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Furkat [3]
2 years ago
9

The force on an object is F⃗ =−17j⃗ . For the vector v⃗ =2i⃗ +3j⃗ , find: (a) The component of F⃗ parallel to v⃗

Physics
1 answer:
Igoryamba2 years ago
8 0

Answer:

(a) \vec F_{\parallel} = -\frac{102}{13}\,i-\frac{103}{13}\,j , (b) \vec F_{\perp} = \frac{102}{13}\,i -\frac{68}{13}\,j, (c) W = -51

Explanation:

The statement is incomplete:

The force on an object is \vec F = -17\,j. For the vector \vec v = 2\,i +3\,j. Find: (a) The component of \vec F parallel to \vec v, (b) The component of \vec F perpendicular to \vec v, and (c) The work W, done by force \vec F through displacement \vec v.

(a) The component of \vec F parallel to \vec v is determined by the following expression:

\vec F_{\parallel} = (\vec F \bullet \hat {v} )\cdot \hat{v}

Where \hat{v} is the unit vector of \vec v, which is determined by the following expression:

\hat{v} = \frac{\vec v}{\|\vec v \|}

Where \|\vec v\| is the norm of \vec v, whose value can be found by Pythagorean Theorem.

Then, if \vec F = -17\,j and \vec v = 2\,i +3\,j, then:

\|\vec v\| =\sqrt{2^{2}+3^{3}}

\|\vec v\|=\sqrt{13}

\hat{v} = \frac{1}{\sqrt{13}} \cdot(2\,i + 3\,j)

\hat{v} = \frac{2}{\sqrt{13}}\,i+ \frac{3}{\sqrt{13}}\,j

\vec F \bullet \hat{v} = (0)\cdot \left(\frac{2}{\sqrt{13}} \right)+(-17)\cdot \left(\frac{3}{\sqrt{13}} \right)

\vec F \bullet \hat{v} = -\frac{51}{\sqrt{13}}

\vec F_{\parallel} = \left(-\frac{51}{\sqrt{13}} \right)\cdot \left(\frac{2}{\sqrt{13}}\,i+\frac{3}{\sqrt{13}}\,j  \right)

\vec F_{\parallel} = -\frac{102}{13}\,i-\frac{153}{13}\,j

(b) Parallel and perpendicular components are orthogonal to each other and the perpendicular component can be found by using the following vectorial subtraction:

\vec F_{\perp} = \vec F - \vec F_{\parallel}

Given that \vec F = -17\,j and \vec F_{\parallel} = -\frac{102}{13}\,i-\frac{153}{13}\,j, the component of \vec F perpendicular to \vec v is:

\vec F_{\perp} = -17\,j -\left(-\frac{102}{13}\,i-\frac{153}{13}\,j  \right)

\vec F_{\perp} = \frac{102}{13}\,i + \left(\frac{153}{13}-17 \right)\,j

\vec F_{\perp} = \frac{102}{13}\,i -\frac{68}{13}\,j

(c) The work done by  \vec F through displacement \vec v is:

W = \vec F \bullet \vec v

W = (0)\cdot (2)+(-17)\cdot (3)

W = -51

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