Question

A load of mass m falls from a height h on to the scale pan hung from a spring as shown in the adjoining figure. If the spring constant is k and mass of the scale pan is zero and the mass m does not bounce relative to the pan, then the amplitude of vibration is

A. `(mg)/(k)`
B. `(mg)/(k)sqrt(((1+2hk)/(mg)))`
C. `(mg)/(k)+(mg)/(k)sqrt(((1+2hk)/(mg)))`
D. `(mg)/(k)sqrt(((1+2hk)/(mg)-(mg)/(k)))`

Answer 1

Correct Answer - B
According to conservation principle
`mgh=(1)/(2)kx_(0)^(2)-mgx_(0)`
where, `x_(0)` is maximum elongation in spring (when particle is in its lowest extreme position).
or `(1)/(2)kx_(0)^(2)-mgx_(0)-mgh=0`
or `x_(0)^(2)-(2mg)/(k)x_(0)-(2mg)/(k)h=0`
`therefore" "x_(0)=((2mg)/(k)pmsqrt([((2mg)/(k))^(2)+4xx(2mg)/(k)h]))/(2)`
`because` Amplitude=elongation in spring for lowest extreme position-elongation in spring for equilibrium position
`=x_(0)-x_(1)=(mg)/(k)sqrt((1+(2hk)/(mg)))`