Suppose a block of mass, m is connected to a spring of force constant, k. The block is moved on a frictionless surface through distance, b with the help of a uniform external force, F so that the spring gets stretched.
What will be the velocity of the block when it completes the distance, b?
Known Quantities:-
Mass - m
Force Constant - k
Distance - b
Force - F
Unknown Quantity: Velocity - v
Note: The mass of the spring and the velocity with which it is moved with the block is considered negligible in both the following methods.
Method 1:-
When the block is moved(consequently, the spring will stretch), the work done by the external force will get stored as Potential Energy(PE) in the spring and also, will appear as Kinetic Energy(KE) of the block.
Initally, the block is at rest, so $KE_i=0$ and at equilibrium position, so $PE_i=0$
$W=\vec{F}\cdot \vec{b} = F b cos0 $
$W=Fb$
$W= \Delta PE(Spring) + \Delta KE(Block)$
$W= (PE_f - PE_i) + (KE_f - KE_i)$
$Fb= \frac{1}{2}kb^2 + \frac{1}{2}mv^2$
$v^2= \frac{2Fb-kb^2}{m}$
$v=\sqrt{\frac{b(2F-kb)}{m}}$
My book solved the problem with this method.
But I tried a different method where I used Newton's equations of motion.
Method 2:-
As the Force is uniform, so the acceleration, a produced in the block will also be uniform. So, we can apply Newton's Equations Of Motion.
$F=ma$
$a=\frac{F}{m}$
Initially, the block was at rest, so intital velocity, $u=o$.
Applying 3rd equation of motion,
$v^2=u^2+2as$
$v=\sqrt{2ab}$
Now, I just want to make sure that is the method correct or not.
If it's correct then, both the methods 1 and 2 should give same velocity.
But, that's the real problem that the values of velocity from method 1 and method 2 are not matching up.
Please, figure me out.
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