Question

A mass of 2 kg falls from a height of 40 cm on a spring of force constant 1960 N/m. The spring is compressed by:
(A) 10 cm
(B) $0.4$ cm
(C) $0.01$ cm
(D) $0.04$ cm

Answer 1

Hint:
When a body falls from a height, its potential energy is transferred to the object it comes in contact with. The energy of a spring depends on the extent to which it is displaced from its mean position.
Formula used: $P = mgh$, where P is the potential energy of the object with mass m when it is at a height h above the ground, g is the acceleration due to gravity.

Complete step by step answer:
When the mass would fall from a height, the initial potential energy possessed by it would become equivalent to the energy of the spring, as it comes down.
In this question, we are provided with the following information:
Mass of the body $m = 2kg$
Height at which the body is initially $h = 40cm = 0.4m$ [As $1m = 100cm$]
Spring constant $k = 1960N/m$
Amount of compression of the spring is x
We know that the potential energy is given as:
$\Rightarrow P = mgh$
Taking in account the compression of the spring, this potential energy will be:
$\Rightarrow P = mg(h + x)$
Also, the energy of the spring is given as:
$\Rightarrow E = \dfrac{1}{2}k{x^2}$
We are aware that these two energies will be equal. Hence,
$\Rightarrow mg(h + x) = \dfrac{1}{2}k{x^2}$
Solving for x, we get:
$\Rightarrow 2mgh + 2mgx = k{x^2}$
$\Rightarrow 2 \times 2 \times 9.8 \times 0.4 + 2 \times 2 \times 9.8x = 1960{x^2}$
Solving it further gives us:
$\Rightarrow 15.68 + 39.2x = 1960{x^2}$
Using the relation for roots of a quadratic equation $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
We input the values accordingly, to get:
$\Rightarrow x = \dfrac{{39.2 \pm \sqrt {{{39.2}^2} - 4 \times 1960 \times - 15.68} }}{{2 \times 1960}}$
$\Rightarrow x = \dfrac{{39.2 \pm \sqrt {124467.84} }}{{3920}} = \dfrac{{39.2 \pm 352.8}}{{3920}}$
Since the compression cannot be negative, we only consider the positive value to get:
$\Rightarrow x = \dfrac{{39.2 + 352.8}}{{3920}} = \dfrac{{392}}{{3920}} = 0.1m$
This is finally equal to 10 cm, and hence the answer is option (A).

Note:
The energy stored in a spring is the elastic potential energy. As this energy depends on the deformation produced in the spring, it finds applications in the real-world in the form of piezoelectric tiles. These tiles are capable of converting the potential energy of the spring to mechanical energy by applying pressure.