Scientists design a new particle accelerator in which protons (mass 1.7 X 10^-27 kg) follow a circular trajectory given by $r=c\cos \left(k{t}^2\right)i+c\sin \left(k{t}^2\right)j$ 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?

Verified step by step guidance

1. 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.
2. 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 \( |\mathbf{r}(t)| = \sqrt{(c \cos(kt^2))^2 + (c \sin(kt^2))^2} \).
3. Simplify the expression for \( |\mathbf{r}(t)| \): \( |\mathbf{r}(t)| = \sqrt{c^2 \cos^2(kt^2) + c^2 \sin^2(kt^2)} \).
4. Use the Pythagorean trigonometric identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \) to simplify further: \( |\mathbf{r}(t)| = \sqrt{c^2} = c \).
5. 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 \( c \).

Key Concepts

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.