EXAMPLES 359 



If the energy of the two currents were potential energy, we should have 



_<5L_ oW _<W__^ 



dx ~ dx' dx dz ' 



so that the force X would be exactly opposite to that observed. 

 On the other hand, if the energy is kinetic energy, we have 



dL _ dT _ dE 



dx dx dx 



so that the value of X agrees with that observed. 



Hence we conclude that the energy of an electric current is wholly kinetic. 



GENERAL EXAMPLES 



1. The friction of an engine is such that one horse power- can run it at 

 250 revolutions per second when it is doing no external work. The inertia 

 of its moving parts is such that when running at 125 revolutions per sec- 

 ond, and acted on by one horse power, its speed is accelerated at the rate 

 of 10 revolutions per second. If the engine is left to itself when running 

 at its full speed of 250 revolutions per second, find how many revolutions 

 it will make before coming to rest. 



2. A square is moving freely about a diagonal with angular velocity o>, 

 when suddenly one of the angular points not in that diagonal becomes fixed. 

 Determine the impulsive pressure on the fixed point, and show that the 

 new angular velocity will be \ u. 



3. Four equal rods, each of length 2 a and mass m, are freely jointed so 

 as to form a rhombus. The system falls from rest with one diagonal ver- 

 tical, and strikes a fixed horizontal inelastic plane. Find the impulse and 

 the subsequent motion. 



4. Two particles connected by a rigid rod move on a smooth vertical 

 circle. Find the time of a small oscillation. 



5. A uniform rod of length I has the two points at distance c from 

 its middle point connected by equal strings of length L to two fixed points 

 at distances 2 c apart in the same horizontal line. 



Find the principal coordinates and the corresponding periods of 

 vibration. 



6. If the rod of the last question receives a horizontal blow of impulse 

 I at one extremity and at right angles to its length, find the subsequent 

 motion. 



7. A rough uniform cylinder of radius a has an inextensible string coiled 

 round its central section. One end of the string is fastened to a fixed point 

 P, and the cylinder is rolled up the string until it is touching P, with the 

 tangent to the cylinder at P vertical. The cylinder is then let go. Find the 

 motion. 



