1) A particle is suspended by a string of length ‘l’. This is projected with such a velocity v along the horizontal such that after the string becomes slack it flies through its initial position. Find V?
Solution:
Fig (49)
Let the velocity be v1 at B where the string becomes slack and the string makes angle q with horizontal, by the law of conservation of energy.
Here centripetal force is provided by component of weight
From equation (2) and (3) we get glSinq = v2-2gl (1+Sinq) ---------- (4) At B the particle becomes a projectile of velocity v at 90-q horizontal Here Ux = v1Sinq and Uy = v1Cosq ax = 0 and ay = -g \lCosq = v1Sinq t
Þ 2Sin3q+3Sin2q-1 = 0 \ Sinq = ½ is acceptable solution
2) System consists of two identical slabs each of mass m linked by compressed weightless spring of stiffness K as shown in figure. The slabs are also connected by a thread which is burnt at a certain moment. a) Find at what value of Dl the initial compression of spring, the lower slab will bounce up after the thread is burned through b) If the upper block is compressed by than what is the max displacement of center of mass in upward direction?
Fig (50) Solution: When the lower block is just about to jump, then Kx = mg
From energy conservation
b) When the lower block is just about to move up, center of mass will be displaced to maximum for this
Work energy theorem gives,
On solving this yields
Only force acting on it is 2mg (both masses taken single system)
