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# VECTORS AND SCALARS - PowerPoint PPT Presentation

VECTORS AND SCALARS  A scalar quantity has only magnitude and is completely specified by a number and a unit. Examples are: mass (2 kg), volume (1.5 L), and frequency (60 Hz). Scalar quantities of the same kind are added by using ordinary arithmetic.

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A scalar quantity has only magnitude and is completely specified by a number and a unit.

Examples are: mass (2 kg), volume (1.5 L), and frequency (60 Hz).

Scalar quantities of the same kind are added by using ordinary arithmetic.

A vector quantity has both magnitude and direction. Examples are displacement (an airplane has flown 200 km to the southwest), velocity (a car is moving at 60 km/h to the north), and force (a person applies an upward force of 25 N to a package). When vector quantities are added, their directions must be taken intoaccount.

A vectoris represented by an arrowed line whose length is proportional to the vector quantity and whose direction indicates the direction of the vector quantity.

The resultant, or sum, of a number of vectors of a particular type (force vectors, for example) is that single vector that would have the same effect as all the original vectors taken together.

R

A vector in two dimensions may be resolved into two component vectors acting along any two mutually perpendicular directions.

2.1 Draw and calculate the components of the vector

F (250 N, 235o)

Fx = F cos 

= 250 cos (235o)

= - 143.4 N

Fy = F sin 

= 250 sin (235o)

= - 204.7 N

Fx

Fy

F

To add two or more vectors A, B, C,… by the component method, follow this procedure:

1. Resolve the initial vectors into components x and y.

2. Add the components in the x direction to give Σxand add the components in the y direction to give Σy.

That is, the magnitudes of Σxand Σy are given by, respectively:

Σx = Ax + Bx + Cx…

Σy = Ay + By + Cy…

3. Calculate the magnitudeanddirectionof the

resultant R from its components by using the Pythagorean theorem:

and

2.2 Three ropes are tied to a stake and the following forces are exerted. Find the resultant force.

A (20 N, 0º)

B (30 N, 150º)

C (40 N, 232º)

A (20 N, 0)

B (30 N, 150)

C (40 N, 232)

x-component

20 cos 0

30 cos 150

40 cos 232

Σx = - 30.6 N

y-component

20 sin 0

30 sin 150

40 sin 232

Σy = -16.5 N

= 34.7 N

Since Σx = (-) and Σy = (-)

therefore: 180 + 28.3 = 208.3

R (34.7 N, 208.3)

2.3 Four coplanar forces act on a body at point O as shown in the figure. Find their resultant with the component method.

A (80 N, 0)

B (100 N, 45)

C (110 N, 150)

D (160 N, 200)

A (80 N, 0)B (100 N, 45)

C (110 N, 150)D (160 N, 200)

x-component y-component

80 0

100 cos 45100 sin 45

110 cos 150110 sin 150

160 cos 200160 sin 200

Σx = - 95 N Σy = 71 N

= 118.6 N

Since Σx = (-) and Σy = (+)

R is in the IIQuadrant, therefore:

180 - 36.7= 143.3

R (118.6 N, 143.3)

= 6.5 N

= 29

Since Σx = (+) and Σy = (-)

R is in the IV Quadrant,

therefore:

360 - 29= 331

R (6.5 N, 331)

Objective:

The purpose of this experiment is to use the force table to experimentally determine the equilibrant forceof two and three other forces. This result is checked by the component method.

A system of forces is in equilibrium when a force called the equilibrant forceis equal and opposite to theirresultant force.

Force Table Set

MASS (kg)

FORCE

(N)

mg = m (9.8 m/s2)

DIRECTION

F1

F2

Equilibrant

FE

Resultant

FR

DATA Table

An object that experiences a push or a pull has a force exerted on it. Notice that it is the object that is considered. The object is called the system. The world around the object that exerts forces on it is called the environment.

system

Forces can act either through the physical contact of two objects (contact forces: push or pull) or at a distance (field forces: magnetic force, gravitational force).

Description of Force

Direction of Force

Applied Force

An applied force is a force that is applied to an object by another object or by a person. If a person is pushing a desk across the room, then there is an applied force acting upon the desk. The applied force is the force exerted on the desk by the person.

In the direction of the pull or push.

FA

Description of Force

Direction of Force

Normal Force

The normal force is the support force exerted upon an object that is in contact with another stable object. For example, if a book is resting upon a surface, then the surface is exerting an upward force upon the book in order to support the weight of the book. The normal force is always perpendicular to the surface

Perpendicular to the surface

FN

Description of Force

Direction of Force

Friction Force

The friction force is the force exerted by a surface as an object moves across it or makes an effort to move across it. The friction force opposes the motion of the object. For example, if a book moves across the surface of a desk, the desk exerts a friction force in the direction opposite to the motion of the book.

Opposite to the motion of the object

FF

Description of Force

Direction of Force

Air Resistance Force

Air resistance is a special type of frictional force that acts upon objects as they travel through the air. Like all frictional forces, the force of air resistance always opposes the motion of the object. This force will frequently be ignored due to its negligible magnitude. It is most noticeable for objects that travel at high speeds (e.g., a skydiver or a downhill skier) or for objects with large surface areas.

Opposite to the motion of the object

FD

Description of Force

Direction of Force

Tensional Force

Tension is the force that is transmitted through a string, rope, or wire when it is pulled tight by forces acting at each end. The tensional force is directed along the wire and pulls equally on the objects on either end of the wire.

In the direction of the pull

FT

Description of Force

Direction of Force

Gravitational Force (also known as Weight)

The force of gravity is the force with which the earth, moon, or other massive body attracts an object towards itself. By definition, this is the weight of the object. All objects upon earth experience a force of gravity that is directed "downward" towards the center of the earth. The force of gravity on an object on earth is always equal to the weight of the object.

Straight downward

Fg

Each force has a specific identifiable, immediate cause called agent. You should be able to name the agent of each force, for example the force of the desk or your hand on your book. The agent can be animate such as a person, or inanimate such as a desk, floor or a magnet. The agent for the force of gravity is Earth's mass. If you can't name an agent, the force doesn't exist.

agent

A free-body-diagram (FBD) is a vector diagram that shows all the forces that act on an object whose motion is being studied.

Directions:

- Choose a coordinate system defining the positive

direction of motion.

- Replace the object by a dot and locate it in the center

of the coordinate system.

- Draw arrows to represent the forces acting on the

system.

FN

FG

FN

FG

FN

FF

FG

FN

FGY

FGX

FG

FN

FF

FG

FT

FG

FT1

FT2

FG

FT1

FT2

FG

FN

FT

FG

FN1

FN2

FG

FN

FF

FA

FG

FN

FA

θ

FF

FG

FT

FG

FN

FF

θ

FA

FG

FG

FD

FG

ARISTOTLE studied motion and divided it into two types: natural motionandviolent motion.

Natural motion: up or down.

Objects would seek their natural resting places.

Natural for heavy things to fall and for very light things to rise.

Violent motion: imposed motion.A result of forces.

GALILEO demolish the notion that a force is necessary to keep an object moving.

He argued that ONLY when friction is present, is a force needed to keep an object moving.

In the absence of air resistance (drag) both objects will fall at the same time.

A ball rolling down the incline rolls up the opposite incline and reaches its initial height.

As the angle of the upward incline is reduced, the ball rolls a greater distance before reaching its initial height.

How far will the ball roll along the horizontal?

Isaac Newton incline and reaches its

(1642-1727)

FIRST LAW OF MOTION

According to Newton's

First Law of Motion:

"If no net force acts on it, a body at rest remains at rest and a body in motion remains in motion at constant speed in a straight line."

NEWTON'S FIRST LAW OF MOTION incline and reaches its

"An object at rest tends to stay at rest and an object

in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force."

There are two parts to this statement:

- one which predicts the behavior ofstationary

objects and

- the other which predicts the behavior of moving objects.

The behavior of all objects can be described by saying that objects tend to "keep on doing what they're doing" (unless acted upon by an unbalanced force).

If at rest, they will continue in this same state of rest.

If in motion with an eastward velocity of 5 m/s, they will continue in this same state of motion (5 m/s, East).

It is the natural tendency of objects to resist changes in their state of motion.

This tendency to resist changes in their state of motion is described asinertia.

Inertiais the resistance an object has to a change in its state of motion.

The elephant at rest tendstoremain at rest.

Tablecloth trick: their state of motion.

Too little force, too little time to overcome "inertia" of tableware.

If the car were to abruptly stop and the seat belts were not being worn, then the passengers in motion would continue in motion.

Now perhaps you will be convinced of the need to wear your seat belt. Remember it's the law - the law of inertia.

If the truck were to abruptly stop and the straps were no longer functioning, then the ladder in motion would continue in motion.

If the motorcycle were to abruptly stop, then the rider in motion would continue in motion. The rider would likely be propelled from the motorcycle and be hurled into the air.

FIRST CONDITION FOR EQUILIBRIUM motion would

A body is in translational equilibrium if and only if the vector sum of the forces acting upon it is zero.

Σ Fx = 0 Σ Fy = 0

60 motion would 0

300

50 N

Example of Free Body Diagram

B

By

A

A

B

Ay

300

600

Ax

Bx

Fg

1. Draw and label a sketch.

2. Draw and label vector force diagram. (FBD)

3. Label x and y components opposite and adjacent to angles.

2.4 motion would A block of weight 50 N hangs from a cord that is knotted to two other cords, A and B fastened to the ceiling. If cord B makes an angle of 60˚ with the ceiling and cord A forms a 30°angle, draw the free body diagram of the knot and find the tensions A and B.

A

B

ΣFx = B cos 60º - A cos 30º = 0

By

Ay

30º

60º

Ax

Bx

= 1.73 A

ΣFy = B sin 60º + A sin 30º - 50 = 0

1.73 A sin 60º + A sin 30º = 50

1.5 A + 0.5 A = 50

A = 25 N

B = 1.73 (25) = 43.3 N

A = 25 NB = 43.3 N

50 N

N1L

2.5 motion would A 200 N block rests on a frictionless inclined plane of slope angle 30º. A cord attached to the block passes over a frictionless pulley at the top of the plane and is attached to a second block. What is the weight of the second block if the system is in equilibrium?

FN

FT

FT

N1L

y

θ

x

FG2

200 N

ΣFy = FT - FG = 0

FT = FG2

FG2 = 100 N

ΣFx = FT - 200 sin 30º= 0

FT = 200 sin 30º

= 100 N

To swing open a door, you exert a force. motion would

The doorknob is near the outer edge of the door. You exert the force on the doorknob at right angles to the door, away from the hinges.

To get the most effect from the least force, you exert the force as far from the axis of rotation (imaginary line through the hinges) as possible.

TORQUE motion would

Torque is a measure of a force's ability to rotate an object.

Each of the motion would 20-N forces has a different torque due to the direction of force.

Direction of Force

20 N

q

20 N

q

20 N

Magnitude of force

The 40-N force produces twice the torque as does the 20-N force.

Location of force

20 N

40 N

The forces nearer the end of the wrench have greater torques.

20 N

20 N

20 N

Torque is Determined by Three Factors:

• The magnitude of the applied force.

• The direction of the applied force.

• The location of the applied force.

The perpendicular distance from the axis of rotation to the line of force is called the lever arm of that force. It is the lever arm that determines the effectiveness of a given force in causing rotational motion. If the line of action ofa force passes through the axis of rotation (A) the lever arm is zero.

6 line of force is called the cm

40 N

Torque

Depends on the magnitude of the applied force and on the length of the lever arm, according to the following equation.

r is measured perpendicular to the line of action of the force F

Units for Torque

Units: Nm

t = Fr

t = (40 N)(0.60 m)

= 24.0 Nm

Applying a Torque line of force is called the

Sign Convention: line of force is called the

Torque will be positive if F tends to produce counterclockwise rotation.

Torque will be negative if F tends to produce clockwise rotation.

ROTATIONAL EQUILIBRIUM line of force is called the

An object is in rotational equilibriumwhen the sum of the forces and torques acting on it is zero.

First Condition of Equilibrium:

Σ Fx = 0 and Σ Fy = 0

(translationalequilibrium)

Second Condition of Equilibrium:

Στ = 0

(rotationalequilibrium)

By choosing the axis of rotation at the point of application of an unknown force, problems may

be simplified.

CENTER OF MASS line of force is called the

The terms "center of mass" and "center of gravity" are used synonymously in a uniform gravity field to represent the unique point in an object or system that can be used to describe the system's response to external forces and torques.

The concept of the center of mass is that of an average of the masses factored by their distances from a reference point. In one plane, that is like the balancing of a seesaw about a pivot point with respect to the torques produced.

Center of Gravity line of force is called the

The center of gravityof an object is the point at which all the weight of an object might be considered as acting for purposes of treating forces and torques that affect the object.

The single support force has line of action that passes through the c. g. in any orientation.

Examples of Center of Gravity line of force is called the

Note: C. of G. is not always inside material.

2.6 line of force is called the A 300 N girl and a 400 N boy stand on a 16 m platform supported by posts A and B as shown. The platform itself weighs 200 N. What are the forces exerted by the supports on the platform?

ΣF = line of force is called the 0

A + B - 300 - 200 - 400 = 0

A + B = 900 N

Selecting B as the hinge

ΣτB = 0

-A(12) +300(10) +200(4) - 400(4) = 0

- A12 + 3000 + 800 - 1600 = 0

= 183. 3 N

B = 900 - 183.3

= 716.7 N

Selecting line of force is called the A as the hinge

ΣτA = 0

- 300(2) - 200(8) + B(12) - 400(16) = 0

- 600 - 1600 + B12 - 6400 = 0

= 716.7 N

A = 900 - 716.7

= 183. 3 N

2.7 line of force is called the A uniform beam of negligible weight is held up by two supports A and B. Given the distances and forces listed, find

the forces exerted by the supports.

ΣF= 0

A - 60 - 40 + B = 0

A + B = 100

ΣτA = 0

= - 60 (3) - 40(9) + B(11) = 0

A

B

3 m

6 m

2 m

= 49.1 N

A = 100 - 49.1 = 50.9 N

60 N

40 N