Scientists design a new particle accelerator in which protons (mass 1.7 X 10 -27 kg) follow a circular trajectory given by r = c cos ( k t 2 ) i + c sin ( k t 2 ) j \mathbf{r} = c \cos(kt^2) \mathbf{i} + c \sin(kt^2) \mathbf{j} where c = 5.0 m and k = 8.0 x 10 4 rad/s 2 are constants and t is the elapsed time. What is the radius of the circle?

Hey, everyone today, we're dealing with the problem using position vectors. So we're being told that a private tech company has built one of the most advanced and powerful particle accelerators ever already uses electricity to quote unquote push the charged particles which are electrons along a circular path, making them go faster and faster. The path to follow is given by a specific equation where R is equal to V cosine Katie squared, I plus V sign Katie squared, J where V is equal to four m and K is equal to seven times 10 to the fourth radiance per second squared. These values are constant and T is the time with this, we're being asked to determine the circles of radius. In other words, we're being asked to determine the magnitude, right. We're going to need the magnitude or the length of this position of this vector that we're given, right. So if we're to do this, if we can use our uh magnitude formula, which uses the square root of the sum of the squares of the terms here. So that will mean the magnitude of R, the radius that we're looking for will therefore be the square root of, excuse me, the square root of a Kassian Katie. Oops K or sorry, not a V. It's V Kassian K T squared plus the sign Katie squared. And we square both of these terms because again, we're taking the square root of the sum of squares. So this is give us the square root of oops. Alright. That little leader squirt of, come on a squirt of V squared, cosign square K T squared plus V squared. That's V squared sine squared K T squared, which we can simplify and get oops V squared into Kassian squared, K T squared plus sine square K T squared. So using the identity that Kassian squared T squared plus sine squared K T squared is equal to one, you two are sort of calculus slash Trigana metric identities. We can therefore just say that the magnitude of our will just be equal to the square root of B squared. And we know that V is equal to four. So this will just simply be the square of four squared, Which would give us four US four, which means The radius of this circle. The magnitude of its position vector is therefore four m or answer choice B I hope this helps. And I look forward to seeing you all in the next one.

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4. 2D Kinematics

Intro to Motion in 2D: Position & Displacement

Physics

4. 2D Kinematics

Intro to Motion in 2D: Position & Displacement

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Problem 60a

Textbook Question

Scientists design a new particle accelerator in which protons (mass 1.7 X 10^-27 kg) follow a circular trajectory given by $r = c \cos (k t^{2}) i + c \sin (k t^{2}) j$ where c = 5.0 m and k = 8.0 x 10⁴ rad/s² are constants and t is the elapsed time. What is the radius of the circle?

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The given trajectory of the proton is described in vector form as $ \mathbf{r}(t) = c \cos(kt^2) \hat{i} + c \sin(kt^2) \hat{j} $, where $ c $ and $ k $ are constants. This represents a circular motion in the $ x $-$ y $ plane.

To determine the radius of the circle, note that the magnitude of the position vector $ \mathbf{r}(t) $ remains constant for circular motion. The magnitude is given by $ |\mathbf{r}(t)| = \sqrt{(c \cos(kt^2))^2 + (c \sin(kt^2))^2} $.

Simplify the expression for $ |\mathbf{r}(t)| $: $ |\mathbf{r}(t)| = \sqrt{c^2 \cos^2(kt^2) + c^2 \sin^2(kt^2)} $.

Use the Pythagorean trigonometric identity $ \cos^2(\theta) + \sin^2(\theta) = 1 $ to simplify further: $ |\mathbf{r}(t)| = \sqrt{c^2} = c $.

Since $ c $ is given as 5.0 m, the radius of the circle is $ c = 5.0 \ \text{m} $. Thus, the radius of the circular trajectory is determined by the constant $ c $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle. In this context, the protons are following a circular trajectory, which means their position can be described using polar coordinates. The radius of the circle is a crucial parameter that defines the distance from the center of the circle to any point on its circumference.

Parametric Equations

The trajectory of the protons is described using parametric equations, which express the coordinates of points as functions of a parameter, in this case, time (t). The equations r = c cos(kt^2) and r = c sin(kt^2) represent the x and y coordinates of the protons' position in a two-dimensional plane, allowing us to analyze their motion over time.

Constants in Motion

In the given equations, c and k are constants that influence the motion of the protons. The constant c represents the maximum radius of the circular path, while k affects the rate at which the angle changes with time. Understanding these constants is essential for determining the radius of the circle and analyzing the dynamics of the particle accelerator.

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Multiple Choice

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Multiple Choice

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