A small boy attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency $ω$ and amplitude a. If at height y from the mean position the body gets detached from the spring. Calculate the value of y so that height obtained by the mass is maximum. The body does not interact with the spring during its subsequent motion after detachment

Open in App

Solution

The total energy of the body will be due to gravitational potential energy, kinetic energy and elastic potential of spring.
The mass will reach the highest point when its mechanical energy [gravitational potential energy and kinetic energy] is maximum. This is possible when elastic potential energy of system is zero.
This means that the mass should detach when the spring is at its natural length.
Let L = Natural length of spring when mass m is hanging at the equilibrium.
Now, $m g = K l l = \frac{m g}{K}$
So, $y = \frac{m g}{K}$
But spring constant $K = m ω^{2}$
$y = \frac{m g}{m ω^{2}} = \frac{g}{ω^{2}}$
where $\frac{g}{ω} < a$

Suggest Corrections

Similar questions

Q. A small body attached to one end of a vertically hanging spring is performing SHM about it's mean position with angular frequency

ω

and amplitude a. If at a height

y^{*}

from the mean position the body gets detached from the spring, calculate the value of

y^{*}

so that the height H attained by the mass is maximum. The body does not interact with the spring during it's subsequent motion after detachment.

(a ω^{2} > g)

Q. A mass

m

is undergoing SHM in the vertical direction about the mean position

y_{0}

with amplitude A and angular frequency

ω

. At a distance

y

from the mean position, the mass detaches from the spring. Assume that the spring contracts and does not obstruct the motion of

m

. Find the distance

y^{*}

(measured from the mean position) such that the height

h

attained by the block is maximum.

(A ω)^{2} > g

A mass m is undergoing SHM in vertical direction about the mean position with amplitude A and angular velocity $ω$ . At a distance y from the mean position, the mass detaches from the spring. Assume that the spring contracts and does not obstruct the motion of m. Find the distance $y_{0}$ (measured from the mean position) such that the height attained by the block is maximum $(A ω^{2} > g)$