Uniform Circular Motion

Answer (c) 90?

Answer (a)

Concept: Angular speed

Step 1: Periods.
Minute hand:
Second hand:

Step 2: Ratio.

Step 1: Understanding the Concept:
Orbital velocity . Given and frequency ratio , so .

Step 2: Detailed Explanation:

Ratio

Step 3: Final Answer:
The ratio of orbital velocities is .

Step 1: Understanding the Concept:
Linear velocity in uniform circular motion is the continuous distance traveled along the circumference of the circle divided by the elapsed time.
It directly relates to the angular frequency, which measures how rapidly the particle cycles through revolutions.
Step 2: Key Formula or Approach:
The fundamental relationship between linear velocity , radius , and angular velocity is given by .
The angular velocity can be calculated from the frequency of revolutions using the formula .
Step 3: Detailed Explanation:
First, carefully identify the given physical values from the problem statement.
Radius .
The particle effectively completes in a total time of .
Convert the given time entirely into standard SI base units (seconds): .
Now, calculate the fundamental frequency , which is the number of revolutions per second:
Next, calculate the corresponding angular velocity :
Finally, apply the linear velocity formula using the radius and angular velocity:
Step 4: Final Answer:
The linear velocity of the particle is exactly .

Step 1: Understanding the Concept:
This statement strictly provides the geometric boundary condition for a particle undergoing circular motion.
Without additional kinematic variables like time, velocity, or frequency, it acts as a descriptive setup rather than a complete solvable question.
Step 2: Key Formula or Approach:
The foundational property of a circular path is its circumference, defined by the formula .
This tells us the total path length covered in one single, complete revolution.
Step 3: Detailed Explanation:
The given radius of the circular path is strictly .
By substituting this radius into the circumference formula, we can determine the distance for one loop:
Since no further operational data is provided in this specific line, this is the maximum extractable physical information.
Step 4: Final Answer:
The particle follows a path where one revolution equals a distance of .

Step 1: Understanding the Concept
In circular motion, power is delivered only by the tangential force, as the centripetal force is always perpendicular to the velocity. Power is the rate of change of kinetic energy or the product of tangential force and velocity.

Step 2: Key Formula or Approach
1. Centripetal acceleration .
2. Tangential acceleration .
3. Power .

Step 3: Detailed Explanation
1. Find velocity ( ): Given . Since :
2. Find tangential acceleration ( ): Differentiate velocity with respect to time:
3. Calculate Power ( ):

Step 4: Final Answer
The power delivered to the particle is .