We read your photo as:

Find the resultant force

Physics

ScanSolve AI Answer

Answer

129 units\sqrt{129} \text{ units}

22.41∘ relative to the 8-unit force22.41^\circ \text{ relative to the 8-unit force}

Step-by-step solution

5 steps
  1. Identify the given forces and the angle between them.

    We have two forces: F1=8F_1 = 8 units (acting horizontally) and F2=5F_2 = 5 units.
    The force F2F_2 is at an angle of 30∘30^\circ from the North axis. Since the North axis is perpendicular to the horizontal force F1F_1, the angle between F1F_1 and F2F_2 is 90∘−30∘=60∘90^\circ - 30^\circ = 60^\circ.
    Let α\alpha be the angle between the two forces, so α=60∘\alpha = 60^\circ.

  2. Calculate the magnitude of the resultant force RR using the Law of Cosines.

    The formula for the magnitude of the resultant of two vectors F1F_1 and F2F_2 with an angle α\alpha between them is:

    R=F12+F22+2F1F2cos⁡(α)R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos(\alpha)}
  3. Substitute the values and calculate RR.

    R=82+52+2(8)(5)cos⁡(60∘)R = \sqrt{8^2 + 5^2 + 2(8)(5) \cos(60^\circ)} R=64+25+80×12R = \sqrt{64 + 25 + 80 \times \frac{1}{2}} R=89+40R = \sqrt{89 + 40} R=129R = \sqrt{129}

    The magnitude of the resultant force is 129 units\boxed{\sqrt{129} \text{ units}}.

  4. Calculate the direction of the resultant force θ\theta using the Law of Sines.

    The angle θ\theta is the angle between the resultant force RR and the horizontal force F1=8F_1 = 8 units. Consider the triangle formed by F1F_1, F2F_2 (shifted), and RR. The angle opposite RR in this triangle is 180∘−α=180∘−60∘=120∘180^\circ - \alpha = 180^\circ - 60^\circ = 120^\circ. The angle θ\theta is opposite the force F2=5F_2 = 5 units.
    Using the Law of Sines:

    F2sin⁡(θ)=Rsin⁡(180∘−α)\frac{F_2}{\sin(\theta)} = \frac{R}{\sin(180^\circ - \alpha)} 5sin⁡(θ)=129sin⁡(120∘)\frac{5}{\sin(\theta)} = \frac{\sqrt{129}}{\sin(120^\circ)} sin⁡(θ)=5sin⁡(120∘)129\sin(\theta) = \frac{5 \sin(120^\circ)}{\sqrt{129}} sin⁡(θ)=5×32129\sin(\theta) = \frac{5 \times \frac{\sqrt{3}}{2}}{\sqrt{129}} sin⁡(θ)=532129\sin(\theta) = \frac{5\sqrt{3}}{2\sqrt{129}}
  5. Calculate the value of θ\theta.

    θ=arcsin⁡(532129)\theta = \arcsin\left(\frac{5\sqrt{3}}{2\sqrt{129}}\right) θ≈arcsin⁡(5×1.732052×11.3578)\theta \approx \arcsin\left(\frac{5 \times 1.73205}{2 \times 11.3578}\right) θ≈arcsin⁡(8.6602522.7156)\theta \approx \arcsin\left(\frac{8.66025}{22.7156}\right) θ≈arcsin⁡(0.38125)\theta \approx \arcsin(0.38125) θ≈22.41∘\theta \approx 22.41^\circ

    The direction of the resultant force is 22.41∘ relative to the 8-unit force\boxed{22.41^\circ \text{ relative to the 8-unit force}}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp