Question

If the motion of a particle is described by \( x = 5 \cos(8\pi t), y = 5 \sin(8\pi t) \) and \( z = 5t \), then the trajectory of the particle is

• A. Circular
• B. Elliptical
• C. Helical
• D. Spiral

Hint:

A helical path combines circular motion in a plane and linear motion along an axis.

Answer 1

The Correct Option is C

Step 1: Understanding the motion equations.
The equations for \(x\) and \(y\) describe a circular motion since \(x = 5 \cos(8\pi t)\) and \(y = 5 \sin(8\pi t)\), which follow a circular path in the \(xy\)-plane. The \(z\)-motion is linear, given by \(z = 5t\), indicating a constant increase in the \(z\)-direction.

Step 2: Analyzing the options.
- (A) Circular: Incorrect. While \(x\) and \(y\) describe circular motion, the \(z\)-motion indicates that the path is not purely circular.
- (B) Elliptical: Incorrect. The motion in the \(xy\)-plane is circular, not elliptical.
- (C) Helical: Correct. The particle follows a helical path, with circular motion in the \(xy\)-plane and linear motion along the \(z\)-axis.
- (D) Spiral: Incorrect. A spiral involves decreasing radius in the \(xy\)-plane, which is not the case here.

Step 3: Conclusion.
The correct answer is (C) Helical because the particle's motion is circular in the \(xy\)-plane and linear in the \(z\)-direction, forming a helical trajectory.