Uniform Circular Motion

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

1. E/ is a constant where E is the total energy of the particle: This statement is TRUE as is a constant.
2. The trajectory of the particle in x-y plane is a circle: This statement is TRUE as .
3. In a_x- a_y plane, trajectory of the particle is an ellipse (a_x, a_y denotes the components of acceleration): This statement is FALSE. The trajectory is a circle.
4. : 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.