Question

While sliding a couch across a floor, Andrea and Jennifer exert forces F → A and F → J on the couch. Andrea’s force is due north with a magnitude of 130.0 N and Jennifer’s force is 32° 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 F → DS should they push so that the couch does not move?

Answer 1

Answer:

Explanation:

Andrea’s force is due north with a magnitude of 130.0 N

F → A =   ( we have considered east as x axis and north as y axis )

Jennifer’s force is 32° east of north

in vector form

F → J = 180 cos 58 i + 180 sin58 j

= 95.38 i + 152.64 j

Resultant force

= 130 j + 95.38 i + 152.64 j

b )  = 95.38 i + 282.64 j

Magnitude of this vector

R² = 95.38² + 282.64²

R = 298.3 N

Tan θ = 95.38 / 282.64

θ = 18.64 °

c ) one must exert an equal and opposite force equal to the resultant

that is 298.3 N to prevent the movement of the

Answer 2

The combined effect of two or more force vectors is found by adding the components of the force vectors together

The values for the component and magnitude of the resultant forces are;

(a) The net force in component form, is 95.39·i + 282.65·j

(b) The magnitude of the net force is approximately 298.3 N

The direction of the net force, θ is approximately 71.35° north of east

(c) The combined force with which David and Stephanie need to push is -95.39·i - 282.65·j

The reason the above values are correct is as follows:

The given forces are;

Andrea's force, $\overset \rightarrow F_A$ = 130.0 N in the direction due north

Jennifer's force, $\overset \rightarrow F_J$ = 180.0 N in the direction 32° east of north

(a) Required:

The net force in component form

Solution;

Andrea's force in component form, $\overset \rightarrow F_A$ = 130.0·j

Jennifer's force in component form, $\overset \rightarrow F_J$ = 180.0 × sin(32°)·i + 180.0 × cos(32°)·j

The net force in component form, $\overset \rightarrow F_R$ = $\overset \rightarrow F_A$ + $\overset \rightarrow F_J$

∴ $\overset \rightarrow F_R$ = (180.0 × sin(32°))·i + (130 + 180.0 × cos(32°)·j

Which gives;

The net force in component form, $\overset \rightarrow F_R$ = 95.39·i + 282.65·j

(b) Required:

The magnitude and direction of the net force

Solution:

The magnitude of the net force, $| \overset \rightarrow F_R | = \mathbf{\sqrt{\overset \rightarrow F_A ^2 + \overset \rightarrow F_J^2 }}$

Therefore

$| \overset \rightarrow F_R | = \sqrt{95.39 ^2 + 282.65^2 } \approx 298.3$

The magnitude of the net force, $| \overset \rightarrow F_R |$ ≈ 298.3 N

The direction of the net force

$The \ direction \ of \ the \ net \ force, \ \theta = arctan \left(\dfrac{F_y}{F_x} \right)$

Therefore;

$The \ direction \ of \ the \ net \ force, \ \theta = arctan \left(\dfrac{282.62}{95.39} \right) \approx 71.35 ^{\circ}$

The direction of the net force, θ ≈ 71.35° north of east

(c) Required:

The combined force with which David and Stephanie need to push so the couch does not move

Solution:

The combined force with which David and Stephanie need to push so the couch does not move is found as follows;

According to Newton's third law of motion, the couch does not move when the net force acting on it is zero

Therefore;

$\overset \rightarrow F_R$ + $\overset \rightarrow F_{DS}$ = 0

$\overset \rightarrow F_{DS}$ = - $\overset \rightarrow F_R$

∴ $\overset \rightarrow F_{DS}$ = -95.39·i - 282.65·j

The combined force with which David and Stephanie need to push, so the couch does not move $\overset \rightarrow F_{DS}$ = -95.39·i - 282.65·j

The magnitude of the combined force $| \overset \rightarrow F_R |$ ≈ 298.3 N

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