Yes, the orbital period does depend on the distance in circular motion. The orbital period refers to the time it takes for an object to complete one full orbit around another object under the influence of gravitational forces.
According to Kepler's third law of planetary motion, the square of the orbital period (T) is directly proportional to the cube of the average distance between the objects (r) raised to a power of 3/2. Mathematically, it can be expressed as:
T^2 = (4π^2 / G) * r^3
where G is the gravitational constant.
This relationship implies that as the distance between two objects increases, the orbital period also increases. In other words, objects that are farther away from the central body take longer to complete an orbit compared to objects closer to it.
This relationship can be understood intuitively by considering the balance between gravitational force and centripetal force. In circular motion, the gravitational force acting between two objects provides the centripetal force required to keep the orbiting object in a circular path. The strength of the gravitational force decreases with distance, and for the centripetal force to remain constant, the velocity of the orbiting object must decrease as the distance increases. Consequently, a longer orbital path requires more time to complete.
This relationship holds true for objects in stable circular orbits, such as planets around the Sun or satellites around the Earth. However, it's important to note that other factors such as the masses of the objects and the presence of other gravitational influences can also affect the precise orbital period. Nonetheless, the dependence of orbital period on distance remains a fundamental principle in circular motion governed by gravitational forces.