91Ó°ÊÓ

Circular motion refers to the movement of an object along a circular path. One of the key concepts here is centripetal acceleration, which is the acceleration that keeps an object moving in a circle. This acceleration is always directed towards the center of the circular path. The formula for centripetal acceleration is given by:

• \( a_n = \frac{v^2}{r} \).
• Here, \( v \) is the velocity of the object.
• \( r \) is the radius of the circular path.

In our scenario, the centripetal acceleration \( a_n \) changes with time, demonstrating how an object can speed up or slow down while still being directed inward, maintaining its circular movement. The formula adapts to: \( a_n = k t^2 \), where \( k \) is a constant and \( t \) is time. Thus, while the path remains circular, the speed can progressively change. Such a change is often indicative of forces at play, allowing the object to sustain this accelerated circular motion.

Kinetic energy is vital in understanding how an object in motion stores energy. It's given by the formula:

• \( K = \frac{1}{2} m v^2 \).
• \( m \) represents mass.
• \( v \) is the velocity of the object.

In scenarios involving circular motion with changing velocity, like our exercise, kinetic energy becomes time-dependent. Since velocity changes over time as \( v = t \sqrt{k r} \), we find that the kinetic energy correspondingly changes to:

• \( K = \frac{1}{2} m (t \sqrt{k r})^2 = \frac{1}{2} m k r t^2 \).

This tells us that as time passes, the kinetic energy of the particle increases quadratically with time. This insight is essential for calculating the work done on or by a particle moving through its path, particularly when integrating over a time period to determine average power.

Power is the rate at which work is done or energy is transferred over time. For circular motion, especially with changing velocities, computing the average power requires understanding how energy transforms with time. The formula linking power to kinetic energy changes is:

• \( P = \frac{dK}{dt} \).

However, to find the average power over a period, we need to integrate kinetic energy and then divide by that time period. In this scenario:

• The integral of kinetic energy from \(0\) to \(t_0\) is \(\int_0^{t_0} \frac{1}{2} m k r t^2 \, dt\), resulting in \(\frac{1}{6} m k r t_0^3\).
• To find the average power, divide this result by \(t_0\) to get \(\frac{1}{6} m k r t_0^2\).

This method ensures that we accurately reflect the energy transformations happening over the entire interval, an essential step for systems where force and motion conditions are not constant. Understanding this concept allows us to determine how efficiently energy is utilized over time.