# Circular Motion and Gravitation

Circular motion is everywhere, from atoms to galaxies, from flagella to Ferris wheels. Two terms are frequently used to describe such motion. In general, we say that an object rotates when the axis of rotation lies within the body, and that it revolves when the axis is outside it. Thus, the Earth rotates on its axis and revolves about the Sun.

When a body rotates on its axis, all the particles of the body revolve – that is, they move in circular paths about the body’s axis of rotation. For example, the particles that make up a compact disc all travel in circles about the hub of the CD player. In fact, as a “particle” on Earth, you are continually in circular motion about the Earth’s rotational axis.

Gravity plays a large role in determining the motions of the planets, since it supplies the force necessary to maintain their nearly circular orbits. Newton’s Law of Gravity describes this fundamental force and will analyze the planetary motion in terms of this and other related basic laws. The same considerations will help you understand the motions of Earth satellites, of which there is one natural one and many artificial ones.

Angular Measure

Motion is described as a time rate of change of position. Angular velocity involves a time rate of change of position, which is expressed by an angle. It is important to be able to relate the angular description of circular motion to the orbital or tangential description, that is, to relate the angular displacement to the arc length s. The arc length is the distance traveled along the circular path and the angle θ is said to subtend (define) the arc length. A unit that is very convenient for relating angle to the arc length is the radian (rad), which is defined as the angle subtending an arc length (s) that is equal to the radius (r).

2π rad=360°

1 rad=360°2π=57.3°

The number of radians subtended by an arbitrary arc length is equal to the number of radii that will go into s, or the number of radians. Thus,

s=rθ

Sample Problems:

1. A spectator standing at the center of a circular running track observes a runner start a practice race 256 m due east of her position. The runner runs on the same track to the finish line, which is located due north of the observer’s position. What is the distance of the run?

2. A sailor measures the length of a distant tanker as an angular distance of 1° 9’ with a divided circle. He knows from the shipping charts that the tanker is 150 m in length. How far is the tanker from the sailor?

3. Convert the following angles from degrees to radians: (a) 10° (b) 27° (c) 451°

4. Convert the following angles from radians to degrees. (a) π25rad (b) π10rad (c) 1.25 rad

5. What is the arc length subtended by an angle of π/4 rad on a circle with a radius of 6.0 m?

6. A jogger on a circular track that has a radius of 0.250 km jogs a distance of 1 km. What angular distance does the jogger cover in (1) radians and (2) degrees?

7. If the arc length is twice the radius of a circle, what angle subtends this arc?

Angular Speed and Velocity

The angular speed is given by the formula

ω=θt or θ= ωt

The unit of angular speed is rad/s. But a more common descriptive unit is rpm (revolutions per minute). This nonstandard unit of rpm is readily converted to rad/s since 1 revolution = 2π.

Relationship Between Tangential and Angular Speeds

A particle moving in a circle has an instantaneous velocity tangential to its circular path. For a constant angular velocity and speed, the particle’s orbital or tangential speed v (the magnitude of the tangential velocity) is also constant.

s=rθ=rωt

The length of the arc or the distance is also given by

s=vt

Combining the last two equations, we get

v=rω

where ω is in rad/s.

Period of Frequency

Some other quantities commonly used to describe the circular motion are the frequency and...

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