅 = 0.05 𝑚 𝑓𝑜𝑟 𝑏𝑜𝑡ℎ 𝑠𝑝ℎ𝑒𝑟𝑒𝑠

Radius of Sphere A= 2R= 0.1m

Radius of Sphere A= R= 0.05m

Distance d=8R= 0.4 m

G= 6.673 x10-11m3/kg. s2

𝜌 = 2690 𝑘𝑔/𝑚3

𝑉 =4

3𝜋 × 𝑟3

𝐹 =𝐺𝑚𝑐𝑚𝑡

𝑑2=𝐺𝜌𝑉

𝑑2=

𝐺𝜌4

3𝜋×0.13

4

3𝜋×0.053

0.42

= 6.61 × 10−9𝑁

Because the spheres are located at 30° with x-axis

𝐹 = 𝐹𝑐𝑜𝑠 𝜃𝑖 + 𝐹𝑠𝑖𝑛 𝜃𝑗

𝐹 = 6.61 × 10−9𝑐𝑜𝑠 30𝑖 + 6.61 × 10−9𝑠𝑖𝑛 30𝑗

𝐹 = − 5.72× 10−9 𝑖 − 3.305 × 10−9𝑗 N

So the forces are 𝐹 = − 5.72× 10−9 −3.305 × 10−9𝑁

Convert the quantities 300 𝐼𝑏. 𝑠 and 52 slug/ft3 to appropriate SI

units.

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Class activity 1//

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Solution//

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Convert each of the following.

(a) 20 Ib. ft to N . m,

(b) 450 lb/ft3 to kN/m3

(c) 15 ft/h to mm/s.

Class activity 2//

Solution//

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Class activity 3//

The two 100-mm-diameter spheres constructed of different metals

are located in deep space. Determine the gravitational force 𝑭 which

the copper sphere exerts on the titanium sphere if (a) 𝑑 = 2𝑚, and

(b) 𝑑 = 4𝑚. 𝜌𝑐 = 8910𝑘𝑔

𝑚3 and 𝜌𝑡 = 3080𝑘𝑔

𝑚3

SOLUTION

𝑟 = 0.05 𝑚 𝑓𝑜𝑟 𝑏𝑜𝑡ℎ 𝑠𝑝ℎ𝑒𝑟𝑒𝑠

G= 6.673 x10-11m3/kg. s2

𝜌𝑐 = 8910 𝑘𝑔/𝑚3

𝜌𝑡 = 3080 𝑘𝑔/𝑚3

𝑉𝑡 = 𝑉𝑐 =4

3𝜋 × 𝑟3 =

4

3𝜋 × 0.053 = 5.236 × 10 −4

a) 𝑑 = 2𝑚

𝐹 =𝐺𝑚𝑐𝑚𝑡

𝑑2=𝐺𝜌𝑐𝜌𝑡𝑉

𝑑2=

6.673×10−11× 8910×3080 5.236 ×10 −4

22= −1.255 × 10−10𝑁

b) 𝑑 = 4𝑚

𝐹 =𝐺𝑚𝑐𝑚𝑡

𝑑2=

𝐺 8910 ×5.236 ×10 −4 3080 ×5.236 ×10 −4

42= −3.14 × 10−11𝑁

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• What is Force?

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Weight a crane can lift.

• External and Internal Effect of Force

• External force: Applied or Reactive force

• Internal force: deformation through the material

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• Force Classifications:

• Contact force •Body force

•Concentrated force

• Distributed force

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• Scalars and Vectors• Scalar: A scalar is any positive or negative physical quantity that can be completely

specified by its magnitude . Examples of scalar quantities include length, mass, and

time.

• Vector: A vector is any physical quantity that requires both a magnitude and a direction

for its complete description. Examples of vectors encountered in statics are force,

position, and moment.

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• Vector Operation

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Multiplication and Division of a

Vector by a Scalar

Vector Addition

• Vector Operation

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• Vector Addition of Forces

Concurrent forces: the line of action of two or more forces intersect at that point.

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Parallelogram law

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The law of parallelogram of forces states that if two vectors acting on a particle at the same

time be represented in magnitude and direction by the two adjacent sides of a parallelogram

drawn from a point their resultant vector is represented in magnitude and direction by the

diagonal of the parallelogram drawn from the same point.

Triangle law (Trigonometry)

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Triangle law of vector addition states that when two vectors are represented as two sides

of the triangle with the order of magnitude and direction, then the third side of the triangle

represents the magnitude and direction of the resultant vector.

• Finding the Components of a Force

➢Scalar notation:

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➢Cartesian Vector notation:

Where 𝑭𝒙 and 𝑭𝒚 are vector

components of F in the x- and y-

directions.

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➢Coplanar Force Resultants:

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➢Angles with their axis:

Unit Vector:

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A vector is a quantity that has both magnitude, as well as

direction. A vector that has a magnitude of 1 is a unit vector.

It is also known as Direction Vector.

Unit Vector =Vector

Magnitude of Vector

EXAMPLE// Find the unit vector Ԧ𝑞 for the given vector,

−2 Ƹ𝑖+4 Ƹ𝑗–4𝑘. Show Unit vector component q.

𝑞 =−2 Ƹ𝑖+4 Ƹ𝑗–4𝑘

(−2)2+(4)2+(−4)2=−2 Ƹ𝑖+4 Ƹ𝑗–4𝑘

36=−2 Ƹ𝑖+4 Ƹ𝑗–4𝑘

6=−2 Ƹ𝑖

6+4 Ƹ𝑗

6−4𝑘

6=−1

3Ƹ𝑖 +

2

3Ƹ𝑗 −

2

3𝑘

The screw eye in Figure shown a

is subjected to two forces, 𝐹1and

𝐹2. Determine the magnitude and

direction of the resultant force.

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Example 1//

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Solution//

Resolve the horizontal 600-lb force in Figure

shown into components acting along the 𝑢

and 𝑣 axes and determine the magnitudes of

these components.

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Example 2//

Lecturer: Jwan Khaleel M.

Solution//

Example 3//“Class activity ( you have 10 minutes to think/solve it by group)”

Determine the components

of the F force acting along the

u and v axes. If you know:

𝜃1= 70°𝜃2= 45°𝜃3= 60°F = 250 N

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The 1800-N force F is applied to the end

of the I-beam. Express 𝑭 as a vector

notation 𝑖 and 𝑗.

Example 4//

𝐹 = −18003

5𝑖 − 1800

4

5𝑗 = (−1080𝑖 − 1440𝑗 )N

Solution:

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The end of the boom O in Fig. a is subjected to three concurrent and coplanar forces. Determine the magnitude and direction of the resultant force.

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Example 5//

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Solution:

Example 6//“Class activity ( you have 10 minutes to think/solve it by group)”

The 500-N force 𝑭 is applied to the vertical pole as shown:

1) Write 𝑭 in terms of the unit vectors 𝒊 and 𝒋 and identify both its vector and scalar components.

2) Determine the scalar components of the force vector 𝑭 along the x′- and y′-axes.

3) Determine the scalar components of 𝑭 along the x-and y′-axes.

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Assignment 1// (solve this problems then submit your answer)

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Determine the magnitude of the resultant force acting on the screw eye and its direction

measured clockwise from the x axis.

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Resolve each force acting on

the post into its x and y

components.

Assignment 2// (solve this problems then submit your answer)

Assignment 3 (solve this problems then submit your answer)

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Express each of the three forces

acting on the column in Cartesian

vector form and compute the

magnitude of the resultant force.

Next lecture:

• Cartesian Vectors

• Addition of Cartesian Vectors

• Position Vectors

• Force Vector Directed Along a Line

• Dot Product

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References:

Engineering Mechanics R.C.

Hibbeler 13th edition (Statics and

Dynamics).

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The end of the lecture

Enjoy your time

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