Question

A massless spring (k= 800 N/m), attached with a mass (500g) is completely immersed in 1 kg of water. The spring is stretched by 2 cm and released so that it starts vibrating. What would be the order of magnitude of the change in the temperature of the water when the vibrations stop completely? (Assume that the water container and spring receive negligible heat and specific heat of mass= 400 J/kg K, specific heat of water= 4184 J/kg K)
A. ${10}^{-3} K$
B. ${10}^{-4} K$
C. ${10}^{-1} K$
D. ${10}^{-5} K$

Answer 1

To solve this problem, use the law of conservation of energy, according to which the loss in the potential energy by the spring is equal to the gain in heat energy by the mass and water. So, find the value of loss of potential energy and then find the gain in heat energy by the mass and water. Equate these two equations and find the magnitude of the change in temperature of the water when the vibrations stop completely.

Complete answer:
Given: k= 800 N/m
Mass of spring (m)= 500g= 0.5 kg
Mass of water (M)= 1 kg
x= 2 cm= 0.02 m
Specific heat of mass ${S}_{mass}$= 400 J/kg K
Specific heat of water ${S}_{water}$= 4184 J/kg K
According to law of conservation of energy,
Loss in potential energy of spring= Gain in heat energy of mass and water …(1)
Potential energy of a spring is given by,
$E= \dfrac {1}{2}k{x}^{2}$
Substituting the values in above equation we get,
$E= \dfrac {1}{2} \times 800 \times {0.02}^{2}$
$\Rightarrow E= 0.16 J$ …(2)
Gain in heat energy of mass and water is given by,
$E= {(mS\Delta T)}_{mass}+ {(MS\Delta T)}_{water}$
Substituting the values in above equation we get,
$E= \Delta T (0.5 \times 400 + 1 \times 4184)$
$\Rightarrow E= 4384 \Delta T$ …(3)
Now, substituting the equation. (2) and (3) in equation. (1) we get,
$0.16= 4384 \Delta T$
$\Rightarrow \Delta T= \dfrac {0.16}{4384}$
$\Rightarrow \Delta T= 3.65 \times {10}^{-5}$
Hence, the order of magnitude of the change in the temperature of the water when the vibrations stop completely would be ${10}^{-5} K$.

So, the correct answer is “Option D”.

Note:
Before solving the problem, students should look at the units whether those are balanced or not. If the units are not mentioned then convert them before applying them. In this question, we have obtained the value of heat energy so we should have an idea about what is heat energy? Heat energy is the energy which is due to small particles like atoms, molecules or the ions in solids, liquids or gases. Transfer of heat energy is possible by three methods. Those methods are Conduction, Convection and Radiation.