Question

Two people, A and B are pulling on a tree with ropes while person C is cutting the tree down. Person A applies a force of 80.0 N [45.0°] on one rope. Person B applies a force of 90.0 N [345°] on the other rope. Calculate the net force on the tree?

Answer 1

Answer on Question #41292 – Physics – Mechanics

Two people, A and B are pulling on a tree with ropes while person C is cutting the tree down. Person A applies a force of 80.0 N [45.0°] on one rope. Person B applies a force of 90.0 N [345°] on the other rope. Calculate the net force on the tree?

Solution:

$\mathrm{F_A} = 80.0 \mathrm{~N}$ – force applied by person A;

$\mathrm{F_B} = 90.0 \mathrm{~N}$ – force applied by person B;

$\mathrm{F_{net}}$ – net force of the tree;

Formula for the net force:

$$\widetilde {\mathrm {F}} _ {\text {net}} = \widetilde {\mathrm {F}} _ {\mathrm {A}} + \widetilde {\mathrm {F}} _ {\mathrm {B}}$$

Projections of forces on the X-axis:

$$\begin{array}{l} x: F _ {n e t X} = F _ {A X} + F _ {B X} = F _ {A} \cos 4 5 {}^ {\circ} + F _ {B} \cos 3 4 5 {}^ {\circ} = 8 0 N \cdot \cos 4 5 {}^ {\circ} + 9 0 N \cdot \cos 3 4 5 {}^ {\circ} \\ = 1 4 3. 5 \mathrm {N} \\ \end{array}$$

Projections of forces on the Y-axis:

$$\begin{array}{l} y: F _ {n e t Y} = F _ {A Y} + F _ {B Y} = F _ {A} \sin 4 5 {}^ {\circ} + F _ {B} \sin 3 4 5 {}^ {\circ} = 8 0 N \cdot \sin 4 5 {}^ {\circ} + 9 0 N \cdot \sin 3 4 5 {}^ {\circ} \\ = 3 3. 3 \mathrm {N} \\ \end{array}$$

From the right triangle ABC ( $\alpha$ – angle that net force makes with X-axis):

$$\begin{array}{l} \tan \alpha = \frac {\mathrm {F} _ {\text {netY}}}{\mathrm {F} _ {\text {netX}}} \Rightarrow \alpha = \arctan \left(\frac {\mathrm {F} _ {\text {netY}}}{\mathrm {F} _ {\text {netX}}}\right) = \arctan \left(\frac {3 3 . 3 \mathrm {N}}{1 4 3 . 5 \mathrm {N}}\right) = 1 3 {}^ {\circ} \\ F _ {n e t} ^ {2} = F _ {n e t X} ^ {2} + F _ {n e t Y} ^ {2} \\ F _ {n e t} = \sqrt {F _ {n e t X} ^ {2} + F _ {n e t Y} ^ {2}} = \sqrt {(1 4 3 . 5 N) ^ {2} + (3 3 . 3 N) ^ {2}} = 1 4 7. 3 N \\ \end{array}$$

Answer: net force on the tree is equal to 147.3N [13°]

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