Math  /  Algebra

Question23. While sliding a couch across a floor, Andrea and Jennifer exert forces FA\vec{F}_{A} and FJ\vec{F}_{J} on the couch. Andrea's force is due north with a magnitude of 130.0 N and Jennifer's force is 3232^{\circ} east of north with a magnitude of 180.0 N . (a) Find the net force in component form. (b) Find the magnitude and direction of the net force. (c) If Andrea and Jennifer's housemates, David and Stephanie, disagree with the move and want to prevent its relocation, with what combined force FDS\vec{F}_{D S} should they push so that the couch does not move?

Studdy Solution

STEP 1

1. Andrea's force, FA\vec{F}_{A}, is directed due north with a magnitude of 130.0 N.
2. Jennifer's force, FJ\vec{F}_{J}, is directed 3232^{\circ} east of north with a magnitude of 180.0 N.
3. We need to find the net force in component form, the magnitude and direction of the net force, and the force required by David and Stephanie to prevent the couch from moving.

STEP 2

1. Break down each force into its components.
2. Calculate the net force in component form.
3. Determine the magnitude and direction of the net force.
4. Calculate the force required by David and Stephanie to prevent the couch from moving.

STEP 3

Break down each force into its components.
Andrea's force, FA\vec{F}_{A}, is directed due north, so its components are: FAx=0 F_{A_x} = 0 FAy=130.0N F_{A_y} = 130.0 \, \text{N}
Jennifer's force, FJ\vec{F}_{J}, is 3232^{\circ} east of north, so its components are: FJx=180.0Nsin(32) F_{J_x} = 180.0 \, \text{N} \cdot \sin(32^{\circ}) FJy=180.0Ncos(32) F_{J_y} = 180.0 \, \text{N} \cdot \cos(32^{\circ})

STEP 4

Calculate the net force in component form.
The net force components are: Fnetx=FAx+FJx=0+180.0sin(32) F_{\text{net}_x} = F_{A_x} + F_{J_x} = 0 + 180.0 \cdot \sin(32^{\circ}) Fnety=FAy+FJy=130.0+180.0cos(32) F_{\text{net}_y} = F_{A_y} + F_{J_y} = 130.0 + 180.0 \cdot \cos(32^{\circ})

STEP 5

Determine the magnitude and direction of the net force.
The magnitude of the net force is: Fnet=(Fnetx)2+(Fnety)2 |\vec{F}_{\text{net}}| = \sqrt{(F_{\text{net}_x})^2 + (F_{\text{net}_y})^2}
The direction (angle θ\theta east of north) is given by: θ=tan1(FnetxFnety) \theta = \tan^{-1}\left(\frac{F_{\text{net}_x}}{F_{\text{net}_y}}\right)

STEP 6

Calculate the force required by David and Stephanie to prevent the couch from moving.
David and Stephanie need to exert a force FDS\vec{F}_{DS} that is equal in magnitude but opposite in direction to Fnet\vec{F}_{\text{net}}.
Thus: FDS=Fnet \vec{F}_{DS} = -\vec{F}_{\text{net}}
This means: FDSx=Fnetx F_{DS_x} = -F_{\text{net}_x} FDSy=Fnety F_{DS_y} = -F_{\text{net}_y}

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