# Angular-velocity Sentence Examples

The rate at which work is done on a particular axle is measured by the product where T is the torque or turning moment exerted on the axle by the motor or mechanism applied to it for this purpose, and is the

**angular velocity**of the axle in radians per second.We therefore have the fundamental theorem that the

**angular velocity**of the body around the centre of attraction varies inversely as the square of its distance, and is therefore at every point proportional to the gravitation of the sun.A small sphere of the fluid, if frozen suddenly, would retain this

**angular velocity**.With v=o, the

**angular velocity**of the cylinder is 2w; in this way the velocity may be calculated of the propagation of ripples and waves on the surface of a vertical whirlpool in a sink.An

**angular velocity**R, which gives components - Ry, Ix of velocity to a body, can be resolved into two shearing velocities, -R parallel to Ox, and R parallel to Oy; and then ik is resolved into 4'1+1'2, such that 4/ 1 -R-Rx 2 and 1//2+IRy2 is constant over the boundary.The components of velocity of the moving origin are denoted by U, V, W, and the components of

**angular velocity**of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u', v', w' the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have u =U +u' - yR +zQ, v =V +v -zP +xR, w=W +w -xQ +yP.As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of

**angular velocity**, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of**angular velocity**S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.The

**angular velocity**of the shaft is proportional to the rate of working.If 0 is the angle of twist, the

**angular velocity**is d0/dt.According to numerous observations made at Cape Thorsden, the apparent

**angular velocity**of arcs increases on the average with their altitude.AdvertisementThe optical apparatus generally consists of a mirror mounted on an axis parallel to the axis of the earth, and rotated with the same

**angular velocity**as the sun.It is easily seen that if the mirror be rotated at the same

**angular velocity**as the sun the right ascensions will remain equal throughout the day, and therefore this device reflects the rays in the direction of the earth's axis; a second fixed mirror reflects them in any other fixed direction.By adjusting the right ascension of the plane ABC and rotating the axis with the

**angular velocity**of the sun, it follows that BC will be the direction of the solar rays throughout the day.Further, by causing the hour circle, and with it the polar axis, to rotate by clockwork or some equivalent mechanical contrivance, at the same

**angular velocity**as the earth on its axis, but in the opposite direction, the telescope will, apart from the effects of refraction, automatically follow a star from rising to setting.Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant

**angular velocity**about the pole, in the direction of diminishing radius vector.AdvertisementThen dU/dt, =w say, is the

**angular velocity**of the body.The

**angular velocity**being constant, the effective force on a particle m at a distance r from Oz is snw2r toward& this axis, and its components are accordingly w2mx, wfmy, 0.If the extraneous forces have zero moment about G the

**angular velocity**0 is constant.The circle is described with the constant

**angular velocity**o.As an example of this latter type, suppose that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical diameter with the

**angular velocity**n; it will appear that under a certain condition the motion of G consequent on a slight disturbance will be oscillatory.AdvertisementNow T = 3/41w1, where w is the

**angular velocity**and I is the moment of inertia about the instantaneous axis.The motion of the body relative to 0 is therefore completely represented if we imagine the momental ellipsoid at 0 to roll without sliding on a plane fixed in space, with an

**angular velocity**proportional at each instant to the radius-vector of the point of contact.The

**angular velocity**(r) about this axis is then constant.As a first application of the equations (2) take the case of a solid constrained to rotate with constant

**angular velocity**to about a fixed axis (1, m, n).Let the

**angular velocity**of the rotation be denoted by a=dO/dt, then the linear velocity of any point A at the distance r from the axis is or; and the path of that point is a circle of the radius r described about the axis.AdvertisementLet -y denote the total

**angular velocity**of the rotation of the cone B about the instantaneous axis, $ its**angular velocity**about the axis OB relatively to the plane AOB, and a the**angular velocity**with which the plane AOB turns round the axis OA.Let V5 denote the velocity of advance at a given instant, which of course is common to all the particles of the body; a the

**angular velocity**of the rotation at the same instant; 2,r = 6.2832 nearly, the circumference of a circle of the radius unity.That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the

**angular velocity**of the turning piece by the perpendicular distance from its axis to a pair of points of contact.Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the

**angular velocity**ratio must be expressible in whole numbers.The

**angular velocity**ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connection of the teeth cuts the Ca line of centres at the pitchpoint.Thus the relative motion of the wheels is unchanged; but I is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the

**angular velocity**cii+a1.Coupling of Parallel AxesOldhams CouplingA coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction with the same mean

**angular velocity**.The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connection n which case the

**angular velocity**ratio at any instant is the recipocal of the ratio of the common perpendiculars let fall from the me of connection upon the respective axes of rotation.Then, from the principles of 60 it is evident that at each instant ai/ai = ai/aa, and consequently that ai; so that the fluctuations of

**angular velocity**ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant.Required the relation between the velocity of translation 02 of W and the

**angular velocity**af of the differential barrel.When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the

**angular velocity**.Its moment is found by multiplying the normal pressure between the rolling surfaces by an arm, whose length depends on the nature of the rolling surfaces, and the work lost in a unit of time in overcoming it is the product of its moment by the

**angular velocity**of the rolling surfaces relatively to each other.Neglecting the mass of the shaft itself, when the shaft rotates with an

**angular velocity**a, the centrifugal force Wae/g will act upon the shaft and cause its axis to deflect from the axis of rotation a distance, y say.Let a small body of the weight w undergo translation in a circulai path of the radius p, with the

**angular velocity**of deflexion a, so that the common linear velocity of all its particles is v=ap. Then the actual energy of that body is WV2/2g = Waip2/2g.The product wp/g, by which the half-square of the

**angular velocity**is multiplied, is called the moment of inertia of the revolving body.Flywheels.A flywheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the periodical variations in its

**angular velocity**.The principles according to which variations of

**angular velocity**store and restore energy are the same as those of 117, only substituting moment of inertia for mass, and angular for linear velocity.Let da be the deviation of

**angular velocity**to be produced in the interval dt, and I the moment of the inertia of the body about an axis through its centre of gravity; then 1/8Id(&) = Iada is the variation of the bodys actual energy.But this theory gives no clue to the results relating to hydrogen, which belongs to a high level, and which Adams has shown to move with an

**angular velocity**decidedly greater than the equatorial**angular velocity**below it, and not to show any sign of falling off towards the poles.Lines of lanthanum and carbon which are believed to belong to a low level showed systematically smaller

**angular velocity**than the average.It is an elementary principle of mechanics that this force varies directly as the product of the distance of the moving body from the centre of motion into the square of its

**angular velocity**.The above gear ratio equation can be used to calculate the

**angular velocity**of the large gear from the small gear.