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Chapter 10: Problem 54

Describe the motion of a particle if the tangential component of acceleration is \(0 .\)

Short Answer

Expert verified
The particle will move with a constant speed. If the path is a straight line, its velocity is also constant; on a circular path, it constantly changes direction due to radial (centripetal) acceleration.

Step by step solution

01

Description of Particle Motion

The particle moves with a constant speed, in a straight line or along the circumference of a circle. For linear motion, it means the velocity is constant and there isn't any acceleration. On a circular path, a non-zero radial acceleration keeps the particle moving in a circle, maintaining the constant speed. The direction of motion continually changes due to the radial acceleration.

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Most popular questions from this chapter

Consider the vector-valued function \(\mathbf{r}(t)=\left(e^{t} \sin t\right) \mathbf{i}+\left(e^{t} \cos t\right) \mathbf{j}\). Show that \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) are always perpendicular to each other.

Use the model for projectile motion, assuming there is no air resistance. A bale ejector consists of two variable-speed belts at the end of a baler. Its purpose is to toss bales into a trailing wagon. In loading the back of a wagon, a bale must be thrown to a position 8 feet above and 16 feet behind the ejector. (a) Find the minimum initial speed of the bale and the corresponding angle at which it must be ejected from the baler. (b) The ejector has a fixed angle of \(45^{\circ} .\) Find the initial speed required for a bale to reach its target.

The graph of the vector-valued function \(\mathbf{r}(t)\) and a tangent vector to the graph at \(t=t_{0}\) are given. (a) Find a set of parametric equations for the tangent line to the graph at \(t=t_{0}\) (b) Use the equations for the tangent line to approximate \(\mathbf{r}\left(t_{0}+\mathbf{0 . 1}\right)\) $$ \mathbf{r}(t)=\left\langle t,-t^{2}, \frac{1}{4} t^{3}\right\rangle, \quad t_{0}=1 $$

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Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a car's speedometer is constant, then the car cannot be accelerating.
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