208 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



48. The bob of a pendulum (weight W) is suspended by a cord from one 

 end of an inextensible rod of negligible mass, which is constrained to move 

 vertically, and the other end of the rod is attached to a cord passing over 

 a smooth pulley and supporting a body of weight W. Prove that the period 

 of small oscillations of the pendulum is the same as when the point of support 

 is at rest, and that when the suspending cord makes an angle 6 with the 

 vertical the tension is 



w f cos0 2 (cos 0- cos a) 

 + (l + cos 2 0) 2 



where a is the amplitude of the oscillations. 



49. A simple pendulum is suspended from the roof of a railway carriage 

 and remains vertical while the train is running uniformly at 30 miles an hour. 

 When the brakes are put on, the pendulum oscillates through an angle of 3. 

 Prove that the train will come to rest after running about 385 yards, the 

 resistance being assumed constant. 



50. Prove that the time of a beat of a circular pendulum of length a 

 oscillating through an angle 2a is equal to the time of complete revolution of 

 a pendulum of length acosec 2 ^a, the height of the line of zero velocity above 

 the lowest point being 2acosec 4 ^a. 



51. The bob of a simple pendulum of length I and mass m is acted on by 

 a horizontal force mpg cos nt, where p is a large number, and In 2 is large 

 compared with g. Show that the pendulum may oscillate about either of 

 two points distant a from the lowest point with an amplitude /3, where 



52. The point of support of a simple pendulum of length I and weight 

 w is attached to a massless spring so that it can move to and fro in a 

 horizontal line ; prove that the time of vibration is 



where W is the weight required to stretch the spring a length I. 



53. A platform is sliding down a smooth spherical hill from rest at the 

 summit. From a point fixed on it a plumb-line is suspended in a tube which 

 is always held perpendicular to the surface of the hill at the point occupied 

 by the platform. Prove that the tension of the cord, when the platform has 

 descended a distance x measured vertically, is w(a &c)/a, where a is the 

 radius of the sphere, and w is the weight of the lead. 



54. A ring slides on a smooth wire bent into the form of a curve in 

 a vertical plane, being attached by an elastic thread to a fixed point in the 

 plane ; it starts from a position in which the thread has its natural length, 

 and the modulus of elasticity is twice the weight of the ring. Prove that it 



