We read your photo as:

A vector diagram shows two vectors, one of 50N pointing upwards and another vector with magnitude R at 120 degrees to the horizontal axis, with a horizontal vector of 12N. Method 1: Using cosine rule.

Physics

ScanSolve AI Answer

Answer

619N6\sqrt{19} N

Step-by-step solution

11 steps

Method 1: Using the cosine rule

  1. State the formula for the magnitude of the resultant force FF.

    F=F12+F22+2F1F2cos⁡θF = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta}

  2. Identify the values

    F1=30 NF_1 = 30~\mathrm{N}, F2=12 NF_2 = 12~\mathrm{N}, θ=120∘\theta = 120^\circ.

  3. Compute cos⁡120∘\cos 120^\circ.

    cos⁡120∘=−12\cos 120^\circ = -\frac{1}{2}

  4. Substitute into the formula.

    F2=(30)2+(12)2+2(30)(12)(−12)F^2 = (30)^2 + (12)^2 + 2(30)(12)\left(-\frac{1}{2}\right)

  5. Calculate each term.

    F2=900+144−360F^2 = 900 + 144 - 360

  6. Simplify.

    F2=1044−360=684F^2 = 1044 - 360 = 684 F=684 NF = \sqrt{684}~\mathrm{N}

  7. Simplify the square root.

    684=36×19684 = 36 \times 19 F=36×19=619 NF = \sqrt{36 \times 19} = 6\sqrt{19}~\mathrm{N}

    Method 2: Using resolution of components (assuming 30 N30~\mathrm{N} vertical)

  8. Vertical component FyF_y.

    Fy=30+12cos⁡120∘=30+12(−12)=30−6=24 NF_y = 30 + 12 \cos 120^\circ = 30 + 12\left(-\frac{1}{2}\right) = 30 - 6 = 24~\mathrm{N}

  9. Horizontal component FxF_x.

    Fx=12sin⁡120∘=12×32=63 NF_x = 12 \sin 120^\circ = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}~\mathrm{N}

  10. Magnitude of resultant.

    F=Fy2+Fx2=(24)2+(63)2F = \sqrt{F_y^2 + F_x^2} = \sqrt{(24)^2 + (6\sqrt{3})^2}

  11. Substitute and simplify.

    F=576+36×3=576+108=684=619 NF = \sqrt{576 + 36 \times 3} = \sqrt{576 + 108} = \sqrt{684} = 6\sqrt{19}~\mathrm{N}

    6\sqrt{19} N

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

More Physics Questions

QA vector has a magnitude of 20 units and makes an angle of 30 degrees with the horizontal. What is…QUsing the tail-to-head method and a scale of 1 cm represents 10 N, draw an accurate vector diagram…QTwo forces, 30 N and 40 N, act at right angles to each other. Using a scale of 1 cm = 5 N, find the…QSolve for the resultant force of the given Force vectors by any of the methods like polygon method…QWhat is magnitude of vector
See all Physics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.