CBSE-XI-Physics
02: Physics and Mathematics
- #1A vector
A→makes an angle of 20° and
B→makes an angle of 110° with the X-axis. The magnitudes of these vectors are 3 m and 4 m respectively. Find the resultant.Ans :
From the above figure, we have:
Angle between `` \stackrel{\to }{\mathrm{A}}`` and `` \stackrel{\to }{\mathrm{B}}`` = 110° - 20° = 90°
`` \left|\stackrel{\to }{\mathrm{A}}\right|=3\mathrm{m}\mathrm{and}\left|\stackrel{\to }{\mathrm{B}}\right|=4\mathrm{m}``
Magnitude of the resultant vector is given by
`` \mathrm{R}=\sqrt{{\mathrm{A}}^{2}+{\mathrm{B}}^{2}+2\mathrm{AB}\mathrm{cos}\mathrm{\theta }}
=\sqrt{{3}^{2}+{4}^{2}+2\times 3\times 4\times \mathrm{cos}90°}
=5\mathrm{m}``
Let β be the angle between `` \stackrel{\to }{\mathrm{R}}\mathrm{and}\stackrel{\to }{\mathrm{A}}``.
`` \mathrm{\beta }={\mathrm{tan}}^{-1}\left(\frac{A\mathrm{sin}90°}{A+B\mathrm{cos}90°}\right)
={\mathrm{tan}}^{-1}\left(\frac{4\mathrm{sin}90°}{3+4\mathrm{cos}90°}\right)
={\mathrm{tan}}^{-1}\frac{4}{3}
={\mathrm{tan}}^{-1}\left(1.333\right)
=53°``
Now, angle made by the resultant vector with the X-axis = 53° + 20° = 73°
∴ The resultant `` \stackrel{\to }{\mathrm{R}}`` is 5 m and it makes an angle of 73° with the x-axis.
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