The radius of the earth's very nearly circular orbit around the sun is 1.5 x 10^11m. Find the magnitude of the earth's (a) velocity

Verified step by step guidance

1. Identify the formula to calculate the orbital velocity of the Earth around the Sun. The formula is v = \frac{2\pi r}{T}, where v is the orbital velocity, r is the radius of the orbit, and T is the orbital period.
2. Substitute the given value for the radius of the Earth's orbit, r = 1.5 \times 10^{11} \text{ m}.
3. Use the known value of the Earth's orbital period around the Sun, which is approximately T = 365.25 \text{ days}. Convert this period into seconds for consistency in units (1 day = 86400 seconds).
4. Calculate T in seconds by multiplying the number of days in a year by the number of seconds in a day: T = 365.25 \times 86400 \text{ s}.
5. Substitute the values of r and T into the orbital velocity formula to find the magnitude of the Earth's velocity.

Key Concepts

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle. In the context of the Earth's orbit, it describes how the planet travels in a nearly circular path around the Sun, maintaining a constant distance from the Sun. This motion is characterized by a constant speed but a changing velocity due to the continuous change in direction.

Gravitational Force

The gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. In the case of the Earth and the Sun, this force provides the necessary centripetal force that keeps the Earth in its orbit. The strength of this force depends on the masses of the two objects and the distance between them, influencing the orbital dynamics.

Orbital Velocity

Orbital velocity is the speed required for an object to maintain a stable orbit around a celestial body. It can be calculated using the formula v = √(GM/r), where G is the gravitational constant, M is the mass of the central body (the Sun), and r is the radius of the orbit. For Earth, this velocity ensures that the gravitational pull from the Sun balances the inertial tendency of the Earth to move in a straight line.