circular motion

uniform circular motion

Motion of a particle moving at a constant speed on a circle. Though the magnitude of the velocity of such an object may be constant, the object is constantly accelerating because its direction is constantly changing. At any given instant its direction is perpendicular to a radius of the circle drawn to the point of location of the object on the circle. The acceleration is strictly a change in direction and is a result of a force directed toward the centre of the circle. This centripetal force causes centripetal acceleration.

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In physics, circular motion is rotation along a circle: a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation, or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. We can talk about circular motion of an object if we ignore its size, so that we have the motion of a point mass in a plane. For example, the center of mass of a body can undergo circular motion.

Examples of circular motion are: an artificial satellite orbiting the Earth in geosynchronous orbit, a stone which is tied to a rope and is being swung in circles (cf. hammer throw), a racecar turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, a gear turning inside a mechanism.

Circular motion is accelerated even if the angular rate of rotation is constant, because the object's velocity vector is constantly changing direction. Such change in direction of velocity involves acceleration of the moving object by a centripetal force, which pulls the moving object towards the center of the circular orbit. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion.

Formulas for uniform circular motion

For motion in a circle of radius R, the circumference of the circle is C = 2π R. If the period for one rotation is T, the angular rate of rotation ω is:

  • omega = frac {2 pi}{T} .

The speed of the object traveling the circle is

  • v, = frac {2 pi R } {T} = omega R

The angle θ swept out in a time t is:

  • theta = 2 pi frac{t}{T} = omega t,

The acceleration due to change in the direction of the velocity is found by noticing that the velocity completely rotates direction in the same time T the object takes for one rotation. Thus, the velocity vector sweeps out a path of length 2π v every T seconds, or:

  • a, = frac {2 pi v }{T} = omega^2 R ,

and is directed radially inward.

The vector relationships are shown in Figure 1. The axis of rotation is shown as a vector Ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of Ω is chosen using the right-hand rule. With this convention for depicting rotation, the velocity is given by a vector cross product as

mathbf{v} = boldsymbol Omega times mathbf r ,

which is a vector perpendicular to both Ω and r (t ), tangential to the orbit, and of magnitude ω R. Likewise, the acceleration is given by

mathbf{a} = boldsymbol Omega times mathbf v ,

which is a vector perpendicular to both Ω and v (t ) of magnitude ω |v| = ω2 R and directed exactly opposite to r (t ).

Constant speed

In the simplest case the speed, mass and radius are constant.

Consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second.

Then consider a body of mass m, moving in a circle of radius r, with an angular velocity of ω.

  • The speed is v = r·ω.
  • The centripetal (inward) acceleration is a = r·ω 2 = r −1·v 2.
  • The centripetal force is F = m·a = r·m·ω 2 = r−1·m·v 2.
  • The momentum of the body is p = m·v = r·m·ω.
  • The moment of inertia is I = r 2·m.
  • The angular momentum is L = r·m·v = r 2·m·ω = I·ω.
  • The kinetic energy is E = 2−1·m·v 2 = 2−1·r 2·m·ω 2 = (2·m)−1·p 2 = 2−1·I·ω 2 = (2·I)−1·L 2 .
  • The circumference of the orbit is 2·π·r.
  • The period of the motion is T = 2·π·ω −1.
  • The frequency is f = T −1 . (Instead of letter f, the frequency is often denoted by the Greek letter ν, which however is almost indistinguishable from the letter v used here for velocity).
  • The quantum number is J = 2·π·L h−1

Variable speed

In the general case, circular motion requires that the total force can be decomposed into the centripetal force required to keep the orbit circular, and a force tangent to the circle, causing a change of speed.

The magnitude of the centripetal force depends on the instantaneous speed.

In the case of an object at the end of a rope, subjected to a force, we can decompose the force into a radial and a lateral component. The radial component is either outward or inward.

Description of circular motion using polar coordinates

During circular motion the body moves on a curve that can be described in polar coordinate system as a fixed distance R from the center of the orbit taken as origin, oriented at an angle θ (t) from some reference direction. See Figure 2. The displacement vector stackrel{vec r}{} is the radial vector from the origin to the particle location:

vec r=R hat u_R (t) ,

where hat u_R (t) is the unit vector parallel to the radius vector at time t and pointing away from the origin. It is handy to introduce the unit vector orthogonal to hat u_R as well, namely hat u_theta. It is customary to orient hat u_theta to point in the direction of travel along the orbit.

The velocity is the time derivative of the displacement:

vec v = frac {d}{dt} vec r(t) = frac {d R}{dt} hat u_R + Rfrac {d hat u_R } {dt} .

Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector hat u_R has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle θ the same as the angle of vec r (t). If the particle displacement rotates through an angle dθ in time dt, so does hat u_R, describing an arc on the unit circle of magnitude dθ. See the unit circle at the left of Figure 2. Hence:

frac {d hat u_R } {dt} = frac {d theta } {dt} hat u_theta ,

where the direction of the change must be perpendicular to hat u_R (or, in other words, along hat u_theta) because any change dhat u_R in the direction of hat u_R would change the size of hat u_R . The sign is positive, because an increase in dθ implies the object and hat u_R have moved in the direction of hat u_theta. Hence the velocity becomes:

vec v = frac {d}{dt} vec r(t) = Rfrac {d hat u_R } {dt} = R frac {d theta } {dt} hat u_theta = R omega hat u_theta .

The acceleration of the body can also be broken into radial and tangential components. The acceleration is the time derivative of the velocity:

vec a = frac {d}{dt} vec v = frac {d}{dt} left(R omega hat u_theta right) .
=R left(frac {d omega}{dt} hat u_theta + omega frac {d hat u_theta}{dt} right) .

The time derivative of hat u_theta is found the same way as for hat u_R . Again, hat u_theta is a unit vector and its tip traces a unit circle with an angle that is π/2 + θ. Hence, an increase in angle dθ by vec r (t) implies hat u_theta traces an arc of magnitude dθ, and as hat u_theta is orthogonal to hat u_R , we have:

frac {d hat u_theta } {dt} = -frac {d theta } {dt} hat u_R = -omega hat u_R ,

where a negative sign is necessary to keep hat u_theta orthogonal to hat u_R . (Otherwise, the angle between hat u_theta and hat u_R would decrease with increase in dθ.) See the unit circle at the left of Figure 2. Consequently the acceleration is:

vec a = R left(frac {d omega}{dt} hat u_theta + omega frac {d hat u_theta}{dt} right)
=R frac {d omega}{dt} hat u_theta - omega^2 R hat u_R .

The centripetal acceleration is the radial component, which is directed radially inward:

vec a_R= -omega ^2R hat u_R ,
while the tangential component changes the magnitude of the velocity:
vec a_{theta}= R frac {d omega}{dt} hat u_theta = frac {d R omega}{dt} hat u_theta =frac {d |vec v|}{dt} hat u_theta .

Description of circular motion using complex numbers

Circular motion can be described using complex numbers. Let the x axis be the real axis and the y axis be the imaginary axis. The position of the body can then be given as z, a complex "vector":
z=x+iy=R(cos theta +i sin theta)=Re^{itheta} ,
where i is the imaginary unit, and
theta =theta (t) ,
is the angle of the complex vector with the real axis and is a function of time t. Since the radius is constant:
dot R =ddot R =0 ,
where a dot indicates time differentiation. With this notation the velocity becomes:
v=dot z = R frac {d}{d theta}left(i thetaright) e^{i theta} = iRdot theta e^{itheta} = iomega cdot Re^{itheta}= iomega z
and the acceleration becomes:
a=dot v =idot omega z +i omega dot z =(idot omega z -omega^2)z
= left(idot omega-omega^2 right) R e^{itheta}
=-omega^2 R e^{itheta} + dot omega e^{ifrac{pi}{2}}R e^{itheta} .

The first term is opposite to the direction of the displacement vector and the second is perpendicular to it, just like the earlier results.

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