Question

A block of mass $ 4kg $ falls from a height of $ 3m. $ on a spring of force constant $ 1500\text{ N/m} $ calculate maximum compression of spring. $ \left( g=9.8\text{ N/kg} \right) $
(a) $ 0.395\text{ m} $
(b) $ 0.686\text{ m} $
(c) $ 0.735\text{ m} $
(d) $ 0.676\text{ m} $

Answer 1

Hint :We will calculate velocity of block falling on spring from height of $ 3\text{ m} $ first then using kinetic energy and spring potential energy formula we will calculate maximum compression of spring.
 $ {{v}^{2}}={{u}^{2}}+2gh $
Where,
 $ v= $ final velocity
 $ u= $ initial velocity
 $ g= $ gravitational force
 $ h= $ height
 $ K.E.=\dfrac{1}{2}m{{v}^{2}} $
Where,
 $ m=\text{mass} $
 $ S.P.E.=\dfrac{1}{2}K{{x}^{2}} $
Where,
 $ K= $ force constant
 $ x= $ maximum compression of spring.

Complete Step By Step Answer:
Given,
 $ m=4kg $
 $ n=3\text{ m} $
 $ K=15\text{ N/m} $
 $ g=9.8\text{ N/kg} $
Let's imagine a spring is there and a block $ 4kg $ is falling on spring from height $ 3m. $

Then there will be some velocity so we will need to calculate the velocity of the block falling from a height of $ 3m. $
We have a gravitational force which is acting downward direction and height of the block is $ 3m. $
So,
 $ {{v}^{2}}={{u}^{2}}+2gh $
Here, initial velocity is zero, so it will become
 $ {{v}^{2}}=2gh $
 $ v=\sqrt{2gh} $
 $ v=\sqrt{2\times 9.8\times 3} $
 $ v=7.66\text{ m/s }...\text{(i)} $
Velocity of block falling from height of $ 3m. $ is $ 7.66\text{ m/s}\text{.} $
When the block will fall in spring then spring will compress and velocity becomes zero.
Means kinetic energy is converted into spring potential energy
So, we can write that,
Kinetic energy is equal to spring potential energy.
 $ K.E.=S.P.E...(ii) $
We know that, kinetic energy formula is
 $ K.E.=\dfrac{1}{2}m{{v}^{2}} $
 $ =\dfrac{1}{2}\times 4\times {{\left( 7.66 \right)}^{2}} $
 $ =117.35\text{ N}\text{.m}...\text{(iii)} $
And spring potential energy formula is
 $ S.P.E.=\dfrac{1}{2}K{{x}^{2}} $
 $ =\dfrac{1}{2}\left( 1500 \right){{x}^{2}} $
 $ S.P.E=750{{x}^{2}}...(iv) $
Putting this value in equation $ (ii) $
 $ K.E.=S.P.E $
 $ 117.35=750{{x}^{2}} $
 $ x=0.395\text{ m}\text{.} $
Maximum compression of spring is $ 0.395\ \text{m}\text{.} $

Note :
(a) Velocity is a ratio of displacement to change in time. Distance and displacement are the main key point of velocity. There will be no velocity if there is no displacement in the object or if the object does not cover any distance.
There are two kinds of velocity
(i) initial velocity: velocity of body at beginning of time given is called initial velocity.
(ii) Final velocity: velocity of body at the end of time given is called final velocity.
Mass is the amount of matter in the body. Mass does not change with time. It only changes in extreme cases like a huge amount of energy is given or taken from a body. Mass and weight are different things in physics SI unit of mass in kilogram (kg)
Read questions properly and remember the formulas and steps of solution.
You can directly calculate answer by using this formula $ x=\sqrt{\dfrac{2mgh}{K}} $