ES 1021 - Assignment 2 Part A PDF

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Tutorial Assignment # 2 - Part (A) Due: 10:20am on Thursday, October 24, 2019 You will receive no credit for items you complete after the assignment is due. Grading Policy

Vector Cross Product Learning Goal: To understand the rules for computing cross products. Let vectors:

,

, and

.

Part A - Cross product of two vectors, B and C Calculate

.

Express the components numerically separated by commas.

Hint 1. The cross product If

and

, then

ANSWER: = 4,5,-17

Correct

Part B - Cross product of two vectors, C and B Calculate

.

Express the components numerically separated by commas. ANSWER: = -4,-5,17

Correct

Part C - Cross product of two vectors, 2B and 3C Calculate

.

Express the components numerically separated by commas. ANSWER: = 24,30,-102

Correct

Typesetting math: 90%

Part D - Vector triple product Calculate

.

Express the components numerically separated by commas. ANSWER: = 15,5,5

Correct

Part E - Scalar triple product Calculate

.

Express your answer numerically. ANSWER: = 55

Correct

Let

and

be different vectors with lengths

and

, respectively.

Part F - Magnitude of the cross product of two perpendicular vectors If

and

are perpendicular, calculate

Express your answer in terms of

and

. .

Hint 1. The angle between perpendicular vectors The angle between perpendicular vectors is equal to

radians or 90 degrees.

Hint 2. Magnitude of the cross product , where

is the length of

,

is the length of

ANSWER: =

Correct

Part G - Magnitude of the cross product of two parallel vectors If

and

are parallel, calculate

Typesetting math: 90%

Express your answer numerically.

.

, and

is the angle between

and

.

Hint 1. The angle between two parallel vectors The angle between parallel vectors is equal to 0.

ANSWER: = 0

Correct

± Moment of a Force: Vector Formulation Learning Goal: To understand the concept of moment of a force and how to calculate it using a vector formulation. A man wishes to spin a merry-go-round in the x–y plane centered at the origin upward and cannot stand on the merry-go-round, he must apply a force in the direction of the unit vector . He pulls with a force = 49.0 from point A [ ].

. Because he must pull

Part A What are the x, y, and z components of the resulting moment acting on the merry-go-round about the origin? Express your answers numerically in pound-feet to three significant figures separated by commas.

Hint 1. How to approach the problem The moment of a force

about a point O can be expressed using the vector cross product as

where represents a position vector drawn from O to any point lying on the line of action of be taking the cross product of the two vectors, not multiplying them.

. Note that you will

Hint 2. Find the components of What are the x, y, and z components of ? Express your answers numerically in feet to three significant figures separated by commas. ANSWER: Typesetting math: 90%

,

,

= 3.00,4.00,0.000

Hint 3. Find the components of the force vector What are the x, y, and z components of the force vector

?

Express your answers numerically in pounds to three significant figures separated by commas.

Hint 1. How to find the components of the force vector Multiply the unit vector follows:

by the force's magnitude

to obtain the components of the force vector

as

ANSWER: ,

,

= -16.3,32.7,32.7

Hint 4. Find an expression for the x, y, and z components of the moment Given a position vector , , and in terms of

,

,

,

Express your answers in terms of

,

, ,

and a force of , and ? ,

,

, and

, what is the expression for

separated by commas.

Hint 1. Finding the cross product of two vectors The cross product of two vectors calculated by using the compact determinant form:

and

Expand using the three minors and multiply by the unit vector from the top row as follows

Typesetting math: 90%

can be

ANSWER: ,

,

=

,

,

ANSWER: ,

,

= 131,-98.0,163

Correct Because a merry-go-round is free to rotate about the z axis, a moment about the x or y axis is easily overlooked. Just because a body is not free to rotate in a direction does not mean that a moment's component does not exist in that direction. For example, a moment about the x or y axis on a poorly designed merry-goround can cause the merry-go-round to wobble up and down.

Part B Find

, the magnitude of the moment, and , the angle between

and the force

.

Express your answers in pound-feet and degrees to three significant figures separated by a comma.

Hint 1. How to approach the problem Find , the magnitude of the moment, and , the magnitude of the position vector. Use magnitude , and the relationship

to find the angle

, , the given force's

between the moment arm and the force.

Hint 2. Find the magnitude of What is the magnitude of the moment

?

Express your answer numerically in pound-feet to four significant figures.

Hint 1. How to find the magnitude of a three-dimensional vector The magnitude of a three-dimensional vector can be determined by taking the square root of the sum of the squares of each component as follows:

ANSWER: = 231.0

Hint 3. Find the magnitude of What is the magnitude of the position vector Express your answer numerically in feet to four significant figures. Typesetting math: 90%

?

Hint 1. How to find the magnitude of a three-dimensional vector The magnitude of a three-dimensional vector can be determined by taking the square root of the sum of the squares of each component as follows

ANSWER: = 5.000

Hint 4. Find an expression for What is an expression for

in terms of ,

Express

, and

in terms of ,

, and

?

.

ANSWER:

=

ANSWER: , = 231,70.5

,

Correct

± Moment of a Force: Scalar Formulation Learning Goal: To understand the concept of moment of a force and how to calculate it using a scalar formulation. The magnitude of the moment of a force with a magnitude around a point O is defined as follows: where d is the force's moment arm. The moment arm is the perpendicular distance from the axis at point O to the force's line of action.

Part A A stool at a restaurant is anchored to the floor. When a customer is in the process of sitting down, a horizontal force with magnitude is exerted at the top of the stool support as shown in the figure. When the customer is seated, a vertical force with magnitude is exerted on the stool support. If the maximum moment magnitude that the stool support can sustain about point A is = 160 , what is the maximum height that the stool can have if the magnitudes of the two forces are = 65.0 and = 140 ? Assume that moments acting counterclockwise about point A are positive whereas moments acting clockwise about A are negative. Express your answer numerically in feet to three significant figures.

Typesetting math: 90%

Hint 1. How to approach the problem The magnitude of the moment of force about a point O is defined as where d is the moment arm and is the force's magnitude. The moment arm is the perpendicular distance from the axis at point O to the force's line of action. The units of moment of force are newton-meters ( ) or pound-feet ( ). For this coordinate system, the moment is positive if it would cause counterclockwise rotation about point O and negative if it would cause clockwise rotation about point O. Hint 2. Find an expression for What is an expression for , the maximum height of the stool in terms of and the forces' magnitudes and ? Express your answer in terms some or all of the variables

,

, the maximum moment magnitude,

, and

Hint 1. Identify the moment due to the force What is ANSWER:

ANSWER:

=

ANSWER: = 2.46 Typesetting math: 90%

, the moment magnitude about point A due to the force

?

.

Correct

Part B As shown, a pipe is anchored to a wall at point A. During the pipe's installation, several forces are applied to the pipe at different locations. If = 12.7 , = 10.0 , = 16.8 , = 16.9 , = 0.350 , = 0.650 , and = 0.700 , what is , the net moment about point A due to these forces? Assume that moments acting counterclockwise about point A are positive whereas moments acting clockwise are negative. Express your answer numerically in pound-feet to three significant figures.

Hint 1. How to approach the problem In this problem, all the moment vectors are collinear because all the forces acting on the object are coplanar. Thus, the resultant moment of the system can be determined by adding the moments of all forces. When adding the moments, keep in mind that a moment acting in the counterclockwise direction is positive whereas a moment acting in the clockwise direction is negative; therefore, the overall sign of indicates the direction of the net moment. Hint 2. Find the moment about point A of What is the moment about point A of the force ? Assume that moments acting in the counterclockwise direction about point A are positive whereas moments acting in the clockwise direction are negative.

Express your answer numerically in pound-feet to three significant figures.

Hint 1. Identify the moment arm of Which distance is the moment arm about A of the force ANSWER: Typesetting math: 90%

?

0.000

ANSWER: = 8.26

Hint 3. Find the moment about point A of What is the moment about point A of the force ? Assume that moments acting in the counterclockwise direction about point A are positive whereas moments acting in the clockwise direction are negative.

Express your answer numerically in pound-feet to three significant figures.

Hint 1. Identify the moment arm of Which distance is the moment arm about A of the force ANSWER:

0.000

ANSWER: Typesetting math: 90%

?

= -6.50

Hint 4. Find the moment about point A of What is the moment about point A of the force ? Assume that moments acting in the counterclockwise direction about point A are positive whereas moments acting in the clockwise direction are negative.

Express your answer numerically in pound-feet to three significant figures.

Hint 1. Identify the moment arm of Which distance is the moment arm about A of the force

?

ANSWER:

0.000

ANSWER: = 0.000

Hint 5. Find the moment about point A of What is the moment about point A of the force ? Assume that moments acting in the counterclockwise direction about point A are positive whereas moments acting in the clockwise direction are negative.

Typesetting math: 90%

Express your answer numerically in pound-feet to three significant figures.

Hint 1. Identify the moment arm of Which distance is the moment arm about A of the force

?

ANSWER:

0.000

ANSWER: = -5.92

ANSWER: = -4.16

Correct

Part C As shown, a server at a restaurant is carrying a tray that has a glass of water resting on it. The glass and tray exert a force with magnitude = 4.80 on the server's hand. If = 1.25 , = 1.10 , = 0.250 , = 165 , and = 120 , what is the moment of force about the shoulder joint at point A? Assume that moments acting in the counterclockwise direction about point A are positive whereas moments acting in the clockwise direction are negative. Express your answer numerically in pound-feet to three significant figures. Typesetting math: 90%

Hint 1. How to approach the problem The magnitude of the moment of force with magnitude

about a point O is defined as

where d is the moment arm. The moment arm is the perpendicular distance from the axis at point O to the force's line of action. The units of the moment are newton-meters ( ) or pound-feet ( ). For this coordinate system, the moment is positive if it would cause counterclockwise rotation about point O and negative if it would cause clockwise rotation about point O. The challenge here is in determining the moment arm. Because the force's line of action is in the vertical direction, the moment arm is a horizontal distance. Hint 2. Find the moment arm about point A of What is , the moment arm associated with the moment about the shoulder joint from force

?

Express your answer numerically in feet to four significant figures.

Hint 1. How to find the moment arm Because the force's line of action is in the vertical direction, the moment arm must be a horizontal distance. The moment arm is the distance in the x direction between the shoulder joint and the point upon which the force acts. This distance is the sum of the x components of the upper arm, lower arm, and hand. You are given the angles between the arm segments and the horizontal. Use the cosine function to find each x component:

Hint 2. Find the x component of the biceps length What is the x component of the biceps segment of the arm, given that the segment's length is 1.25 and that the angle between the segment and the positive x axis is 165 ? If necessary, indicate the direction of the x component by including a negative sign.

Typesetting math: 90%

Express your answer numerically in feet to four significant figures. ANSWER: = -1.207

Hint 3. Find the x component of the forearm's length What is the x component of the forearm, given that the forearm's length is 1.10 and that the angle between the forearm and the negative x axis is 120 ? If necessary, indicate the direction of the x component by including a negative sign.

Express your answer numerically in feet to four significant figures. ANSWER: = 0.5500

Hint 4. Find the x component of the hand's length What is the x component of the hand, given that the hand's length is 0.250 direction of the x component by including a negative sign.

Typesetting math: 90%

? If necessary, indicate the

Express your answer numerically in feet to four significant figures. ANSWER: = 0.2500

ANSWER: = 0.4074

ANSWER: = 1.96

Correct

± Principle of Moments Learning Goal: To apply the principle of moments and the principle of transmissibility. As shown, a rope is attached to a high shed that is to be relocated. A man pulls on the end of the rope at point A; the rope is attached to the shed at point B. As the man pulls on the rope, it creates an angle with the horizontal. The end of the rope is located at from the shed and off the ground. The man pulls on the rope with a force of magnitude .

Typesetting math: 90%

Part A What is , the contribution to the moment about point O made by the x component of the force at point A? What is , the contribution to the moment about point O made by the y component of the force at point A? What is the total moment due to the force about point O? Assume that moments acting counterclockwise about point O are positive whereas moments acting clockwise are negative. Express your answers numerically in pound-feet to three significant figures separated by commas.

Hint 1. How to approach the problem The principle of moments simplifies the process of calculating a moment by separating a force into its components. To calculate the total moment, add the moments that result from the separate components: Here, and are the x and y components of the force . Assume that moments acting counterclockwise about point O are positive whereas moments acting clockwise are negative. Hint 2. Find an expression for What is the contribution to the total moment about point O from the x component of the force at point A in terms of the force , the angle , and the distances and ?

Express your answer in terms some or all of the variables

Typesetting math:1.90% Hint Find

an expression for

, , , and .

What is , the x component of the force, in terms of the force's magnitude Express your answer in terms of and .

and the angle ?

ANSWER: =

Hint 2. Find the moment arm of the x component of the force What is the moment arm for

, the x component of the force

?

ANSWER:

ANSWER: =

Hint 3. Find an expression for What is the contribution to the total moment about point O from the y component of the force at point A in terms of

the force

, the angle , and the distances

and ?

Express your answer in terms of some or all of the variables

, , , and .

Hint 1. Find an expression for What is the y component of the force Express your answer in terms of ANSWER: Typesetting math: 90%

in terms of the force's magnitude and .

and the angle ?

=

Hint 2. Find the moment arm of the y component of the force What is the moment arm for

, the y component of the force

?

ANSWER:

ANSWER: =

Hint 4. Find \sin\theta and \cos\theta The values of

and

can be calculated using the given lengths. What are

and

Express your answers numerically to four significant figures separated by a comma.

Hint 1. Find an expression for What is

in terms of the known lengths , , and ?

Express your answer in terms of , , and .

Hint 1. Right-angle trigonometry Recall that the functions sine, cosine, and tangent relate the sides of a right triangle.

Therefore

,

,

, and

ANSWER:

=

Hint 2. Find an expression for Typesetting math: 90%

What is

in terms of the known lengths , , and ?

.

?

Express your answer in terms of , , and . Hint 1. Right-angle trigonometry Recall that the functions sine, cosine, and tangent relate the sides of a right triangle.

Therefore

,

,

, and

.

ANSWER:

=

ANSWER: ,

= 0.5190,0.8548

ANSWER: ,

,

= 165,254,419

All attempts used; correct answer displayed

Part B Using the principle of transmissibility, the force can be slid along its line of action from point A to any other point along its line of action. What po...