Find the components of the vertical force Bold Upper FFequals=left angle 0 comma negative 4 right angle0,−4 in the directions parallel to and normal to the plane that makes an angle of StartFraction pi Over 3 EndFraction π 3 with the positive​ x-axis. Show that the total force is the sum of the two component forces.

Answers

Answer 1
Answer:

Answer:

F_p = < - √(3) , -3 >\n\nF_o = < √(3) , -1 >

Step-by-step explanation:

- A plane is oriented in a Cartesian coordinate system such that it makes an angle of ( π / 3 ) with the positive x - axis.

- A force ( F ) is directed along the y-axis as a vector < 0 , - 4 >

- We are to determine the the components of force ( F ) parallel and normal to the defined plane.

- We will denote two unit vectors: ( u_p ) parallel to plane and ( u_o ) orthogonal to the defined plane. We will define the two unit vectors in ( x - y ) plane as follows:

- The unit vector ( u_p ) parallel to the defined plane makes an angle of ( 30° ) with the positive y-axis and an angle of ( π / 3 = 60° ) with the x-axis. We will find the projection of the vector onto the x and y axes as follows:

                         u_o = < cos ( 60° ) , cos ( 30° ) >

                         u_o = < (1)/(2) ,  (√(3) )/(2) >

- Similarly, the unit vector ( u_o ) orthogonal to plane makes an angle of ( π / 3 ) with the positive x - axis and angle of ( π / 6 ) with the y-axis in negative direction. We will find the projection of the vector onto the x and y axes as follows:

                        u_p = < cos ( (\pi )/(6)  ) , - cos ( (\pi )/(3) ) >\n\nu_p = < (√(3) )/(2)  , -(1)/(2)  >\n

- To find the projection of force ( F ) along and normal to the plane we will apply the dot product formulation:

- The Force vector parallel to the plane ( F_p ) would be:

                          F_p = u_p(F . u_p)\n\nF_p = < (1)/(2) , (√(3) )/(2) > [  < 0 , - 4 > . < (1)/(2) , (√(3) )/(2) > ]\n\nF_p = < (1)/(2) , (√(3) )/(2) > [ -2√(3)  ]\n\nF_p = < -√(3)  , -3 >\n

- Similarly, to find the projection of force ( F_o ) normal to the plane we again employ the dot product formulation with normal unit vector (  u_o  ) as follows:

                         F_o = u_o ( F . u_o )\n\nF_o = < (√(3) )/(2) , - (1)/(2) > [ < 0 , - 4 > . < (√(3) )/(2) , - (1)/(2) > ] \n\nF_o = < (√(3) )/(2) , - (1)/(2) > [ 2 ] \n\nF_o = < √(3) , - 1 >

- To prove that the projected forces ( F_o ) and ( F_p ) are correct we will apply the vector summation of the two orthogonal vector which must equal to the original vector < 0 , - 4 >

                       F = F_o + F_p\n\n< 0 , - 4 > = < √(3), -1 > + < -√(3), -3 >  \n\n< 0 , - 4 > = < √(3) - √(3) , -1 - 3 > \n\n< 0 , - 4 > = < 0 , - 4 >  .. proven                    


Related Questions

Find the domain of f and f −1 and its domain. f(x) = ln(ex − 3). (a) Find the domain of f. (Enter your answer using interval notation.) (−2,[infinity]) (b) Find f −1. f −1(x) = x+ln(3)
Look at the picture below
A city has two water towers. One tower holds 7.35 x 10 with an exponent of 5, gallons of water and the other tower holds 9.78x 10 with an exponent of 5, gallons of water. What is the combined water capacity of the two towers?
Solve for x 4 - (x + 2) < -3(x + 4)
Solve this equation 3x+6=-2-2x

The residents of a certain dormitory have collected the following data: People who live in the dorm can be classified as either involved in a relationship or uninvolved. Among involved people, 10 percent experience a breakup of their relationship every month. Among uninvolved people, 15 percent will enter into a relationship every month. What is the steady-state fraction of residents who are uninvolved

Answers

Answer:

The steady state proportion for the U (uninvolved) fraction is 0.4.

Step-by-step explanation:

This can be modeled as a Markov chain, with two states:

U: uninvolved

M: matched

The transitions probability matrix is:

\begin{pmatrix} &U&M\nU&0.85&0.15\nM&0.10&0.90\end{pmatrix}

The steady state is that satisfies this product of matrixs:

[\pi] \cdot [P]=[\pi]

being π the matrix of steady-state proportions and P the transition matrix.

If we multiply, we have:

(\pi_U,\pi_M)*\begin{pmatrix}0.85&0.15\n0.10&0.90\end{pmatrix}=(\pi_U,\pi_M)

Now we have to solve this equations

0.85\pi_U+0.10\pi_M=\pi_U\n\n0.15\pi_U+0.90\pi_M=\pi_M

We choose one of the equations and solve:

0.85\pi_U+0.10\pi_M=\pi_U\n\n\pi_M=((1-0.85)/0.10)\pi_U=1.5\pi_U\n\n\n\pi_M+\pi_U=1\n\n1.5\pi_U+\pi_U=1\n\n\pi_U=1/2.5=0.4 \n\n \pi_M=1.5\pi_U=1.5*0.4=0.6

Then, the steady state proportion for the U (uninvolved) fraction is 0.4.

Determine the maximized area of a rectangle that has a perimeter equal to 56m by creating and solving a quadratic equation. What is the length and width?

Answers

Answer:

Area of rectangle = 196\,m^2

Length of rectangle = 14 m

Width of rectangle = 14 m

Step-by-step explanation:

Given:

Perimeter of rectangle is 56 m

To find: the maximized area of a rectangle and the length and width

Solution:

A function y=f(x) has a point of maxima at x=x_0 if f''(x_0)<0

Let x, y denotes length and width of the rectangle.

Perimeter of rectangle = 2( length + width )

=2(x+y)

Also, perimeter of rectangle is equal to 56 m.

So,

56=2(x+y)\nx+y=28\ny=28-x

Let A denotes area of rectangle.

A = length × width

A=xy\n=x(28-x)\n=28x-x^2

Differentiate with respect to x

(dA)/(dx)=28-2x

Put (dA)/(dx)=0

28-2x=0\n2x=28\nx=14

Also,

(d^2A)/(dx^2)=-2<0

At x = 14, (d^2A)/(dx^2)=-2<0

So, x = 14 is a point of maxima

So,

y=28-x=28-14=14

Area of rectangle:

A=xy=14(14)=196\,m^2

Length of rectangle = 14 m

Width of rectangle = 14 m

PLEASE HELP RLLY NEED IT GUYS

Answers

Answer:

156.95

Step-by-step explanation:

length x width x height

Find the value of Z.

A. 3
B. 6\sqrt{2}
C. 2
D. 2\sqrt{2}

Answers

So you would first find the length of y.

To do that, use the Pythagorean's Theorem:

a²+b²=c²

in this case, c = 3, and b = 1.

a² + 1² = 3²
a² + 1 = 9
a² = 8
a = √8

Now to find z, use the Pythagorean's Theorem again:

a² + b² = c²

where √8 = a and 8 = b, and z = c.

√8² + 8² = z²
8 + 64 = z²
72 = z²
z = √72

To simplify this, take out the largest perfect square, or 36:

z = √36√2
z = 6√2

So your answer would be B.

The range of the function f(x) = x + 5 is {7, 9}. What is the function’s domain?

Answers

the range is the y valuewe can rewrite the given function f(x)=x+5 to y=x+5and our given range are y=7 and y=9so sub in ur y values into the equation to find the x value which are the domain...y=x+5

The circumference of a circle is 28 in.What is the diameter of the circle?

Responses


28 over pi, in.


14 over pi, in.


square root of 28 over pi end root, in.

14π−−√ in.



I think it is 28/pi but I would like to make sure

Answers

The diameter οf the circle is 28/π inches οr apprοximately 8.89 inches (rοunded tο twο decimal places).

The fοrmula fοr the circumference (C) οf a circle is given by:

C = 2πr

where r is the radius οf the circle.

If the circumference οf the circle is 28 inches, we can sοlve fοr the radius by dividing bοth sides οf the equatiοn by 2π:

C/2π = r

Substituting the given value οf C = 28, we get:

r = 28/2π

r = 14/π

Finally, tο find the diameter (d) οf the circle, we multiply the radius by 2:

d = 2r

Substituting the value οf r = 14/π, we get:

d = 2(14/π) = 28/π

Therefοre, the diameter οf the circle is 28/π inches οr apprοximately 8.89 inches (rοunded tο twο decimal places).

Learn more about  diameter

brainly.com/question/13624974

#SPJ1