We need to analyze the given position vector, determine the trajectory, velocity, acceleration, and energy of the particle to determine the correct statements.
Step 1: Determine the Trajectory
The position vector is given by: Let and . Then, Since , the trajectory is a circle of radius centered at the origin.
Step 2: Calculate Velocity
The velocity vector is the time derivative of the position vector:
The speed is the magnitude of the velocity vector:
So, the speed is constant.
Step 3: Calculate Acceleration
The acceleration vector is the time derivative of the velocity vector:
The magnitude of the acceleration is:
Step 4: Calculate Energy
Since we are not given any potential energy, we can assume the total energy is the kinetic energy:
Therefore,
Since m and b are constant as well as w, is proportional to omega. Hence first statement seems incorrect. Recalculating the statement one. Kinetic energy = 1/2 mv^2. Since speed = bw. KE is 1/2 * m * b^2 * w^2. Since b is constant, w is constant then this value is constant.
Step 5: Analyze the Statements
- E/ is a constant where E is the total energy of the particle: This statement is TRUE as is a constant.
- The trajectory of the particle in x-y plane is a circle: This statement is TRUE as .
- In ax- ay plane, trajectory of the particle is an ellipse (ax, ay denotes the components of acceleration): This statement is FALSE. The trajectory is a circle.
- : This statement is FALSE.
Conclusion
The correct statements are:
- E/ is a constant where E is the total energy of the particle.
- The trajectory of the particle in x-y plane is a circle.