Question

consider the force shown in (figure 1). figure 1 of 1 part a express the force as a cartesian vector. enter the components of the force separated by commas. express your answers in newtons to three significant figures. view available hint(s) f_x, f_y, f_z = n

Explanation:

Step1: Find the component in the y-z plane

First, we find the magnitude of the force component in the \( y\text{-}z \) plane. The angle between the force vector and the \( x \)-axis is \( 60^\circ \), so the component in the \( y\text{-}z \) plane (\( F_{yz} \)) is given by \( F \sin(60^\circ) \).
\[
F_{yz}=F\sin(60^\circ)=500\sin(60^\circ)=500\times\frac{\sqrt{3}}{2}\approx433.01\,\text{N}
\]

Step2: Find \( F_y \) and \( F_z \)

Now, we find the \( y \)-component and \( z \)-component from \( F_{yz} \). The angle between \( F_{yz} \) and the \( y \)-axis is \( 45^\circ \) (assuming the angle with the \( z \)-axis is complementary, but here we use the given \( 45^\circ \) for \( y \)-component). Wait, actually, looking at the diagram, the angle between the projection on \( y\text{-}z \) plane and \( y \)-axis is \( 45^\circ \), so:
\[
F_y = F_{yz}\cos(45^\circ)=433.01\times\frac{\sqrt{2}}{2}\approx306\,\text{N}
\]
\[
F_z = F_{yz}\sin(45^\circ)=433.01\times\frac{\sqrt{2}}{2}\approx306\,\text{N}
\]
Wait, no, maybe the angle with \( z \)-axis is \( 45^\circ \). Wait, let's re - examine. The force makes \( 60^\circ \) with \( x \)-axis, so the angle between the force and \( x \)-axis is \( 60^\circ \), so the component along \( x \)-axis is \( F_x = F\cos(60^\circ) \)
\[
F_x = 500\cos(60^\circ)=500\times0.5 = 250\,\text{N}
\]
Then the component in \( y\text{-}z \) plane is \( F_{yz}=F\sin(60^\circ)=500\times\frac{\sqrt{3}}{2}\approx433.01\,\text{N} \)
Now, if the angle between the \( y\text{-}z \) plane projection and \( y \)-axis is \( 45^\circ \), then:
\[
F_y=F_{yz}\cos(45^\circ)=433.01\times\frac{\sqrt{2}}{2}\approx306\,\text{N}
\]
\[
F_z = F_{yz}\sin(45^\circ)=433.01\times\frac{\sqrt{2}}{2}\approx306\,\text{N}
\]
Wait, but maybe the angle with \( z \)-axis is \( 45^\circ \), so \( F_z = F_{yz}\cos(45^\circ) \) and \( F_y=F_{yz}\sin(45^\circ) \), but the result is the same because \( \sin(45^\circ)=\cos(45^\circ) \)

Step3: Final components

So, \( F_x = 250\,\text{N} \), \( F_y\approx306\,\text{N} \), \( F_z\approx306\,\text{N} \) (rounded to three significant figures)

Answer:

\( 250, 306, 306 \)