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 (kt^2) î + c sin (kt^2) ĵ, where c = 5.0 m and k = 8.0 x 10^4 rad/s^2 are constants and t is the elapsed time. a. What is the radius of the circle?

Verified step by step guidance

1. Identify the parametric equations of the trajectory: The given equations x = c \cos(kt^2) and y = c \sin(kt^2) represent the x and y coordinates of the proton's position as functions of time.
2. Recognize the form of the trajectory: The equations resemble the parametric form of a circle, x = r \cos(\theta) and y = r \sin(\theta), where \theta is the angle, suggesting that the trajectory is circular.
3. Determine the radius of the circle: In the standard parametric equations of a circle, the coefficient of \cos and \sin terms represents the radius. Here, the coefficient is 'c'.
4. Substitute the given value of c: Since c = 5.0 m, this value is the radius of the circle.
5. Conclude that the radius of the circle in which the protons move is 5.0 meters.

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 defined by a parametric equation. Understanding the properties of circular motion, such as radius, angular velocity, and centripetal acceleration, is essential for analyzing the motion of the protons in the accelerator.

Parametric Equations

Parametric equations express the coordinates of a point as functions of a variable, often time. In this case, the position of the protons is given by r = c cos(kt^2) î + c sin(kt^2) ĵ, where 'c' determines the radius and 'k' affects the angular frequency. Recognizing how these equations describe the trajectory helps in identifying the radius of the circular path.

Radius of Curvature

The radius of curvature is a measure of how sharply a curve bends at a given point. For circular motion, the radius remains constant, which is crucial for determining the path of the protons. In this scenario, the radius can be directly identified from the parametric equations, where 'c' represents the fixed distance from the center to the path of the protons.