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Chapter 11: Problem 3

Write Newton's Second Law of Motion in vector form.

Short Answer

Expert verified
Answer: Newton's Second Law of Motion in vector form is given by the equation \(\vec{F} = m \cdot \vec{a}\), where \(\vec{F}\) represents the force vector, m represents the mass, and \(\vec{a}\) represents the acceleration vector.

Step by step solution

01

Recall Newton's Second Law of Motion

Newton's Second Law of Motion states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. It is given by the equation F = m * a, where F represents the force, m represents the mass, and a represents the acceleration of the object.
02

Represent force, mass, and acceleration as vectors

In order to write Newton's Second Law in vector form, we need to express force, mass, and acceleration as vectors. The force and acceleration are vector quantities because they have both magnitude and direction. The mass is a scalar quantity since it has only magnitude. A vector is denoted as a bold letter or with an arrow on top of the symbol. Thus, force vector is represented as **F** or \(\vec{F}\), and acceleration vector is represented as **a** or \(\vec{a}\).
03

Write Newton's Second Law in vector form

We can now write Newton's Second Law of Motion in its vector form by replacing the force and acceleration scalar symbols with their vector representations: \(\vec{F} = m \cdot \vec{a}\) This equation represents Newton's Second Law of Motion in vector form, which states that the force vector acting on an object is equal to the mass of the object multiplied by its acceleration vector.

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Most popular questions from this chapter

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle 5 t^{-4}-t^{2}, t^{6}-4 t^{3}, 2 / t\right\rangle$$

Suppose the vector-valued function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) is smooth on an interval containing the point \(t_{0} .\) The line tangent to \(\mathbf{r}(t)\) at \(t=t_{0}\) is the line parallel to the tangent vector \(\mathbf{r}^{\prime}\left(t_{0}\right)\) that passes through \(\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right) .\) For each of the following functions, find an equation of the line tangent to the curve at \(t=t_{0} .\) Choose an orientation for the line that is the same as the direction of \(\mathbf{r}^{\prime}\). $$\mathbf{r}(t)=\langle\sqrt{2 t+1}, \sin \pi t, 4\rangle ; t_{0}=4$$

Evaluate the following definite integrals. $$\int_{1}^{4}\left(6 t^{2} \mathbf{i}+8 t^{3} \mathbf{j}+9 t^{2} \mathbf{k}\right) d t$$

Compute the following derivatives. $$\frac{d}{d t}\left(\left(t^{3} \mathbf{i}+6 \mathbf{j}-2 \sqrt{t} \mathbf{k}\right) \times\left(3 t \mathbf{i}-12 t^{2} \mathbf{j}-6 t^{-2} \mathbf{k}\right)\right)$$

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle e^{4 t}, 2 e^{-4 t}+1,2 e^{-t}\right\rangle$$
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