momental Sentence Examples

• (9) Turning the axes to make them coincide with the principal axes of the area A, thus making f f xydA = o, xh = - a 2 cos a, y h = - b 2 sin a, (io) where ffx2dA=Aa2, ffy 2 dA= Ab 2, (II) a and b denoting the semi-axes of the momental ellipse of the area.

• This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental ellipse at the C.G.

• within which the C.P. must lie when the area is immersed completely; the boundary of the core is therefore the locus of the antipodes with respect to the momental ellipse of water lines which touch the boundary of the area.

• An inclining couple due to moving a weight about in a ship will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis at F of the momental ellipse of the water-line area A.

• of the wedge of immersion and emersion, will be the C.P. with respect to FF' of the two parts of the water-line area, so that b 1 b 2 will be conjugate to FF' with respect to the momental ellipse at F.

• of the fluid, equal to the weight vertically upward through the movement of a weight P through a distance c will cause the ship to heel through an angle 0 about an axis FF' through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes a 2 -hV/A, b 2 - hV/A, (I) h denoting the vertical height BG between C.G.

• Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at 0.

• A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g.

• If all the masses lie in a plane (1=0) we have, in the notation of (25), c2 = o, and therefore A = Mb, B = Ma, C = M (a +b), so that the equation of the momental ellipsoid takes the form b2x2+a y2+(a2+b2) z1=s4.

• which may be called the momental ellipse at 0.

• It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.

• The relation between these axes may be expressed by means of the momental ellipsoid at 0.

• If p be the radius-vector 0J of the momental ellipsoid Ax+By+Czf=Me4 (I)

• 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.

• It has been shown by Dc Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B+C>A, the curve has no points of inflexion.

• (9) Turning the axes to make them coincide with the principal axes of the area A, thus making f f xydA = o, xh = - a 2 cos a, y h = - b 2 sin a, (io) where ffx2dA=Aa2, ffy 2 dA= Ab 2, (II) a and b denoting the semi-axes of the momental ellipse of the area.

• This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental ellipse at the C.G.

• within which the C.P. must lie when the area is immersed completely; the boundary of the core is therefore the locus of the antipodes with respect to the momental ellipse of water lines which touch the boundary of the area.

• An inclining couple due to moving a weight about in a ship will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis at F of the momental ellipse of the water-line area A.

• of the wedge of immersion and emersion, will be the C.P. with respect to FF' of the two parts of the water-line area, so that b 1 b 2 will be conjugate to FF' with respect to the momental ellipse at F.

• of the fluid, equal to the weight vertically upward through the movement of a weight P through a distance c will cause the ship to heel through an angle 0 about an axis FF' through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes a 2 -hV/A, b 2 - hV/A, (I) h denoting the vertical height BG between C.G.

• Since they are essentially positive the quadric is an ellipsoid; it is called the momental ellipsoid at 0.

• A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g.

• If A = B = C, the momental ellipsoid becomes a sphere; all axes through 0 are then principal axes, and the moment of inertia is the same for each.

• If all the masses lie in a plane (1=0) we have, in the notation of (25), c2 = o, and therefore A = Mb, B = Ma, C = M (a +b), so that the equation of the momental ellipsoid takes the form b2x2+a y2+(a2+b2) z1=s4.

• which may be called the momental ellipse at 0.

• It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.

• The relation between these axes may be expressed by means of the momental ellipsoid at 0.

• If p be the radius-vector 0J of the momental ellipsoid Ax+By+Czf=Me4 (I)

• We have seen (~ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that ~ (2)

• 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.

• It has been shown by Dc Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B+C>A, the curve has no points of inflexion.

• We have seen (~ 18) that this vector coincides in direction with the perpendicular OH to the tangent plane of the momental ellipsoid at J; also that ~ (2)