If the system is at rest, then the virial function vanishes because X, Y, Z separately vanish for every particle. The above equation expresses that the virial function is equal to twice the kinetic energy. The part of this function which depends on internal forces admits of being simplified. 17. Suppose the forces between any two particles at a distance r to be R. Let it be considered positive when it is repulsive. Then there are two terms in the above sum which arise from the action between two particles at 1, y1, z1, and x, y, z, respectively. These terms are Now p2 = (x, − x ̧)2 + (Y2 − y,)2 + (2, − 2 ̧)3. Whencè Σ (Xx + Yy + Zz) = +ΣRr. If then the forces in any system are partly external, symbolized by X, Y, Z; and partly internal actions symbolized by R, the equation becomes 1 1 |_ Σ (Xx + Yy + Zz) + ¦ ΣRr = kinetic energy. The proposition will hold good for any direction and for any number of particles. If the action R between the particles is attractive the term ΣRr is negative. 18. This equation is evidently of very general application. It gives, through Rr, a measure of the internal forces in bodies, whatever the nature of these forces may be, and whether the particles of the body are at rest or in motion. Its most important application hitherto has been to the kinetic theory of gases. This is beyond our limits; but we will shew how to apply it in one or two very simple cases. Suppose a thin ring of mass m and radius a, composed of a number of particles, to be rotating with angular velocity w in its plane about its centre. Let R be the tension. Let ds be the arc between two adjacent particles. Then by the equation and But R is the same for all points of the ring, These equations can easily be verified by other methods. Again, suppose a network of light cords in equilibrium under any external forces X, Y, Z acting at points x, y, z, and let T be the tension along a cord of length r, then Σ (Xx + Yy + Zz) = ΣTr. EXAMPLES. 1. Prove that a flywheel of radius a rotating with velocity has in it energy enough to raise a mass equal to its own to a height a2w" 2g 2. A cannon-ball of mass M raises by its recoil a mass M to a height of h feet. If the mass of the cannon-ball is m, shew that its velocity of projection is 3. A weight is attached to an elastic string which is fastened to a point. Apply the principle of energy to determine its motion when it falls from rest, the string being initially vertical and unstretched. 4. A nut slides smoothly on its screw. If this be placed in a vertical position and the nut be allowed to run down, prove that its angular velocity when it has descended a space h, will be 2gh k2 + a2 tan2 a) in which a is the radius of the screw-cylinder, a is the inclination of its tangent to the horizon, and k is the radius of gyration of the nut about the axis. 1 n 5. A thin uniform smooth tube of length 2a is balancing horizontally about its middle point which is fixed; a uniform rod whose mass is th of that of the tube and whose length is 2a, is placed end to end in a line with the tube, and then shot into it with such a horizontal velocity that its middle point shall only just reach that of the tube; prove that if v is the velocity of projection of the rod, the angular velocity of the tube and rod when their middle points coincide is 3v2 6. A circular ring is free to move on a smooth horizontal plane on which it lies; and a uniform rod has its extremities connected with and moveable on the smooth arc of the ring; the system being set in motion on the plane, shew that the angular velocity of the rod is constant; and describe the paths of the centres of the rod and ring. 7. A narrow smooth semicircular tube is fixed in a vertical plane with its vertex upwards, and a heavy flexible string passing through it hangs at rest; shew that if the string be cut at one of the ends of the tube, the velocity which the longer portion of the string will have attained when it is just leaving the tube will be 7 being the length of the longer portion, and a the radius of the tube. 8. A particle is suspended so as to oscillate in a cycloid whose vertex is at the lowest point; if it begin to move from a point distant a from the lowest point measured along the curve, and the medium in which it moves give a small resistance kv2 to the acceleration, prove that before it next comes to rest energy will have been dissipated, which is of its original value. 8ka 3 9. A fine circular tube carrying within it a heavy particle is set revolving about a vertical diameter. Shew that the difference of the squares of the absolute velocities of the particle at any two given points of the tube, equidistant from the axis, is the same for all initial velocities of the particle and the tube. 10. A rough cylinder of radius a loaded so that its centre of gravity is at a distance h from its axis is placed on a board of n times its mass which can move on a smooth horizontal plane. Find the time of a small oscillation, and prove that if be the length of the simple equivalent pendulum where k is the radius of gyration of the cylinder about a horizontal axis through its centre of gravity. 11. A mass M of fluid is running round a circular channel of radius a, with velocity u; another equal mass is running round a channel of radius b, with velocity v; the radius of the one channel is made to increase and the other to diminish till each has the original value of the other. Shew that the work required to produce the change is 12. A smooth thin tube in the shape of a quadrant of a circle, of radius a, is fixed in a vertical plane with its lowest radius horizontal. A heavy uniform inextensible P. G. 11 па string, of length 2 is held wholly within the tube and then let go. Find the velocity during the subsequent motion. 13. A uniform imperfectly elastic beam, of length 2a moving parallel to itself impinges on a fixed obstacle. Prove that the kinetic energy after will be to that before impact as 3c +e'a to 3c2+a; c being the distance from the middle point to the point of impact, and e the modulus of elasticity. 14. A plane body is struck by a blow in its own plane. Prove that the work done by the blow will be greater if the body be free than if a point of the body were fixed. 15. Which of the systems described in the Problems at the end of Lessons XI. and XII. are conservative ? 16. If A, B, C be the moments of inertia round three axes at right angles of a uniform cylinder rolling with constant velocity along a plane; prove that 17. Verify the virial equation in the case of a uniform chain hanging in the common catenary. |