Question

A particle of mass $m$ is moving along the curve traced out by the vector function $\mathbf{r}=\mathbf{r}(t) .$ (a) Find the force acting on the particle at any time $t$. (b) Find the magnitude of the acceleration of the particle. $\mathbf{r}(t)=-\cos (3 t) \mathbf{i}+2 \sin (3 t) \mathbf{j}$

Name: Problem 26, Chapter 11: Calculus, Early Transcendentals
Uploaded: 2021-08-07T00:26:45-08:00
Duration: 2 min 18 s
Channel: Monica Miller
Description: A particle of mass m is moving along the curve traced out by the vector function 𝐫=𝐫(t) . (a) Find the force acting on the particle at any time t. (b) Find the magnitude of the acceleration of the particle. 𝐫(t)=-cos(3 t) 𝐢+2 sin(3 t) 𝐣

   A particle of mass $m$ is moving along the curve traced out by the vector function $\mathbf{r}=\mathbf{r}(t) .$
(a) Find the force acting on the particle at any time $t$.
(b) Find the magnitude of the acceleration of the particle.
$\mathbf{r}(t)=-\cos (3 t) \mathbf{i}+2 \sin (3 t) \mathbf{j}$

Calculus, Early Transcendentals

Michael Sullivan,… 1st Edition

Chapter 11, Problem 26

Instant Answer

Solved by Expert Monica Miller

08/07/2021

Step 1

The derivative of $\cos(3t)$ is $-3\sin(3t)$ and the derivative of $2\sin(3t)$ is $6\cos(3t)$. So, the velocity vector is given by: \[\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = 3\sin(3t)\mathbf{i} - 6\cos(3t)\mathbf{j}\] Show more…

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A particle of mass $m$ is moving along the curve traced out by the vector function $\mathbf{r}=\mathbf{r}(t) .$ (a) Find the force acting on the particle at any time $t$. (b) Find the magnitude of the acceleration of the particle. $\mathbf{r}(t)=-\cos (3 t) \mathbf{i}+2 \sin (3 t) \mathbf{j}$

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Key Concepts

Differentiation of Vector Functions

This concept involves taking derivatives of vector functions to determine rates of change, such as obtaining the velocity vector by differentiating the position vector with respect to time and the acceleration vector by differentiating the velocity vector. Component-wise differentiation is used in these processes.

Computing Magnitude of a Vector

Calculating the magnitude of a vector involves taking the square root of the sum of the squares of its components. This determines the overall size of a vector, such as the acceleration vector, regardless of its direction, and is key in analyzing motion quantitatively.

Newton's Second Law

This fundamental principle in mechanics states that the net force acting on a particle is equal to the mass of the particle multiplied by its acceleration. It provides the foundation for relating motion, described through acceleration, to the forces acting on an object.

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