07 Curvilinear Normal Tangential Concept Overview
NORMAL AND TANGENTIAL COMPONENTS | CONCEPT OVERVIEW The TOPIC of NORMAL AND TANGENTIAL COMPONENTS can be referenced under the main SUBJECT of DYNAMICS, and more specifically in the section titled PARTICLE KINEMATICS, on PAGE 72 and 73 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
CONCEPT INTRO: Recall from our previous reviews that when analyzing a PARTICLE in motion within a THREE DIMENSIONAL RECTANGULAR space, the POSITION can be defined with the POSITION VECTOR: π = π₯π + π¦π + π§π This particle will have an INSTANTEOUS VELOCITY modeled as:
π£=
ππ = π₯π + π¦π + π§π ππ‘
As well as an INSTANTEOUS ACCELERATION expressed as: ππ£ π . π π= = = π₯π + π¦π + π§π ππ‘ ππ‘ .
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For more review in this area of PARTICLE MOTION, reference the PARTICLE KINEMATICS CONCEPT OVERVIEW and associated PROBLEM SET. Other relevant information regarding these formulas can also be referenced under the SUBJECT of DYNAMICS on page 72 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Although we most likely will be working in a TWO DIMENSIONAL SPACE, the POSITION, r, VELOCITY, v, and ACCELERATION, a, are all THREE DIMENSIONAL CARTESIAN VECTORS. However, even if we are working in a THREE DIMENSIONAL SPACE initially, we will observe shortly that the ANALYSIS of the NORMAL and TANGENTIAL COMPONENTS will be done entirely in the context of a TWO DIMENSIONAL SPACE. Itβs important to NOTE also that the VELOCITY of the PARTICLE at any point is TANGENT to the PATH of MOTION. Under STRAIGHT LINE MOTION, this DIRECTION is OBVIOUS. However, under CURVILINEAR MOTION, this DIRECTION may not always be evident right off the bat, but will ALWAYS be TANGENT to the PATH at that particular moment in time. So letβs develop our knowledge around PARTICLE CURVILINEAR MOTION a little bit more.
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SEGMENTING A CURVED PATH: When working with any CURVED PATH, we are able to break it up in to equal segments of curves having equal length. As these segments get smaller and smaller, each segment approaches an ARC, which is a SEGMENT OF A CIRCLEβ¦and a circle, as we know, always falls within a TWO DIMENSIONAL PLANE. Given a random CURVILINEAR PATH, each segment along that PATH will have its own unique ARC, tied back to UNIQUE GEOMETRY of some TWO DIMENSIONAL CIRCLE. Thatβs an IMPORTANT NOTE, each ARC will not be the same, but have some UNIQUE GEOMETRY attached to it. Working within a THREE DIMENSIONAL SPACE and analyzing some PARTICLE undergoing CURVILINEAR MOTION, each curve will have a number of different ARCS associated with it, all of which fall within their own TWO DIMENSIONAL PLANE. To reiterate that POINT, any PARTICLE undergoing CURVILINEAR MOTION in a THREE DIMENSIONAL SPACE will be composed of a series of ARCS that layout in their own TWO DIMENSIONAL PLANEβ¦and with this, can be measured individually, fairly quickly and painlessly with TWO DIMENSIONAL GEOMETRY. This plane that each individual ARC along the PATH is falling within is referred to as the OSCULATING PLANE.
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An OSCULATING PLANE refers to the PLANE that contains the small ARC PATH and changes at a different location and different period of time in a PARTICLES overall MOTION. This is not a theory that is outright covered in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. However, it is concept that must be understood due to the clarity it gives to us when analyzing PARTICLE CURVILINEAR MOTION. This is a significant definition because that means when we are working with a PARTICLE undergoing THREE DIMENSIONAL MOTION, we can break down that motion in to smaller sections and analyze each using TWO DIMENSIONAL GEOMETRIC RELATIONSHIPS. Fundamentally, we know that a CIRCLE always has some CENTERPOINT with an associated RADIUS. In the context of an ARC of a CIRCLE, the CENTERPOINT is referred to as the CENTER OF CURVATURE and the RADIUS is referred to as the RADIUS OF CURVATURE.
NORMAL AND TANGENTIAL AXIS: At any point along a CURVED PATH, we can define a set of AXES coming directly from the OBJECT, or PARTICLE, itself.
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These AXES are referred to as the TANGENTIAL AXIS (T-AXIS), which is TANGENT to the ARC, and the NORMAL AXIS (N-AXIS), which is pointing towards the CENTER OF CURVATURE and is NORMAL to the ARC. To illustrate this, generally we will have:
With this set of AXES established at the POSITION of the PARTICLE, we can define the MOTION VECTORS of the PARTICLE using the NORMAL and TANGENTIAL COMPONENTS instead of the X, Y, and Z RECTANGULAR COMPONENTS. For a PARTICLE undergoing CURVILINEAR MOTION, in a short period of TIME, it changes POSITION from π( π₯2 , π¦2 , π§2 ) to π( π₯. , π¦. , π§. ), and the DISTANCE TRAVELED is the LENGTH of the ARC on this PATH, or otherwise, ππ . To reiterate: β’ At any given time, we can set up a pair of AXES, the TANGENTIAL and NORMAL AXIS, at the POSITION of the PARTICLE.
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β’ The TANGENTIAL AXIS runs TANGENT to the CURVE at the POSITION of the PARTICLE and always points in the DIRECTION of the MOTION, represented by the UNIT VECTOR, π7 . β’ The NORMAL AXIS runs NORMAL to the CURVE at the POSITION of the PARTICLE and always points inward towards the CENTER OF CURVATURE, represented by the UNIT VECTOR, π8 .
VELOCITY: We have learned that the VELOCITY VECTOR at the POSITION of a PARTICLE in MOTION is always TANGENT to and POINTING in the DIRECTION of MOTION. With that being said, and knowing the characteristics of the TANGENTIAL and NORMAL AXIS, this means we can write the VELOCITY as an expression using TANGENTIAL COMPONENTS as: π£ = π£(π‘)π7 The NCEES doesnβt do the best at explaining the differentiation of the VELOCITY variables being referenced in this formula, which is found on page 73 of the NCEES Reference Handbook, but π£(π‘) is exactly what we have defined it being up to this point, the TIME derivative of DISPLACEMENT at this particle point, or in formulaic terms:
π£(π‘) =
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ACCELERATION: ACCELERATION is defined as the RATE OF CHANGE in the VELOCITY of a PARTICLE in motion. In NORMAL and TANGENTIAL COMPONENT TERMS it will always be composed of both components. In formulaic terms: π = π7 π7 + π8 π7 The TANGENTIAL ACCLERATION is equal to:
π7 = π£ =
ππ£ ππ‘
The NORMAL ACCLERATION is equal to: π£7 . π8 = π Where: π£7 = MAGNITUDE of VELOCITY at this POSITION π = INSTANTANEOUS RADIUS OF CURVATURE
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Since the TANGENTIAL ACCELERATION and the NORMAL ACCELERATION components are PERPENDICULAR to one another, we can express the overall MAGNITUDE of the PARTICLE ACCELERATION at this POSITION as:
π=
π7 . + π8 .
Before we jump in to some application, here are some IMPORTANT NOTES when working with the NORMAL and TANGENTIAL COMPONENTS of a PARTICLE undergoing CURVILINEAR MOTION. Unlike your standard rectangular coordinates, where the AXES are fixed to the earth, the NORMAL and TANGENTIAL AXIS are DYNAMIC and are CONSTANTLY ADJUSTING to the GEOMETRY of the CURVE and the POSITION of the PARTICLEβ¦they move with the PARTICLE. One benefit in using N-T COMPONENTS is that the VELOCITY always has ONE SINGLE COMPONENT at the POSITION of the PARTICLE that is TANGENT to the CURVE and in the DIRECTION of MOTION. The TWO COMPONENTS of ACCELERATION have distinct meanings, π7 only describes the CHANGE in the MAGNITUDE of the VELOCITY, or the CHANGE in SPEED, while π8 only describes the CHANGE in DIRECTION of the VELOCITY. And because the TANGENTIAL ACCELERATION fully describes the CHANGE in the VELOCITY, we can apply the three BASIC KINEMATIC EQUATIONS along the TANGENTIAL direction as we would normally in RECTILINEAR MOTION.
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Recall that these BASIC KINEMATIC EQUATIONS are:
π£=
ππ ππ‘
π7 =
ππ£ ππ‘
π7 ππ = π£ ππ£
Similarly, as defined under RECTILINEAR MOTION, because the TANGENTIAL ACCELERATION fully describes the CHANGE in the VELOCITY, we can also apply the following formulas: π£ = π£: + π7 π‘ 1 π = π : + π£: π‘ + π7 π‘ . 2 π£ . = π£: . + 2 π7 π β π :
The NORMAL ACCELERATION, π8 , is also referred to as the CENTRIPETAL ACCELERATION and always points towards the CONCAVE side of the PATH of MOTION. If the CURVED PATH can be modeled by the function π¦ = π(π₯), then the RADIUS OF CURVATURE at any point along that CURVE can be defined using the formula: 1 + (ππ¦/ππ₯). π= π. π¦ ππ₯ .
@/.
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NORMAL AND TANGENTIAL COMPONENTS | CONCEPT OVERVIEW The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
PROBLEM: An object is travelling along a curved path as illustrated. If at the point shown, its speed is 28.2 m/s, which is increasing at a rate of 8 m/s . , the direction of the velocity, counterclockwise to the positive x-axis, is closest to:
A. 14.9Β° B. 23.1Β° C. 56.3Β° D. 76.2Β°
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SOLUTION: The TOPIC of NORMAL AND TANGENTIAL COMPONENTS can be referenced under the main SUBJECT of DYNAMICS, and more specifically in the section titled PARTICLE KINEMATICS, on PAGE 72 and 73 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem, we are given information about some object that is travelling along a CURVED PATH. We are told that at a particular point, the PARTICLE has the following characteristics: π£ = 28.2 m/s π = 8 m/s . With this DATA defined, we are asked to determine the DIRECTION of the VELOCITY, COUNTERCLOCKWISE to the POSITIVE X-AXIS. Letβs start our analysis by setting up some measures at the point in time that we are observing the motion of this PARTICLE. From our studies, we know that at any point along a CURVED PATH, we can define a set of AXES coming directly from the OBJECT, or PARTICLE, itself. These AXES are referred to as the TANGENTIAL AXIS (T-AXIS), which is TANGENT to the ARC, and the NORMAL AXIS (N-AXIS), which is pointing towards the CENTER OF CURVATURE and is NORMAL to the ARC.
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To illustrate this for our scenario, we have:
The TANGENTIAL AXIS runs TANGENT to the CURVE at the POSITION of the PARTICLE and always points in the DIRECTION of the MOTION. The NORMAL AXIS runs NORMAL to the CURVE at the POSITION of the PARTICLE and always points inward towards the CENTER OF CURVATURE.
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We arenβt concerned with the NORMAL AXIS at this point because we are looking for the ANGLE of the VELOCITY VECTOR which is always TANGENT to the CURVE and POINTING in the DIRECTION of MOTION, illustrated as:
To reiterate, the VELOCITY of the PARTICLE at any point is TANGENT to the PATH of MOTION and will always run along the T-AXIS. From here, it is simple DERIVATIVE and TRIGONOMETRY work. We know that when given a CURVE defined by some function π(π₯), that we can determine the SLOPE of that CURVE at any point by DEFINING its FIRST DERIVATIVE. The FIRST DERIVATIVE of a function tells us the RATE OF CHANGE at any point, which results in a LINE that is TANGENT with some unique SLOPE.
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This SLOPE is what we are after since it will allow us to determine the ANGLE in which the VELOCITY VECTOR of this PARTICLE makes at this point in its MOTION.
Letβs jump in to some calculations. We are told that this PARTICLE follows a PATH of MOTION defined by the FUNCTION:
π¦=
1 @ π₯. 4
Taking the FIRST DERIVATIVE, we get:
π¦L =
ππ¦ 3 2 = π₯. ππ₯ 8
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Analyzing this function at the point of π₯ = 16, we find that:
π¦L =
2 3 (16). = 1.5 8
This tells us that our PARTICLE has a VELOCITY VECTOR that is TANGENT to the CURVED PATH with a SLOPE of 1.5. Turning now to our knowledge of TRIGONOMETRIC IDENTITIES, and specifically the TANGENT FUNCTION, we know that given a RIGHT TRIANGLE with the standard geometry of X and Y, we can define any ANGLE as:
tan(π) =
π¦ π₯
The X-VALUE represents the RUN of our TANGENT LINE at this point and the YVALUE represents the RISE. The RISE over RUN also represents the SLOPE of a LINE, which we have definedβ¦so with that:
tan(π) =
π¦ = 1.5 π₯
Rearranging to isolate our ANGLE, we get: π = tanQ2 (1.5)
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Or: π = 56.3Β° Therefore, the DIRECTION of the VELOCITY VECTOR, COUNTERCLOCKWISE to the POSITIVE X-AXIS, is 56.3Β°. The correct answer choice is C. ππ. πΒ°
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CONCEPT INTRO: Recall from our previous reviews that when analyzing a PARTICLE in motion within a THREE DIMENSIONAL RECTANGULAR space, the POSITION can be defined with the POSITION VECTOR: π = π₯π + π¦π + π§π This particle will have an INSTANTEOUS VELOCITY modeled as:
π£=
ππ = π₯π + π¦π + π§π ππ‘
As well as an INSTANTEOUS ACCELERATION expressed as: ππ£ π . π π= = = π₯π + π¦π + π§π ππ‘ ππ‘ .
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For more review in this area of PARTICLE MOTION, reference the PARTICLE KINEMATICS CONCEPT OVERVIEW and associated PROBLEM SET. Other relevant information regarding these formulas can also be referenced under the SUBJECT of DYNAMICS on page 72 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Although we most likely will be working in a TWO DIMENSIONAL SPACE, the POSITION, r, VELOCITY, v, and ACCELERATION, a, are all THREE DIMENSIONAL CARTESIAN VECTORS. However, even if we are working in a THREE DIMENSIONAL SPACE initially, we will observe shortly that the ANALYSIS of the NORMAL and TANGENTIAL COMPONENTS will be done entirely in the context of a TWO DIMENSIONAL SPACE. Itβs important to NOTE also that the VELOCITY of the PARTICLE at any point is TANGENT to the PATH of MOTION. Under STRAIGHT LINE MOTION, this DIRECTION is OBVIOUS. However, under CURVILINEAR MOTION, this DIRECTION may not always be evident right off the bat, but will ALWAYS be TANGENT to the PATH at that particular moment in time. So letβs develop our knowledge around PARTICLE CURVILINEAR MOTION a little bit more.
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SEGMENTING A CURVED PATH: When working with any CURVED PATH, we are able to break it up in to equal segments of curves having equal length. As these segments get smaller and smaller, each segment approaches an ARC, which is a SEGMENT OF A CIRCLEβ¦and a circle, as we know, always falls within a TWO DIMENSIONAL PLANE. Given a random CURVILINEAR PATH, each segment along that PATH will have its own unique ARC, tied back to UNIQUE GEOMETRY of some TWO DIMENSIONAL CIRCLE. Thatβs an IMPORTANT NOTE, each ARC will not be the same, but have some UNIQUE GEOMETRY attached to it. Working within a THREE DIMENSIONAL SPACE and analyzing some PARTICLE undergoing CURVILINEAR MOTION, each curve will have a number of different ARCS associated with it, all of which fall within their own TWO DIMENSIONAL PLANE. To reiterate that POINT, any PARTICLE undergoing CURVILINEAR MOTION in a THREE DIMENSIONAL SPACE will be composed of a series of ARCS that layout in their own TWO DIMENSIONAL PLANEβ¦and with this, can be measured individually, fairly quickly and painlessly with TWO DIMENSIONAL GEOMETRY. This plane that each individual ARC along the PATH is falling within is referred to as the OSCULATING PLANE.
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An OSCULATING PLANE refers to the PLANE that contains the small ARC PATH and changes at a different location and different period of time in a PARTICLES overall MOTION. This is not a theory that is outright covered in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. However, it is concept that must be understood due to the clarity it gives to us when analyzing PARTICLE CURVILINEAR MOTION. This is a significant definition because that means when we are working with a PARTICLE undergoing THREE DIMENSIONAL MOTION, we can break down that motion in to smaller sections and analyze each using TWO DIMENSIONAL GEOMETRIC RELATIONSHIPS. Fundamentally, we know that a CIRCLE always has some CENTERPOINT with an associated RADIUS. In the context of an ARC of a CIRCLE, the CENTERPOINT is referred to as the CENTER OF CURVATURE and the RADIUS is referred to as the RADIUS OF CURVATURE.
NORMAL AND TANGENTIAL AXIS: At any point along a CURVED PATH, we can define a set of AXES coming directly from the OBJECT, or PARTICLE, itself.
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These AXES are referred to as the TANGENTIAL AXIS (T-AXIS), which is TANGENT to the ARC, and the NORMAL AXIS (N-AXIS), which is pointing towards the CENTER OF CURVATURE and is NORMAL to the ARC. To illustrate this, generally we will have:
With this set of AXES established at the POSITION of the PARTICLE, we can define the MOTION VECTORS of the PARTICLE using the NORMAL and TANGENTIAL COMPONENTS instead of the X, Y, and Z RECTANGULAR COMPONENTS. For a PARTICLE undergoing CURVILINEAR MOTION, in a short period of TIME, it changes POSITION from π( π₯2 , π¦2 , π§2 ) to π( π₯. , π¦. , π§. ), and the DISTANCE TRAVELED is the LENGTH of the ARC on this PATH, or otherwise, ππ . To reiterate: β’ At any given time, we can set up a pair of AXES, the TANGENTIAL and NORMAL AXIS, at the POSITION of the PARTICLE.
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β’ The TANGENTIAL AXIS runs TANGENT to the CURVE at the POSITION of the PARTICLE and always points in the DIRECTION of the MOTION, represented by the UNIT VECTOR, π7 . β’ The NORMAL AXIS runs NORMAL to the CURVE at the POSITION of the PARTICLE and always points inward towards the CENTER OF CURVATURE, represented by the UNIT VECTOR, π8 .
VELOCITY: We have learned that the VELOCITY VECTOR at the POSITION of a PARTICLE in MOTION is always TANGENT to and POINTING in the DIRECTION of MOTION. With that being said, and knowing the characteristics of the TANGENTIAL and NORMAL AXIS, this means we can write the VELOCITY as an expression using TANGENTIAL COMPONENTS as: π£ = π£(π‘)π7 The NCEES doesnβt do the best at explaining the differentiation of the VELOCITY variables being referenced in this formula, which is found on page 73 of the NCEES Reference Handbook, but π£(π‘) is exactly what we have defined it being up to this point, the TIME derivative of DISPLACEMENT at this particle point, or in formulaic terms:
π£(π‘) =
ππ ππ‘ Made with
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ACCELERATION: ACCELERATION is defined as the RATE OF CHANGE in the VELOCITY of a PARTICLE in motion. In NORMAL and TANGENTIAL COMPONENT TERMS it will always be composed of both components. In formulaic terms: π = π7 π7 + π8 π7 The TANGENTIAL ACCLERATION is equal to:
π7 = π£ =
ππ£ ππ‘
The NORMAL ACCLERATION is equal to: π£7 . π8 = π Where: π£7 = MAGNITUDE of VELOCITY at this POSITION π = INSTANTANEOUS RADIUS OF CURVATURE
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Since the TANGENTIAL ACCELERATION and the NORMAL ACCELERATION components are PERPENDICULAR to one another, we can express the overall MAGNITUDE of the PARTICLE ACCELERATION at this POSITION as:
π=
π7 . + π8 .
Before we jump in to some application, here are some IMPORTANT NOTES when working with the NORMAL and TANGENTIAL COMPONENTS of a PARTICLE undergoing CURVILINEAR MOTION. Unlike your standard rectangular coordinates, where the AXES are fixed to the earth, the NORMAL and TANGENTIAL AXIS are DYNAMIC and are CONSTANTLY ADJUSTING to the GEOMETRY of the CURVE and the POSITION of the PARTICLEβ¦they move with the PARTICLE. One benefit in using N-T COMPONENTS is that the VELOCITY always has ONE SINGLE COMPONENT at the POSITION of the PARTICLE that is TANGENT to the CURVE and in the DIRECTION of MOTION. The TWO COMPONENTS of ACCELERATION have distinct meanings, π7 only describes the CHANGE in the MAGNITUDE of the VELOCITY, or the CHANGE in SPEED, while π8 only describes the CHANGE in DIRECTION of the VELOCITY. And because the TANGENTIAL ACCELERATION fully describes the CHANGE in the VELOCITY, we can apply the three BASIC KINEMATIC EQUATIONS along the TANGENTIAL direction as we would normally in RECTILINEAR MOTION.
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Recall that these BASIC KINEMATIC EQUATIONS are:
π£=
ππ ππ‘
π7 =
ππ£ ππ‘
π7 ππ = π£ ππ£
Similarly, as defined under RECTILINEAR MOTION, because the TANGENTIAL ACCELERATION fully describes the CHANGE in the VELOCITY, we can also apply the following formulas: π£ = π£: + π7 π‘ 1 π = π : + π£: π‘ + π7 π‘ . 2 π£ . = π£: . + 2 π7 π β π :
The NORMAL ACCELERATION, π8 , is also referred to as the CENTRIPETAL ACCELERATION and always points towards the CONCAVE side of the PATH of MOTION. If the CURVED PATH can be modeled by the function π¦ = π(π₯), then the RADIUS OF CURVATURE at any point along that CURVE can be defined using the formula: 1 + (ππ¦/ππ₯). π= π. π¦ ππ₯ .
@/.
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NORMAL AND TANGENTIAL COMPONENTS | CONCEPT OVERVIEW The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
PROBLEM: An object is travelling along a curved path as illustrated. If at the point shown, its speed is 28.2 m/s, which is increasing at a rate of 8 m/s . , the direction of the velocity, counterclockwise to the positive x-axis, is closest to:
A. 14.9Β° B. 23.1Β° C. 56.3Β° D. 76.2Β°
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SOLUTION: The TOPIC of NORMAL AND TANGENTIAL COMPONENTS can be referenced under the main SUBJECT of DYNAMICS, and more specifically in the section titled PARTICLE KINEMATICS, on PAGE 72 and 73 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem, we are given information about some object that is travelling along a CURVED PATH. We are told that at a particular point, the PARTICLE has the following characteristics: π£ = 28.2 m/s π = 8 m/s . With this DATA defined, we are asked to determine the DIRECTION of the VELOCITY, COUNTERCLOCKWISE to the POSITIVE X-AXIS. Letβs start our analysis by setting up some measures at the point in time that we are observing the motion of this PARTICLE. From our studies, we know that at any point along a CURVED PATH, we can define a set of AXES coming directly from the OBJECT, or PARTICLE, itself. These AXES are referred to as the TANGENTIAL AXIS (T-AXIS), which is TANGENT to the ARC, and the NORMAL AXIS (N-AXIS), which is pointing towards the CENTER OF CURVATURE and is NORMAL to the ARC.
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To illustrate this for our scenario, we have:
The TANGENTIAL AXIS runs TANGENT to the CURVE at the POSITION of the PARTICLE and always points in the DIRECTION of the MOTION. The NORMAL AXIS runs NORMAL to the CURVE at the POSITION of the PARTICLE and always points inward towards the CENTER OF CURVATURE.
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We arenβt concerned with the NORMAL AXIS at this point because we are looking for the ANGLE of the VELOCITY VECTOR which is always TANGENT to the CURVE and POINTING in the DIRECTION of MOTION, illustrated as:
To reiterate, the VELOCITY of the PARTICLE at any point is TANGENT to the PATH of MOTION and will always run along the T-AXIS. From here, it is simple DERIVATIVE and TRIGONOMETRY work. We know that when given a CURVE defined by some function π(π₯), that we can determine the SLOPE of that CURVE at any point by DEFINING its FIRST DERIVATIVE. The FIRST DERIVATIVE of a function tells us the RATE OF CHANGE at any point, which results in a LINE that is TANGENT with some unique SLOPE.
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This SLOPE is what we are after since it will allow us to determine the ANGLE in which the VELOCITY VECTOR of this PARTICLE makes at this point in its MOTION.
Letβs jump in to some calculations. We are told that this PARTICLE follows a PATH of MOTION defined by the FUNCTION:
π¦=
1 @ π₯. 4
Taking the FIRST DERIVATIVE, we get:
π¦L =
ππ¦ 3 2 = π₯. ππ₯ 8
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Analyzing this function at the point of π₯ = 16, we find that:
π¦L =
2 3 (16). = 1.5 8
This tells us that our PARTICLE has a VELOCITY VECTOR that is TANGENT to the CURVED PATH with a SLOPE of 1.5. Turning now to our knowledge of TRIGONOMETRIC IDENTITIES, and specifically the TANGENT FUNCTION, we know that given a RIGHT TRIANGLE with the standard geometry of X and Y, we can define any ANGLE as:
tan(π) =
π¦ π₯
The X-VALUE represents the RUN of our TANGENT LINE at this point and the YVALUE represents the RISE. The RISE over RUN also represents the SLOPE of a LINE, which we have definedβ¦so with that:
tan(π) =
π¦ = 1.5 π₯
Rearranging to isolate our ANGLE, we get: π = tanQ2 (1.5)
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Or: π = 56.3Β° Therefore, the DIRECTION of the VELOCITY VECTOR, COUNTERCLOCKWISE to the POSITIVE X-AXIS, is 56.3Β°. The correct answer choice is C. ππ. πΒ°
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