Circular Motion and Law of Gravitation

Circular Motion

Circular motion is motion in a circular path. A polar coordinate system is more appropriate than cartesian coordinate system for this kind of motion. Because if cartesian coordinate system is used, the value of the x-coordinate and y-coordinate of the position of the particle will change constantly. But if polar coordinate system is used, the r-coordinate which is distance between the center (origin) and the particle remains constant. Only the θ-coordinate which is the angle between the position vector of the particle and the positive x-axis changes. Thus, if polar coordinate system is used only the angle between the position vector of the particle and the positive x-axis has to be considered to describe the motion.

There are two units for the measurement of an angle: a degree and a radian. A degree is defined to be ( 1 ⁄ 360 ) th of a complete circle. That is, there are 360 degrees in one revolution. A degree is abbreviated as deg or as °. A radian is defined to be the degree measure of a central angle that subtends an arc-length equal to the radius of the circle. The radian is the SI unit of measurement for an angle. A radian is abbreviated as rad. Generally, if θ is a central angle in radians, r is the radius of the circle and s is the arc-length subtended by the central angle, then

θ = s ⁄ r

Since a radian is a ratio between lengths, it is unitless. The number of radians in a complete circle may be obtained by dividing the circumference of a circle by the radius. There are 2π radians in a complete circle. A relationship between a degree and a radian can be obtained using the fact that one revolution is equal to 360 degrees or 2π radians.

rad = ( 180 ⁄ π ) deg

deg = ( π ⁄ 180 ) rad

Uniform Circular Motion

Uniform circular motion is motion in a circular path with a constant speed. The time taken for one complete revolution is called the period ( abbreviated as T ) of the motion. Its unit of measurement is second. The number of cycles executed per second is called the frequency ( abbreviated as f ) of the motion. Its unit of measurement is 1 ⁄ s which is defined to be Hertz abbreviated as Hz. Frequency and period are inverses of each other.

f = 1 ⁄ T

The number of radians executed per second is called the angular speed ( abbreviated as ω ) of the object. Its unit of measurement is rad ⁄ s. Since there are 2π radians in a cycle, angular speed is equal to 2π times frequency.

ω = 2πf = 2π ⁄ T

The speed of the object ( v ) may be obtained as the ratio between the circumference of the circular path and the period of the motion.

v = 2πr ⁄ T = 2πrf = ωr

Acceleration of a Uniform Circular Motion

Uniform circular motion is an accelerated motion even though the speed is constant because direction is changing constantly. An acceleration caused by a change in direction only is called centripetal or radial acceleration. Direction of centripetal acceleration is always towards the center of the circular path. As the particle is displaced by a small arc-length |Δs|, the position vector and the velocity vectors of the particle rotate by the same small angle. The triangle formed by the length of the initial position vector, final position vector and the arc-length |Δs|; and the triangle formed by the length of the initial velocity, final velocity vector and the vector joining the tips of this vectors, which is change in velocity Δv, are similar triangles. Thus corresponding sides of these two triangles are proportional.

|Δv| ⁄ v = |Δs| ⁄ r

|Δv| = v|Δs| ⁄ r

Dividing both sides by the time interval Δt during which this displacement took place

|Δv| ⁄ Δt = ( v ⁄ r )( |Δs| ⁄ Δt )

But |Δv| ⁄ Δt is equal to the magnitude of centripetal or radial acceleration ac . And |Δs| ⁄ Δt is equal to the speed of the object v.

ac = v 2 ⁄ r

The force responsible for centrapetal acceleration is called centripetal force Fc . It is related with centripetal acceleration by Newton's second law.

Fc = mac = mv 2 ⁄ r

Uniformly Accelerated Circular Motion

Uniformly accelerated circular motion is motion in a circular path where the speed of the object is changing with time at a constant rate.

Motion Variables of a circular motion

Angular position ( θ ) of a particle is angle formed between the position vector of the particle and the positive x-axis. Its unit of measurement is the radian.

Angular displacement ( Δθ ) of a particle is defined to be change in the angular position of the particle. Its unit of measurement is radian.

Δθ = θf - θi

Average angular velocity ( ωav ) is defined to be change in angular displacement per a unit time. Its unit is rad ⁄ s.

ωav = ( θf - θi ) ⁄ Δt

Instantaneous angular velocity ( ω ) is angular velocity at a given instant of time.

Average angular acceleration ( αav ) is change in angular velocity per a unit time. Its unit is rad ⁄ s 2.

αav = ( ωf - ωi ) ⁄ Δt

Instantaneous angular acceleration ( α ) is angular acceleration at a given instant of time.

Net Acceleration of a Uniformly Accelerated Circular Motion

A uniformly accelerated circular motion has two kinds of acceleration. One is acceleration due to change of direction which is called centripetal or radial acceleration ( ac ). Its direction is towards the center. The other is acceleration due to change of speed and is called tangential acceleration ( at ). Its direction is tangent to the circle. Therefore centripetal acceleration and tangential acceleration are perpendicular to each other. The net acceleration is the vector sum of the centripetal and tangential accelerations. Since centripetal and tangential acceleration are perpendicular to each other, an expression for the magnitude of net acceleration ( anet ) can be obtained by applying pythagorean theorem.

anet = √( ac 2 + at 2 )

Relationship between Linear and Angular Variables

Distance or arc-length s, angular position θ and radius r are related by s = rθ. Taking the changes of both sides of the equations, a relationship between linear displacement and angular displacement is obtained.

Δs = rΔθ

Dividing both sides of this equation by the interval of time Δt during which these displacements took place, the equation Δs ⁄ Δt = rΔθ ⁄ Δt is obtained . But Δs ⁄ Δt is equal to the linear speed v and Δθ ⁄ Δt is equal to the angular speed ω.

v = rω

Taking the change of both sides of this equation and then dividing by Δt the equation Δv ⁄ Δt = rΔω ⁄ Δt is obtained. But Δv ⁄ Δt is equal to the tangential acceleration at and Δω ⁄ Δt is equal to the angular acceleration α.

at = rα

Equations of a Uniformly Accelerated Circular Motion

Uniformly accelerated circular motion is motion with a constant angular acceleration. Equations of a uniformly accelerated motion are similar to the equations of a uniformly accelerated linear motion.

ωf = ωi + αt

Δθ = ωi t + αt 2 ⁄ 2

ωf 2 = ωi 2 + 2αΔθ

Δθ = ( ωf + ωi )t ⁄ 2

These equations involve five variables: initial angular velocity ( ωi ), final angular velocity ( ωf ), angular displacement ( Δθ ), angular acceleration ( α ) and time ( t ). Only two of these equations are independent. Thus, if three of these variables are known, the other two can be calculated with the help of these equations.